id	sid	tid	token	lemma	pos
ejde-502	1	1	electronic	electronic	ADJ
ejde-502	1	2	journal	journal	NOUN
ejde-502	1	3	of	of	ADP
ejde-502	1	4	differential	differential	ADJ
ejde-502	1	5	equations	equation	NOUN
ejde-502	1	6	,	,	PUNCT
ejde-502	1	7	vol	vol	NOUN
ejde-502	1	8	.	.	PUNCT
ejde-502	1	9	2023	2023	NUM
ejde-502	1	10	(	(	PUNCT
ejde-502	1	11	2023	2023	NUM
ejde-502	1	12	)	)	PUNCT
ejde-502	1	13	,	,	PUNCT
ejde-502	1	14	no	no	INTJ
ejde-502	1	15	.	.	NOUN
ejde-502	1	16	72	72	NUM
ejde-502	1	17	,	,	PUNCT
ejde-502	1	18	pp	pp	ADJ
ejde-502	1	19	.	.	PUNCT
ejde-502	2	1	1–21	1–21	PROPN
ejde-502	2	2	.	.	PUNCT
ejde-502	3	1	issn	issn	PROPN
ejde-502	3	2	:	:	PUNCT
ejde-502	3	3	1072	1072	NUM
ejde-502	3	4	-	-	SYM
ejde-502	3	5	6691	6691	NUM
ejde-502	3	6	.	.	PUNCT
ejde-502	4	1	url	url	PROPN
ejde-502	4	2	:	:	PUNCT
ejde-502	4	3	https://ejde.math.txstate.edu	https://ejde.math.txstate.edu	PROPN
ejde-502	4	4	,	,	PUNCT
ejde-502	4	5	https://ejde.math.unt.edu	https://ejde.math.unt.edu	PROPN
ejde-502	4	6	doi	doi	PROPN
ejde-502	4	7	:	:	PUNCT
ejde-502	4	8	10.58997	10.58997	NUM
ejde-502	4	9	/	/	SYM
ejde-502	4	10	ejde.2023.72	ejde.2023.72	NOUN
ejde-502	4	11	qualitative	qualitative	ADJ
ejde-502	4	12	properties	property	NOUN
ejde-502	4	13	of	of	ADP
ejde-502	4	14	solutions	solution	NOUN
ejde-502	4	15	to	to	ADP
ejde-502	4	16	a	a	DET
ejde-502	4	17	reaction	reaction	NOUN
ejde-502	4	18	-	-	PUNCT
ejde-502	4	19	diffusion	diffusion	NOUN
ejde-502	4	20	equation	equation	NOUN
ejde-502	4	21	with	with	ADP
ejde-502	4	22	weighted	weight	VERB
ejde-502	4	23	strong	strong	ADJ
ejde-502	4	24	reaction	reaction	NOUN
ejde-502	4	25	razvan	razvan	PROPN
ejde-502	4	26	gabriel	gabriel	PROPN
ejde-502	4	27	iagar	iagar	PROPN
ejde-502	4	28	,	,	PUNCT
ejde-502	4	29	ana	ana	PROPN
ejde-502	4	30	i.	i.	PROPN
ejde-502	4	31	muñoz	muñoz	PROPN
ejde-502	4	32	,	,	PUNCT
ejde-502	4	33	ariel	ariel	PROPN
ejde-502	4	34	sánchez	sánchez	PROPN
ejde-502	4	35	abstract	abstract	ADV
ejde-502	4	36	.	.	PUNCT
ejde-502	5	1	we	we	PRON
ejde-502	5	2	study	study	VERB
ejde-502	5	3	the	the	DET
ejde-502	5	4	existence	existence	NOUN
ejde-502	5	5	and	and	CCONJ
ejde-502	5	6	qualitative	qualitative	ADJ
ejde-502	5	7	properties	property	NOUN
ejde-502	5	8	of	of	ADP
ejde-502	5	9	solutions	solution	NOUN
ejde-502	5	10	to	to	ADP
ejde-502	5	11	the	the	DET
ejde-502	5	12	cauchy	cauchy	ADJ
ejde-502	5	13	problem	problem	NOUN
ejde-502	5	14	associated	associate	VERB
ejde-502	5	15	to	to	ADP
ejde-502	5	16	the	the	DET
ejde-502	5	17	quasilinear	quasilinear	NOUN
ejde-502	5	18	reaction	reaction	NOUN
ejde-502	5	19	-	-	PUNCT
ejde-502	5	20	diffusion	diffusion	NOUN
ejde-502	5	21	equation	equation	NOUN
ejde-502	5	22	∂tu	∂tu	ADV
ejde-502	5	23	=	=	SYM
ejde-502	5	24	∆um	∆um	PROPN
ejde-502	5	25	+	+	CCONJ
ejde-502	5	26	(	(	PUNCT
ejde-502	5	27	1	1	NUM
ejde-502	5	28	+	+	CCONJ
ejde-502	5	29	|x|)σup	|x|)σup	ADJ
ejde-502	5	30	,	,	PUNCT
ejde-502	5	31	posed	pose	VERB
ejde-502	5	32	for	for	ADP
ejde-502	5	33	(	(	PUNCT
ejde-502	5	34	x	x	NOUN
ejde-502	5	35	,	,	PUNCT
ejde-502	5	36	t	t	PROPN
ejde-502	5	37	)	)	PUNCT
ejde-502	5	38	∈	∈	PROPN
ejde-502	5	39	rn	rn	PROPN
ejde-502	5	40	×	×	PROPN
ejde-502	5	41	(	(	PUNCT
ejde-502	5	42	0,∞	0,∞	NUM
ejde-502	5	43	)	)	PUNCT
ejde-502	5	44	,	,	PUNCT
ejde-502	5	45	where	where	SCONJ
ejde-502	5	46	m	m	VERB
ejde-502	5	47	>	>	X
ejde-502	5	48	1	1	NUM
ejde-502	5	49	,	,	PUNCT
ejde-502	5	50	p	p	PROPN
ejde-502	5	51	∈	∈	PROPN
ejde-502	5	52	(	(	PUNCT
ejde-502	5	53	0	0	NUM
ejde-502	5	54	,	,	PUNCT
ejde-502	5	55	1	1	NUM
ejde-502	5	56	)	)	PUNCT
ejde-502	5	57	and	and	CCONJ
ejde-502	5	58	σ	σ	X
ejde-502	5	59	>	>	X
ejde-502	5	60	0	0	X
ejde-502	5	61	.	.	PUNCT
ejde-502	6	1	initial	initial	ADJ
ejde-502	6	2	data	datum	NOUN
ejde-502	6	3	are	be	AUX
ejde-502	6	4	taken	take	VERB
ejde-502	6	5	to	to	PART
ejde-502	6	6	be	be	AUX
ejde-502	6	7	bounded	bound	VERB
ejde-502	6	8	,	,	PUNCT
ejde-502	6	9	non	non	ADJ
ejde-502	6	10	-	-	ADJ
ejde-502	6	11	negative	negative	ADJ
ejde-502	6	12	and	and	CCONJ
ejde-502	6	13	compactly	compactly	ADV
ejde-502	6	14	supported	support	VERB
ejde-502	6	15	.	.	PUNCT
ejde-502	7	1	in	in	ADP
ejde-502	7	2	the	the	DET
ejde-502	7	3	range	range	NOUN
ejde-502	7	4	when	when	SCONJ
ejde-502	7	5	m	m	VERB
ejde-502	7	6	+	+	ADP
ejde-502	7	7	p	p	X
ejde-502	7	8	≥	≥	NOUN
ejde-502	7	9	2	2	NUM
ejde-502	7	10	,	,	PUNCT
ejde-502	7	11	we	we	PRON
ejde-502	7	12	prove	prove	VERB
ejde-502	7	13	existence	existence	NOUN
ejde-502	7	14	of	of	ADP
ejde-502	7	15	local	local	ADJ
ejde-502	7	16	solutions	solution	NOUN
ejde-502	7	17	with	with	ADP
ejde-502	7	18	a	a	DET
ejde-502	7	19	finite	finite	ADJ
ejde-502	7	20	speed	speed	NOUN
ejde-502	7	21	of	of	ADP
ejde-502	7	22	propagation	propagation	NOUN
ejde-502	7	23	of	of	ADP
ejde-502	7	24	their	their	PRON
ejde-502	7	25	supports	support	NOUN
ejde-502	7	26	for	for	ADP
ejde-502	7	27	compactly	compactly	ADV
ejde-502	7	28	supported	support	VERB
ejde-502	7	29	initial	initial	ADJ
ejde-502	7	30	conditions	condition	NOUN
ejde-502	7	31	.	.	PUNCT
ejde-502	8	1	we	we	PRON
ejde-502	8	2	also	also	ADV
ejde-502	8	3	show	show	VERB
ejde-502	8	4	in	in	ADP
ejde-502	8	5	this	this	DET
ejde-502	8	6	case	case	NOUN
ejde-502	8	7	that	that	SCONJ
ejde-502	8	8	,	,	PUNCT
ejde-502	8	9	for	for	ADP
ejde-502	8	10	a	a	DET
ejde-502	8	11	given	give	VERB
ejde-502	8	12	compactly	compactly	ADV
ejde-502	8	13	supported	support	VERB
ejde-502	8	14	initial	initial	ADJ
ejde-502	8	15	condition	condition	NOUN
ejde-502	8	16	,	,	PUNCT
ejde-502	8	17	there	there	PRON
ejde-502	8	18	exist	exist	VERB
ejde-502	8	19	infinitely	infinitely	ADV
ejde-502	8	20	many	many	ADJ
ejde-502	8	21	solutions	solution	NOUN
ejde-502	8	22	to	to	ADP
ejde-502	8	23	the	the	DET
ejde-502	8	24	cauchy	cauchy	PROPN
ejde-502	8	25	problem	problem	NOUN
ejde-502	8	26	,	,	PUNCT
ejde-502	8	27	by	by	ADP
ejde-502	8	28	prescribing	prescribe	VERB
ejde-502	8	29	the	the	DET
ejde-502	8	30	evolution	evolution	NOUN
ejde-502	8	31	of	of	ADP
ejde-502	8	32	their	their	PRON
ejde-502	8	33	interface	interface	NOUN
ejde-502	8	34	.	.	PUNCT
ejde-502	9	1	in	in	ADP
ejde-502	9	2	the	the	DET
ejde-502	9	3	complementary	complementary	ADJ
ejde-502	9	4	range	range	NOUN
ejde-502	9	5	m+	m+	NOUN
ejde-502	9	6	p	p	X
ejde-502	9	7	<	<	X
ejde-502	9	8	2	2	NUM
ejde-502	9	9	,	,	PUNCT
ejde-502	9	10	we	we	PRON
ejde-502	9	11	obtain	obtain	VERB
ejde-502	9	12	new	new	ADJ
ejde-502	9	13	aronson	aronson	PROPN
ejde-502	9	14	-	-	PUNCT
ejde-502	9	15	bénilan	bénilan	PROPN
ejde-502	9	16	estimates	estimate	NOUN
ejde-502	9	17	satisfied	satisfy	VERB
ejde-502	9	18	by	by	ADP
ejde-502	9	19	solutions	solution	NOUN
ejde-502	9	20	to	to	ADP
ejde-502	9	21	the	the	DET
ejde-502	9	22	cauchy	cauchy	PROPN
ejde-502	9	23	problem	problem	NOUN
ejde-502	9	24	,	,	PUNCT
ejde-502	9	25	which	which	PRON
ejde-502	9	26	are	be	AUX
ejde-502	9	27	of	of	ADP
ejde-502	9	28	independent	independent	ADJ
ejde-502	9	29	interest	interest	NOUN
ejde-502	9	30	as	as	ADP
ejde-502	9	31	a	a	DET
ejde-502	9	32	priori	priori	ADJ
ejde-502	9	33	bounds	bound	NOUN
ejde-502	9	34	for	for	ADP
ejde-502	9	35	the	the	DET
ejde-502	9	36	solutions	solution	NOUN
ejde-502	9	37	.	.	PUNCT
ejde-502	10	1	we	we	PRON
ejde-502	10	2	apply	apply	VERB
ejde-502	10	3	these	these	DET
ejde-502	10	4	estimates	estimate	NOUN
ejde-502	10	5	to	to	PART
ejde-502	10	6	establish	establish	VERB
ejde-502	10	7	infinite	infinite	ADJ
ejde-502	10	8	speed	speed	NOUN
ejde-502	10	9	of	of	ADP
ejde-502	10	10	propagation	propagation	NOUN
ejde-502	10	11	of	of	ADP
ejde-502	10	12	the	the	DET
ejde-502	10	13	supports	support	NOUN
ejde-502	10	14	of	of	ADP
ejde-502	10	15	solutions	solution	NOUN
ejde-502	10	16	if	if	SCONJ
ejde-502	10	17	m	m	VERB
ejde-502	10	18	+	+	NOUN
ejde-502	10	19	p	p	X
ejde-502	10	20	<	<	X
ejde-502	10	21	2	2	NUM
ejde-502	10	22	,	,	PUNCT
ejde-502	10	23	that	that	ADV
ejde-502	10	24	is	is	ADV
ejde-502	10	25	,	,	PUNCT
ejde-502	10	26	u(x	u(x	PROPN
ejde-502	10	27	,	,	PUNCT
ejde-502	10	28	t	t	PROPN
ejde-502	10	29	)	)	PUNCT
ejde-502	10	30	>	>	X
ejde-502	10	31	0	0	PUNCT
ejde-502	11	1	for	for	ADP
ejde-502	11	2	any	any	DET
ejde-502	11	3	x	x	SYM
ejde-502	11	4	∈	∈	PROPN
ejde-502	11	5	rn	rn	PROPN
ejde-502	11	6	,	,	PUNCT
ejde-502	11	7	t	t	PROPN
ejde-502	11	8	>	>	X
ejde-502	11	9	0	0	NUM
ejde-502	11	10	,	,	PUNCT
ejde-502	11	11	even	even	ADV
ejde-502	11	12	in	in	ADP
ejde-502	11	13	the	the	DET
ejde-502	11	14	case	case	NOUN
ejde-502	11	15	when	when	SCONJ
ejde-502	11	16	the	the	DET
ejde-502	11	17	initial	initial	ADJ
ejde-502	11	18	condition	condition	NOUN
ejde-502	11	19	u0	u0	NOUN
ejde-502	11	20	is	be	AUX
ejde-502	11	21	compactly	compactly	ADV
ejde-502	11	22	supported	support	VERB
ejde-502	11	23	.	.	PUNCT
ejde-502	12	1	1	1	X
ejde-502	12	2	.	.	X
ejde-502	12	3	introduction	introduction	NOUN
ejde-502	12	4	this	this	DET
ejde-502	12	5	article	article	NOUN
ejde-502	12	6	concerns	concern	VERB
ejde-502	12	7	the	the	DET
ejde-502	12	8	qualitative	qualitative	ADJ
ejde-502	12	9	theory	theory	NOUN
ejde-502	12	10	of	of	ADP
ejde-502	12	11	the	the	DET
ejde-502	12	12	weak	weak	ADJ
ejde-502	12	13	solutions	solution	NOUN
ejde-502	12	14	to	to	ADP
ejde-502	12	15	the	the	DET
ejde-502	12	16	cauchy	cauchy	ADJ
ejde-502	12	17	problem	problem	NOUN
ejde-502	12	18	for	for	ADP
ejde-502	12	19	the	the	DET
ejde-502	12	20	reaction	reaction	NOUN
ejde-502	12	21	-	-	PUNCT
ejde-502	12	22	diffusion	diffusion	NOUN
ejde-502	12	23	equation	equation	NOUN
ejde-502	12	24	∂tu	∂tu	ADV
ejde-502	12	25	=	=	SYM
ejde-502	12	26	∆um	∆um	PROPN
ejde-502	12	27	+	+	CCONJ
ejde-502	12	28	(	(	PUNCT
ejde-502	12	29	1	1	NUM
ejde-502	12	30	+	+	CCONJ
ejde-502	12	31	|x|)σup	|x|)σup	ADJ
ejde-502	12	32	,	,	PUNCT
ejde-502	12	33	(	(	PUNCT
ejde-502	12	34	x	x	X
ejde-502	12	35	,	,	PUNCT
ejde-502	12	36	t	t	PROPN
ejde-502	12	37	)	)	PUNCT
ejde-502	12	38	∈	∈	PROPN
ejde-502	12	39	rn	rn	PROPN
ejde-502	12	40	×	×	PROPN
ejde-502	12	41	(	(	PUNCT
ejde-502	12	42	0,∞	0,∞	NUM
ejde-502	12	43	)	)	PUNCT
ejde-502	12	44	,	,	PUNCT
ejde-502	12	45	n	n	X
ejde-502	12	46	≥	≥	NOUN
ejde-502	12	47	1	1	NUM
ejde-502	12	48	,	,	PUNCT
ejde-502	12	49	(	(	PUNCT
ejde-502	12	50	1.1	1.1	NUM
ejde-502	12	51	)	)	PUNCT
ejde-502	12	52	supplemented	supplement	VERB
ejde-502	12	53	with	with	ADP
ejde-502	12	54	the	the	DET
ejde-502	12	55	initial	initial	ADJ
ejde-502	12	56	condition	condition	NOUN
ejde-502	12	57	u(x	u(x	NOUN
ejde-502	12	58	,	,	PUNCT
ejde-502	12	59	0	0	NUM
ejde-502	12	60	)	)	PUNCT
ejde-502	12	61	=	=	SYM
ejde-502	12	62	u0(x	u0(x	NOUN
ejde-502	12	63	)	)	PUNCT
ejde-502	12	64	,	,	PUNCT
ejde-502	12	65	x	x	PUNCT
ejde-502	12	66	∈	∈	PROPN
ejde-502	12	67	rn	rn	PROPN
ejde-502	12	68	.	.	PUNCT
ejde-502	13	1	(	(	PUNCT
ejde-502	13	2	1.2	1.2	NUM
ejde-502	13	3	)	)	PUNCT
ejde-502	13	4	the	the	DET
ejde-502	13	5	exponents	exponent	NOUN
ejde-502	13	6	in	in	ADP
ejde-502	13	7	(	(	PUNCT
ejde-502	13	8	1.1	1.1	NUM
ejde-502	13	9	)	)	PUNCT
ejde-502	13	10	are	be	AUX
ejde-502	13	11	considered	consider	VERB
ejde-502	13	12	throughout	throughout	ADP
ejde-502	13	13	the	the	DET
ejde-502	13	14	paper	paper	NOUN
ejde-502	13	15	to	to	PART
ejde-502	13	16	belong	belong	VERB
ejde-502	13	17	to	to	ADP
ejde-502	13	18	the	the	DET
ejde-502	13	19	range	range	NOUN
ejde-502	13	20	m	m	VERB
ejde-502	13	21	>	>	X
ejde-502	13	22	1	1	NUM
ejde-502	13	23	,	,	PUNCT
ejde-502	13	24	0	0	PUNCT
ejde-502	13	25	<	<	X
ejde-502	13	26	p	p	X
ejde-502	13	27	<	<	X
ejde-502	13	28	1	1	NUM
ejde-502	13	29	,	,	PUNCT
ejde-502	13	30	0	0	NUM
ejde-502	13	31	<	<	X
ejde-502	13	32	σ	σ	X
ejde-502	13	33	<	<	X
ejde-502	13	34	∞	∞	PROPN
ejde-502	13	35	,	,	PUNCT
ejde-502	13	36	(	(	PUNCT
ejde-502	13	37	1.3	1.3	NUM
ejde-502	13	38	)	)	PUNCT
ejde-502	13	39	although	although	SCONJ
ejde-502	13	40	we	we	PRON
ejde-502	13	41	also	also	ADV
ejde-502	13	42	give	give	VERB
ejde-502	13	43	alternative	alternative	ADJ
ejde-502	13	44	proofs	proof	NOUN
ejde-502	13	45	or	or	CCONJ
ejde-502	13	46	even	even	ADV
ejde-502	13	47	slight	slight	ADJ
ejde-502	13	48	improvements	improvement	NOUN
ejde-502	13	49	of	of	ADP
ejde-502	13	50	known	know	VERB
ejde-502	13	51	results	result	NOUN
ejde-502	13	52	with	with	ADP
ejde-502	13	53	σ	σ	PROPN
ejde-502	13	54	=	=	SYM
ejde-502	13	55	0	0	X
ejde-502	13	56	.	.	PUNCT
ejde-502	14	1	we	we	PRON
ejde-502	14	2	consider	consider	VERB
ejde-502	14	3	bounded	bound	VERB
ejde-502	14	4	,	,	PUNCT
ejde-502	14	5	compactly	compactly	ADV
ejde-502	14	6	supported	support	VERB
ejde-502	14	7	,	,	PUNCT
ejde-502	14	8	non	non	ADJ
ejde-502	14	9	-	-	ADJ
ejde-502	14	10	negative	negative	ADJ
ejde-502	14	11	and	and	CCONJ
ejde-502	14	12	non	non	ADJ
ejde-502	14	13	-	-	ADJ
ejde-502	14	14	trivial	trivial	ADJ
ejde-502	14	15	initial	initial	ADJ
ejde-502	14	16	conditions	condition	NOUN
ejde-502	14	17	,	,	PUNCT
ejde-502	14	18	more	more	ADV
ejde-502	14	19	precisely	precisely	ADV
ejde-502	14	20	u0	u0	ADJ
ejde-502	14	21	∈	∈	PROPN
ejde-502	14	22	l∞(rn	l∞(rn	PROPN
ejde-502	14	23	)	)	PUNCT
ejde-502	14	24	,	,	PUNCT
ejde-502	14	25	suppu0	suppu0	ADJ
ejde-502	14	26	⊆	⊆	NUM
ejde-502	14	27	b(0	b(0	NOUN
ejde-502	14	28	,	,	PUNCT
ejde-502	14	29	r	r	NOUN
ejde-502	14	30	)	)	PUNCT
ejde-502	14	31	,	,	PUNCT
ejde-502	14	32	u0(x	u0(x	NOUN
ejde-502	14	33	)	)	PUNCT
ejde-502	14	34	≥	≥	NOUN
ejde-502	14	35	0	0	NUM
ejde-502	14	36	,	,	PUNCT
ejde-502	14	37	∀x	∀x	X
ejde-502	14	38	∈	∈	PROPN
ejde-502	14	39	rn	rn	PROPN
ejde-502	14	40	,	,	PUNCT
ejde-502	14	41	u0	u0	VERB
ejde-502	14	42	6≡	6≡	NUM
ejde-502	14	43	0	0	NUM
ejde-502	14	44	,	,	PUNCT
ejde-502	14	45	(	(	PUNCT
ejde-502	14	46	1.4	1.4	NUM
ejde-502	14	47	)	)	PUNCT
ejde-502	14	48	2020	2020	NUM
ejde-502	14	49	mathematics	mathematic	NOUN
ejde-502	14	50	subject	subject	ADJ
ejde-502	14	51	classification	classification	NOUN
ejde-502	14	52	.	.	PUNCT
ejde-502	15	1	35b44	35b44	NUM
ejde-502	15	2	,	,	PUNCT
ejde-502	15	3	35b45	35b45	NUM
ejde-502	15	4	,	,	PUNCT
ejde-502	15	5	35k57	35k57	NUM
ejde-502	15	6	,	,	PUNCT
ejde-502	15	7	35k59	35k59	NUM
ejde-502	15	8	.	.	PUNCT
ejde-502	16	1	key	key	ADJ
ejde-502	16	2	words	word	NOUN
ejde-502	16	3	and	and	CCONJ
ejde-502	16	4	phrases	phrase	NOUN
ejde-502	16	5	.	.	PUNCT
ejde-502	17	1	reaction	reaction	NOUN
ejde-502	17	2	-	-	PUNCT
ejde-502	17	3	diffusion	diffusion	NOUN
ejde-502	17	4	equations	equation	NOUN
ejde-502	17	5	;	;	PUNCT
ejde-502	17	6	weighted	weight	VERB
ejde-502	17	7	reaction	reaction	NOUN
ejde-502	17	8	;	;	PUNCT
ejde-502	17	9	strong	strong	ADJ
ejde-502	17	10	reaction	reaction	NOUN
ejde-502	17	11	;	;	PUNCT
ejde-502	17	12	aronson	aronson	PROPN
ejde-502	17	13	-	-	PUNCT
ejde-502	17	14	bénilan	bénilan	PROPN
ejde-502	17	15	estimates	estimate	NOUN
ejde-502	17	16	.	.	PUNCT
ejde-502	18	1	©	©	ADP
ejde-502	18	2	2023	2023	NUM
ejde-502	18	3	.	.	PUNCT
ejde-502	19	1	this	this	DET
ejde-502	19	2	work	work	NOUN
ejde-502	19	3	is	be	AUX
ejde-502	19	4	licensed	license	VERB
ejde-502	19	5	under	under	ADP
ejde-502	19	6	a	a	DET
ejde-502	19	7	cc	cc	NOUN
ejde-502	19	8	by	by	ADP
ejde-502	19	9	4.0	4.0	NUM
ejde-502	19	10	license	license	NOUN
ejde-502	19	11	.	.	PUNCT
ejde-502	20	1	submitted	submit	VERB
ejde-502	20	2	june	june	PROPN
ejde-502	20	3	13	13	NUM
ejde-502	20	4	,	,	PUNCT
ejde-502	20	5	2023	2023	NUM
ejde-502	20	6	.	.	PUNCT
ejde-502	21	1	published	publish	VERB
ejde-502	21	2	october	october	PROPN
ejde-502	21	3	23	23	NUM
ejde-502	21	4	,	,	PUNCT
ejde-502	21	5	2023	2023	NUM
ejde-502	21	6	.	.	PUNCT
ejde-502	22	1	1	1	NUM
ejde-502	22	2	2	2	NUM
ejde-502	22	3	r.	r.	PROPN
ejde-502	22	4	g.	g.	PROPN
ejde-502	22	5	iagar	iagar	PROPN
ejde-502	22	6	,	,	PUNCT
ejde-502	22	7	a.	a.	NOUN
ejde-502	22	8	i.	i.	PROPN
ejde-502	22	9	muñoz	muñoz	PROPN
ejde-502	22	10	,	,	PUNCT
ejde-502	22	11	a.	a.	NOUN
ejde-502	22	12	sánchez	sánchez	PROPN
ejde-502	22	13	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	22	14	and	and	CCONJ
ejde-502	22	15	further	further	ADJ
ejde-502	22	16	regularity	regularity	NOUN
ejde-502	22	17	assumptions	assumption	NOUN
ejde-502	22	18	(	(	PUNCT
ejde-502	22	19	such	such	ADJ
ejde-502	22	20	as	as	ADP
ejde-502	22	21	continuity	continuity	NOUN
ejde-502	22	22	)	)	PUNCT
ejde-502	22	23	will	will	AUX
ejde-502	22	24	be	be	AUX
ejde-502	22	25	specified	specify	VERB
ejde-502	22	26	at	at	ADP
ejde-502	22	27	the	the	DET
ejde-502	22	28	points	point	NOUN
ejde-502	22	29	where	where	SCONJ
ejde-502	22	30	they	they	PRON
ejde-502	22	31	are	be	AUX
ejde-502	22	32	needed	need	VERB
ejde-502	22	33	.	.	PUNCT
ejde-502	23	1	in	in	ADP
ejde-502	23	2	nonlinear	nonlinear	ADJ
ejde-502	23	3	diffusion	diffusion	NOUN
ejde-502	23	4	problems	problem	NOUN
ejde-502	23	5	it	it	PRON
ejde-502	23	6	is	be	AUX
ejde-502	23	7	standard	standard	ADJ
ejde-502	23	8	to	to	PART
ejde-502	23	9	consider	consider	VERB
ejde-502	23	10	data	datum	NOUN
ejde-502	23	11	belonging	belong	VERB
ejde-502	23	12	to	to	ADP
ejde-502	23	13	the	the	DET
ejde-502	23	14	space	space	NOUN
ejde-502	23	15	l1	l1	PROPN
ejde-502	23	16	loc(rn	loc(rn	PRON
ejde-502	23	17	)	)	PUNCT
ejde-502	23	18	and	and	CCONJ
ejde-502	23	19	we	we	PRON
ejde-502	23	20	are	be	AUX
ejde-502	23	21	sure	sure	ADJ
ejde-502	23	22	that	that	SCONJ
ejde-502	23	23	some	some	PRON
ejde-502	23	24	of	of	ADP
ejde-502	23	25	our	our	PRON
ejde-502	23	26	results	result	NOUN
ejde-502	23	27	can	can	AUX
ejde-502	23	28	be	be	AUX
ejde-502	23	29	extended	extend	VERB
ejde-502	23	30	to	to	ADP
ejde-502	23	31	this	this	DET
ejde-502	23	32	weaker	weak	ADJ
ejde-502	23	33	space	space	NOUN
ejde-502	23	34	than	than	ADP
ejde-502	23	35	l∞(rn	l∞(rn	PROPN
ejde-502	23	36	)	)	PUNCT
ejde-502	23	37	.	.	PUNCT
ejde-502	24	1	however	however	ADV
ejde-502	24	2	,	,	PUNCT
ejde-502	24	3	for	for	ADP
ejde-502	24	4	simplicity	simplicity	NOUN
ejde-502	24	5	and	and	CCONJ
ejde-502	24	6	also	also	ADV
ejde-502	24	7	as	as	ADP
ejde-502	24	8	in	in	ADP
ejde-502	24	9	the	the	DET
ejde-502	24	10	range	range	NOUN
ejde-502	24	11	of	of	ADP
ejde-502	24	12	exponents	exponent	NOUN
ejde-502	24	13	(	(	PUNCT
ejde-502	24	14	1.3	1.3	NUM
ejde-502	24	15	)	)	PUNCT
ejde-502	24	16	finite	finite	ADJ
ejde-502	24	17	time	time	NOUN
ejde-502	24	18	blow	blow	NOUN
ejde-502	24	19	-	-	PUNCT
ejde-502	24	20	up	up	NOUN
ejde-502	24	21	of	of	ADP
ejde-502	24	22	bounded	bounded	ADJ
ejde-502	24	23	solutions	solution	NOUN
ejde-502	24	24	is	be	AUX
ejde-502	24	25	expected	expect	VERB
ejde-502	24	26	(	(	PUNCT
ejde-502	24	27	as	as	SCONJ
ejde-502	24	28	established	establish	VERB
ejde-502	24	29	,	,	PUNCT
ejde-502	24	30	for	for	ADP
ejde-502	24	31	example	example	NOUN
ejde-502	24	32	,	,	PUNCT
ejde-502	24	33	for	for	ADP
ejde-502	24	34	self	self	NOUN
ejde-502	24	35	-	-	PUNCT
ejde-502	24	36	similar	similar	ADJ
ejde-502	24	37	solutions	solution	NOUN
ejde-502	24	38	in	in	ADP
ejde-502	24	39	recent	recent	ADJ
ejde-502	24	40	works	work	NOUN
ejde-502	24	41	such	such	ADJ
ejde-502	24	42	as	as	ADP
ejde-502	24	43	[	[	X
ejde-502	24	44	18	18	NUM
ejde-502	24	45	,	,	PUNCT
ejde-502	24	46	19	19	NUM
ejde-502	24	47	,	,	PUNCT
ejde-502	24	48	16	16	NUM
ejde-502	24	49	]	]	PUNCT
ejde-502	24	50	)	)	PUNCT
ejde-502	24	51	,	,	PUNCT
ejde-502	24	52	we	we	PRON
ejde-502	24	53	decided	decide	VERB
ejde-502	24	54	to	to	PART
ejde-502	24	55	avoid	avoid	VERB
ejde-502	24	56	possible	possible	ADJ
ejde-502	24	57	pointwise	pointwise	NOUN
ejde-502	24	58	singularities	singularity	NOUN
ejde-502	24	59	at	at	ADP
ejde-502	24	60	finite	finite	ADJ
ejde-502	24	61	points	point	NOUN
ejde-502	24	62	and	and	CCONJ
ejde-502	24	63	thus	thus	ADV
ejde-502	24	64	require	require	VERB
ejde-502	24	65	(	(	PUNCT
ejde-502	24	66	1.4	1.4	NUM
ejde-502	24	67	)	)	PUNCT
ejde-502	24	68	.	.	PUNCT
ejde-502	25	1	a	a	DET
ejde-502	25	2	deep	deep	ADJ
ejde-502	25	3	study	study	NOUN
ejde-502	25	4	of	of	ADP
ejde-502	25	5	(	(	PUNCT
ejde-502	25	6	1.1	1.1	NUM
ejde-502	25	7	)	)	PUNCT
ejde-502	25	8	in	in	ADP
ejde-502	25	9	the	the	DET
ejde-502	25	10	range	range	NOUN
ejde-502	25	11	p	p	X
ejde-502	25	12	>	>	X
ejde-502	25	13	1	1	NUM
ejde-502	25	14	has	have	AUX
ejde-502	25	15	been	be	AUX
ejde-502	25	16	performed	perform	VERB
ejde-502	25	17	by	by	ADP
ejde-502	25	18	andreucci	andreucci	NOUN
ejde-502	25	19	and	and	CCONJ
ejde-502	25	20	dibenedetto	dibenedetto	NOUN
ejde-502	25	21	in	in	ADP
ejde-502	25	22	[	[	X
ejde-502	25	23	1	1	NUM
ejde-502	25	24	]	]	PUNCT
ejde-502	25	25	for	for	ADP
ejde-502	25	26	weights	weight	NOUN
ejde-502	25	27	with	with	ADP
ejde-502	25	28	any	any	DET
ejde-502	25	29	σ	σ	NUM
ejde-502	25	30	∈	∈	PROPN
ejde-502	25	31	r	r	NOUN
ejde-502	25	32	,	,	PUNCT
ejde-502	25	33	that	that	ADV
ejde-502	25	34	is	is	ADV
ejde-502	25	35	,	,	PUNCT
ejde-502	25	36	both	both	CCONJ
ejde-502	25	37	positive	positive	ADJ
ejde-502	25	38	and	and	CCONJ
ejde-502	25	39	negative	negative	ADJ
ejde-502	25	40	.	.	PUNCT
ejde-502	26	1	properties	property	NOUN
ejde-502	26	2	such	such	ADJ
ejde-502	26	3	as	as	ADP
ejde-502	26	4	existence	existence	NOUN
ejde-502	26	5	of	of	ADP
ejde-502	26	6	local	local	ADJ
ejde-502	26	7	weak	weak	ADJ
ejde-502	26	8	solutions	solution	NOUN
ejde-502	26	9	under	under	ADP
ejde-502	26	10	optimal	optimal	ADJ
ejde-502	26	11	growth	growth	NOUN
ejde-502	26	12	conditions	condition	NOUN
ejde-502	26	13	on	on	ADP
ejde-502	26	14	the	the	DET
ejde-502	26	15	initial	initial	ADJ
ejde-502	26	16	data	datum	NOUN
ejde-502	26	17	,	,	PUNCT
ejde-502	26	18	estimates	estimate	NOUN
ejde-502	26	19	,	,	PUNCT
ejde-502	26	20	regularity	regularity	NOUN
ejde-502	26	21	of	of	ADP
ejde-502	26	22	them	they	PRON
ejde-502	26	23	and	and	CCONJ
ejde-502	26	24	harnack	harnack	NOUN
ejde-502	26	25	-	-	PUNCT
ejde-502	26	26	type	type	NOUN
ejde-502	26	27	inequalities	inequality	NOUN
ejde-502	26	28	are	be	AUX
ejde-502	26	29	obtained	obtain	VERB
ejde-502	26	30	.	.	PUNCT
ejde-502	27	1	the	the	DET
ejde-502	27	2	authors	author	NOUN
ejde-502	27	3	also	also	ADV
ejde-502	27	4	specify	specify	VERB
ejde-502	27	5	in	in	ADP
ejde-502	27	6	[	[	X
ejde-502	27	7	1	1	X
ejde-502	27	8	]	]	PUNCT
ejde-502	27	9	that	that	SCONJ
ejde-502	27	10	some	some	PRON
ejde-502	27	11	of	of	ADP
ejde-502	27	12	their	their	PRON
ejde-502	27	13	results	result	NOUN
ejde-502	27	14	can	can	AUX
ejde-502	27	15	be	be	AUX
ejde-502	27	16	extended	extend	VERB
ejde-502	27	17	to	to	ADP
ejde-502	27	18	the	the	DET
ejde-502	27	19	range	range	NOUN
ejde-502	27	20	0	0	PUNCT
ejde-502	27	21	<	<	X
ejde-502	27	22	p	p	X
ejde-502	27	23	<	<	X
ejde-502	27	24	1	1	NUM
ejde-502	27	25	but	but	CCONJ
ejde-502	27	26	only	only	ADV
ejde-502	27	27	when	when	SCONJ
ejde-502	27	28	σ	σ	X
ejde-502	27	29	<	<	X
ejde-502	27	30	0	0	PUNCT
ejde-502	27	31	and	and	CCONJ
ejde-502	27	32	leave	leave	VERB
ejde-502	27	33	open	open	ADJ
ejde-502	27	34	the	the	DET
ejde-502	27	35	case	case	NOUN
ejde-502	27	36	σ	σ	X
ejde-502	27	37	>	>	X
ejde-502	27	38	0	0	PUNCT
ejde-502	28	1	with	with	ADP
ejde-502	28	2	0	0	NUM
ejde-502	28	3	<	<	X
ejde-502	28	4	p	p	X
ejde-502	28	5	<	<	X
ejde-502	28	6	1	1	NUM
ejde-502	28	7	.	.	PUNCT
ejde-502	28	8	we	we	PRON
ejde-502	28	9	find	find	VERB
ejde-502	28	10	this	this	DET
ejde-502	28	11	latter	latter	ADJ
ejde-502	28	12	case	case	NOUN
ejde-502	28	13	very	very	ADV
ejde-502	28	14	interesting	interesting	ADJ
ejde-502	28	15	to	to	PART
ejde-502	28	16	study	study	VERB
ejde-502	28	17	due	due	ADP
ejde-502	28	18	to	to	ADP
ejde-502	28	19	the	the	DET
ejde-502	28	20	merging	merging	NOUN
ejde-502	28	21	of	of	ADP
ejde-502	28	22	the	the	DET
ejde-502	28	23	non	non	ADJ
ejde-502	28	24	-	-	ADJ
ejde-502	28	25	lipschitz	lipschitz	ADJ
ejde-502	28	26	reaction	reaction	NOUN
ejde-502	28	27	term	term	NOUN
ejde-502	28	28	(	(	PUNCT
ejde-502	28	29	which	which	PRON
ejde-502	28	30	does	do	AUX
ejde-502	28	31	not	not	PART
ejde-502	28	32	produce	produce	VERB
ejde-502	28	33	finite	finite	ADJ
ejde-502	28	34	time	time	NOUN
ejde-502	28	35	blow	blow	NOUN
ejde-502	28	36	-	-	PUNCT
ejde-502	28	37	up	up	NOUN
ejde-502	28	38	by	by	ADP
ejde-502	28	39	itself	itself	PRON
ejde-502	28	40	)	)	PUNCT
ejde-502	28	41	with	with	ADP
ejde-502	28	42	the	the	DET
ejde-502	28	43	unbounded	unbounded	ADJ
ejde-502	28	44	weight	weight	NOUN
ejde-502	28	45	.	.	PUNCT
ejde-502	29	1	the	the	DET
ejde-502	29	2	non	non	ADJ
ejde-502	29	3	-	-	ADJ
ejde-502	29	4	weighted	weighted	ADJ
ejde-502	29	5	equation	equation	NOUN
ejde-502	29	6	ut	ut	PROPN
ejde-502	29	7	=	=	PROPN
ejde-502	29	8	∆um	∆um	PROPN
ejde-502	29	9	+	+	X
ejde-502	29	10	up	up	ADP
ejde-502	29	11	,	,	PUNCT
ejde-502	29	12	(	(	PUNCT
ejde-502	29	13	1.5	1.5	NUM
ejde-502	29	14	)	)	PUNCT
ejde-502	29	15	corresponding	correspond	VERB
ejde-502	29	16	to	to	ADP
ejde-502	29	17	σ	σ	X
ejde-502	29	18	=	=	SYM
ejde-502	29	19	0	0	NUM
ejde-502	30	1	in	in	ADP
ejde-502	30	2	(	(	PUNCT
ejde-502	30	3	1.1	1.1	NUM
ejde-502	30	4	)	)	PUNCT
ejde-502	30	5	is	be	AUX
ejde-502	30	6	nowadays	nowadays	ADV
ejde-502	30	7	quite	quite	ADV
ejde-502	30	8	well	well	ADV
ejde-502	30	9	understood	understand	VERB
ejde-502	30	10	(	(	PUNCT
ejde-502	30	11	at	at	ADP
ejde-502	30	12	least	least	ADJ
ejde-502	30	13	in	in	ADP
ejde-502	30	14	dimension	dimension	NOUN
ejde-502	30	15	n	n	NOUN
ejde-502	30	16	=	=	SYM
ejde-502	30	17	1	1	NUM
ejde-502	30	18	)	)	PUNCT
ejde-502	30	19	after	after	ADP
ejde-502	30	20	a	a	DET
ejde-502	30	21	series	series	NOUN
ejde-502	30	22	of	of	ADP
ejde-502	30	23	works	work	NOUN
ejde-502	30	24	by	by	ADP
ejde-502	30	25	de	de	X
ejde-502	30	26	pablo	pablo	PROPN
ejde-502	30	27	and	and	CCONJ
ejde-502	30	28	vázquez	vázquez	PROPN
ejde-502	31	1	[	[	X
ejde-502	31	2	26	26	NUM
ejde-502	31	3	,	,	PUNCT
ejde-502	31	4	27	27	NUM
ejde-502	31	5	,	,	PUNCT
ejde-502	31	6	28	28	NUM
ejde-502	31	7	,	,	PUNCT
ejde-502	31	8	25	25	NUM
ejde-502	31	9	]	]	PUNCT
ejde-502	31	10	and	and	CCONJ
ejde-502	31	11	the	the	DET
ejde-502	31	12	outcome	outcome	NOUN
ejde-502	31	13	of	of	ADP
ejde-502	31	14	them	they	PRON
ejde-502	31	15	is	be	AUX
ejde-502	31	16	very	very	ADV
ejde-502	31	17	interesting	interesting	ADJ
ejde-502	31	18	,	,	PUNCT
ejde-502	31	19	despite	despite	SCONJ
ejde-502	31	20	the	the	DET
ejde-502	31	21	fact	fact	NOUN
ejde-502	31	22	that	that	SCONJ
ejde-502	31	23	we	we	PRON
ejde-502	31	24	are	be	AUX
ejde-502	31	25	dealing	deal	VERB
ejde-502	31	26	with	with	ADP
ejde-502	31	27	an	an	DET
ejde-502	31	28	ill	ill	ADV
ejde-502	31	29	-	-	PUNCT
ejde-502	31	30	posed	pose	VERB
ejde-502	31	31	problem	problem	NOUN
ejde-502	31	32	.	.	PUNCT
ejde-502	32	1	it	it	PRON
ejde-502	32	2	is	be	AUX
ejde-502	32	3	shown	show	VERB
ejde-502	32	4	that	that	SCONJ
ejde-502	32	5	solutions	solution	NOUN
ejde-502	32	6	exist	exist	VERB
ejde-502	32	7	always	always	ADV
ejde-502	32	8	and	and	CCONJ
ejde-502	32	9	they	they	PRON
ejde-502	32	10	are	be	AUX
ejde-502	32	11	global	global	ADJ
ejde-502	32	12	in	in	ADP
ejde-502	32	13	time	time	NOUN
ejde-502	32	14	if	if	SCONJ
ejde-502	32	15	u0	u0	ADJ
ejde-502	32	16	satisfies	satisfy	VERB
ejde-502	32	17	a	a	DET
ejde-502	32	18	growth	growth	NOUN
ejde-502	32	19	condition	condition	NOUN
ejde-502	32	20	similar	similar	ADJ
ejde-502	32	21	to	to	ADP
ejde-502	32	22	the	the	DET
ejde-502	32	23	one	one	NOUN
ejde-502	32	24	required	require	VERB
ejde-502	32	25	for	for	ADP
ejde-502	32	26	existence	existence	NOUN
ejde-502	32	27	in	in	ADP
ejde-502	32	28	the	the	DET
ejde-502	32	29	porous	porous	ADJ
ejde-502	32	30	medium	medium	NOUN
ejde-502	32	31	equation	equation	NOUN
ejde-502	32	32	,	,	PUNCT
ejde-502	32	33	namely	namely	ADV
ejde-502	32	34	u0(x	u0(x	NOUN
ejde-502	32	35	)	)	PUNCT
ejde-502	32	36	=	=	SYM
ejde-502	32	37	o(|x|2/(m−1	o(|x|2/(m−1	NOUN
ejde-502	32	38	)	)	PUNCT
ejde-502	32	39	)	)	PUNCT
ejde-502	33	1	[	[	X
ejde-502	33	2	27	27	NUM
ejde-502	33	3	]	]	PUNCT
ejde-502	33	4	and	and	CCONJ
ejde-502	33	5	that	that	SCONJ
ejde-502	33	6	there	there	PRON
ejde-502	33	7	is	be	VERB
ejde-502	33	8	always	always	ADV
ejde-502	33	9	a	a	DET
ejde-502	33	10	minimal	minimal	ADJ
ejde-502	33	11	and	and	CCONJ
ejde-502	33	12	a	a	DET
ejde-502	33	13	maximal	maximal	ADJ
ejde-502	33	14	solution	solution	NOUN
ejde-502	33	15	,	,	PUNCT
ejde-502	33	16	both	both	PRON
ejde-502	33	17	constructed	construct	VERB
ejde-502	33	18	via	via	ADP
ejde-502	33	19	approximations	approximation	NOUN
ejde-502	33	20	.	.	PUNCT
ejde-502	34	1	the	the	DET
ejde-502	34	2	most	most	ADV
ejde-502	34	3	interesting	interesting	ADJ
ejde-502	34	4	and	and	CCONJ
ejde-502	34	5	surprising	surprising	ADJ
ejde-502	34	6	features	feature	NOUN
ejde-502	34	7	of	of	ADP
ejde-502	34	8	this	this	DET
ejde-502	34	9	problem	problem	NOUN
ejde-502	34	10	with	with	ADP
ejde-502	34	11	σ	σ	PROPN
ejde-502	34	12	=	=	SYM
ejde-502	34	13	0	0	NUM
ejde-502	34	14	are	be	AUX
ejde-502	34	15	related	relate	VERB
ejde-502	34	16	with	with	ADP
ejde-502	34	17	the	the	DET
ejde-502	34	18	uniqueness	uniqueness	NOUN
ejde-502	34	19	.	.	PUNCT
ejde-502	35	1	this	this	DET
ejde-502	35	2	property	property	NOUN
ejde-502	35	3	depends	depend	VERB
ejde-502	35	4	strongly	strongly	ADV
ejde-502	35	5	on	on	ADP
ejde-502	35	6	two	two	NUM
ejde-502	35	7	aspects	aspect	NOUN
ejde-502	35	8	:	:	PUNCT
ejde-502	35	9	the	the	DET
ejde-502	35	10	sign	sign	NOUN
ejde-502	35	11	of	of	ADP
ejde-502	35	12	the	the	DET
ejde-502	35	13	critical	critical	ADJ
ejde-502	35	14	exponent	exponent	NOUN
ejde-502	35	15	m+	m+	NOUN
ejde-502	35	16	p−	p−	NOUN
ejde-502	35	17	2	2	NUM
ejde-502	35	18	and	and	CCONJ
ejde-502	35	19	the	the	DET
ejde-502	35	20	positivity	positivity	NOUN
ejde-502	35	21	or	or	CCONJ
ejde-502	35	22	not	not	PART
ejde-502	35	23	of	of	ADP
ejde-502	35	24	the	the	DET
ejde-502	35	25	initial	initial	ADJ
ejde-502	35	26	condition	condition	NOUN
ejde-502	35	27	u0	u0	ADJ
ejde-502	35	28	.	.	PUNCT
ejde-502	36	1	more	more	ADV
ejde-502	36	2	precisely	precisely	ADV
ejde-502	36	3	we	we	PRON
ejde-502	36	4	have	have	VERB
ejde-502	36	5	the	the	DET
ejde-502	36	6	following	following	NOUN
ejde-502	36	7	:	:	PUNCT
ejde-502	36	8	•	•	INTJ
ejde-502	36	9	if	if	SCONJ
ejde-502	36	10	m	m	VERB
ejde-502	36	11	+	+	NOUN
ejde-502	36	12	p	p	DET
ejde-502	36	13	−	−	PROPN
ejde-502	36	14	2	2	NUM
ejde-502	36	15	≥	≥	NOUN
ejde-502	36	16	0	0	NUM
ejde-502	36	17	,	,	PUNCT
ejde-502	36	18	uniqueness	uniqueness	NOUN
ejde-502	36	19	of	of	ADP
ejde-502	36	20	solutions	solution	NOUN
ejde-502	36	21	to	to	ADP
ejde-502	36	22	the	the	DET
ejde-502	36	23	cauchy	cauchy	ADJ
ejde-502	36	24	problem	problem	NOUN
ejde-502	36	25	(	(	PUNCT
ejde-502	36	26	1.5)-(1.2	1.5)-(1.2	NOUN
ejde-502	36	27	)	)	PUNCT
ejde-502	36	28	holds	hold	VERB
ejde-502	36	29	if	if	SCONJ
ejde-502	36	30	and	and	CCONJ
ejde-502	36	31	only	only	ADV
ejde-502	36	32	if	if	SCONJ
ejde-502	36	33	suppu0(x	suppu0(x	NOUN
ejde-502	36	34	)	)	PUNCT
ejde-502	36	35	=	=	SYM
ejde-502	37	1	rn	rn	PROPN
ejde-502	38	1	[	[	X
ejde-502	38	2	26	26	NUM
ejde-502	38	3	]	]	PUNCT
ejde-502	38	4	.	.	PUNCT
ejde-502	39	1	if	if	SCONJ
ejde-502	39	2	u0	u0	ADJ
ejde-502	39	3	is	be	AUX
ejde-502	39	4	compactly	compactly	ADV
ejde-502	39	5	supported	support	VERB
ejde-502	39	6	,	,	PUNCT
ejde-502	39	7	then	then	ADV
ejde-502	39	8	its	its	PRON
ejde-502	39	9	support	support	NOUN
ejde-502	39	10	has	have	VERB
ejde-502	39	11	the	the	DET
ejde-502	39	12	property	property	NOUN
ejde-502	39	13	of	of	ADP
ejde-502	39	14	finite	finite	ADJ
ejde-502	39	15	propagation	propagation	NOUN
ejde-502	39	16	and	and	CCONJ
ejde-502	39	17	interfaces	interface	NOUN
ejde-502	39	18	appear	appear	VERB
ejde-502	39	19	.	.	PUNCT
ejde-502	40	1	moreover	moreover	ADV
ejde-502	40	2	,	,	PUNCT
ejde-502	40	3	the	the	DET
ejde-502	40	4	extent	extent	NOUN
ejde-502	40	5	of	of	ADP
ejde-502	40	6	the	the	DET
ejde-502	40	7	non	non	ADJ
ejde-502	40	8	-	-	ADJ
ejde-502	40	9	uniqueness	uniqueness	ADJ
ejde-502	40	10	property	property	NOUN
ejde-502	40	11	in	in	ADP
ejde-502	40	12	this	this	DET
ejde-502	40	13	case	case	NOUN
ejde-502	40	14	is	be	AUX
ejde-502	40	15	addressed	address	VERB
ejde-502	40	16	in	in	ADP
ejde-502	40	17	[	[	X
ejde-502	40	18	28	28	NUM
ejde-502	40	19	]	]	PUNCT
ejde-502	40	20	where	where	SCONJ
ejde-502	40	21	it	it	PRON
ejde-502	40	22	is	be	AUX
ejde-502	40	23	shown	show	VERB
ejde-502	40	24	that	that	SCONJ
ejde-502	40	25	giving	give	VERB
ejde-502	40	26	a	a	DET
ejde-502	40	27	rather	rather	ADV
ejde-502	40	28	general	general	ADJ
ejde-502	40	29	compactly	compactly	ADV
ejde-502	40	30	supported	support	VERB
ejde-502	40	31	initial	initial	ADJ
ejde-502	40	32	condition	condition	NOUN
ejde-502	40	33	u0	u0	ADJ
ejde-502	40	34	and	and	CCONJ
ejde-502	40	35	a	a	DET
ejde-502	40	36	function	function	NOUN
ejde-502	40	37	of	of	ADP
ejde-502	40	38	time	time	NOUN
ejde-502	40	39	ξ(t	ξ(t	NOUN
ejde-502	40	40	)	)	PUNCT
ejde-502	40	41	advancing	advance	VERB
ejde-502	40	42	faster	fast	ADV
ejde-502	40	43	than	than	ADP
ejde-502	40	44	the	the	DET
ejde-502	40	45	interface	interface	NOUN
ejde-502	40	46	of	of	ADP
ejde-502	40	47	the	the	DET
ejde-502	40	48	(	(	PUNCT
ejde-502	40	49	unique	unique	ADJ
ejde-502	40	50	)	)	PUNCT
ejde-502	40	51	minimal	minimal	ADJ
ejde-502	40	52	solution	solution	NOUN
ejde-502	40	53	constructed	construct	VERB
ejde-502	40	54	in	in	ADP
ejde-502	40	55	[	[	X
ejde-502	40	56	27	27	NUM
ejde-502	40	57	]	]	PUNCT
ejde-502	40	58	,	,	PUNCT
ejde-502	40	59	there	there	PRON
ejde-502	40	60	is	be	VERB
ejde-502	40	61	always	always	ADV
ejde-502	40	62	a	a	DET
ejde-502	40	63	solution	solution	NOUN
ejde-502	40	64	to	to	ADP
ejde-502	40	65	the	the	DET
ejde-502	40	66	cauchy	cauchy	ADJ
ejde-502	40	67	problem	problem	NOUN
ejde-502	40	68	with	with	ADP
ejde-502	40	69	data	data	NOUN
ejde-502	40	70	u0	u0	ADJ
ejde-502	40	71	and	and	CCONJ
ejde-502	40	72	interface	interface	NOUN
ejde-502	40	73	at	at	ADP
ejde-502	40	74	time	time	NOUN
ejde-502	40	75	t	t	PROPN
ejde-502	40	76	>	>	X
ejde-502	40	77	0	0	PUNCT
ejde-502	40	78	given	give	VERB
ejde-502	40	79	by	by	ADP
ejde-502	40	80	ξ(t	ξ(t	NOUN
ejde-502	40	81	)	)	PUNCT
ejde-502	40	82	.	.	PUNCT
ejde-502	41	1	this	this	PRON
ejde-502	41	2	is	be	AUX
ejde-502	41	3	a	a	DET
ejde-502	41	4	very	very	ADV
ejde-502	41	5	strong	strong	ADJ
ejde-502	41	6	and	and	CCONJ
ejde-502	41	7	sharp	sharp	ADJ
ejde-502	41	8	non	non	ADJ
ejde-502	41	9	-	-	ADJ
ejde-502	41	10	uniqueness	uniqueness	ADJ
ejde-502	41	11	property	property	NOUN
ejde-502	41	12	,	,	PUNCT
ejde-502	41	13	giving	give	VERB
ejde-502	41	14	rise	rise	NOUN
ejde-502	41	15	in	in	ADP
ejde-502	41	16	fact	fact	NOUN
ejde-502	41	17	to	to	ADP
ejde-502	41	18	an	an	DET
ejde-502	41	19	infinity	infinity	NOUN
ejde-502	41	20	of	of	ADP
ejde-502	41	21	solutions	solution	NOUN
ejde-502	41	22	.	.	PUNCT
ejde-502	42	1	•	•	INTJ
ejde-502	42	2	if	if	SCONJ
ejde-502	42	3	m	m	VERB
ejde-502	42	4	+	+	NOUN
ejde-502	43	1	p	p	X
ejde-502	43	2	−	−	PROPN
ejde-502	43	3	2	2	NUM
ejde-502	43	4	<	<	X
ejde-502	43	5	0	0	NUM
ejde-502	43	6	,	,	PUNCT
ejde-502	43	7	things	thing	NOUN
ejde-502	43	8	change	change	VERB
ejde-502	43	9	radically	radically	ADV
ejde-502	43	10	because	because	SCONJ
ejde-502	43	11	of	of	ADP
ejde-502	43	12	the	the	DET
ejde-502	43	13	infinite	infinite	ADJ
ejde-502	43	14	speed	speed	NOUN
ejde-502	43	15	of	of	ADP
ejde-502	43	16	propagation	propagation	NOUN
ejde-502	43	17	:	:	PUNCT
ejde-502	43	18	even	even	ADV
ejde-502	43	19	if	if	SCONJ
ejde-502	43	20	u0	u0	ADJ
ejde-502	43	21	is	be	AUX
ejde-502	43	22	compactly	compactly	ADV
ejde-502	43	23	supported	support	VERB
ejde-502	43	24	it	it	PRON
ejde-502	43	25	is	be	AUX
ejde-502	43	26	shown	show	VERB
ejde-502	43	27	in	in	ADP
ejde-502	43	28	[	[	X
ejde-502	43	29	26	26	NUM
ejde-502	43	30	]	]	PUNCT
ejde-502	43	31	that	that	SCONJ
ejde-502	43	32	u(x	u(x	PROPN
ejde-502	43	33	,	,	PUNCT
ejde-502	43	34	t	t	PROPN
ejde-502	43	35	)	)	PUNCT
ejde-502	43	36	>	>	X
ejde-502	43	37	0	0	PUNCT
ejde-502	44	1	for	for	ADP
ejde-502	44	2	any	any	DET
ejde-502	44	3	x	x	SYM
ejde-502	44	4	∈	∈	PROPN
ejde-502	44	5	rn	rn	PROPN
ejde-502	44	6	and	and	CCONJ
ejde-502	44	7	t	t	PROPN
ejde-502	44	8	>	>	X
ejde-502	44	9	0	0	PROPN
ejde-502	44	10	,	,	PUNCT
ejde-502	44	11	thus	thus	ADV
ejde-502	44	12	we	we	PRON
ejde-502	44	13	have	have	VERB
ejde-502	44	14	a	a	DET
ejde-502	44	15	property	property	NOUN
ejde-502	44	16	known	know	VERB
ejde-502	44	17	as	as	ADP
ejde-502	44	18	quasi	quasi	NOUN
ejde-502	44	19	-	-	NOUN
ejde-502	44	20	uniqueness	uniqueness	ADJ
ejde-502	44	21	:	:	PUNCT
ejde-502	44	22	there	there	PRON
ejde-502	44	23	exists	exist	VERB
ejde-502	44	24	a	a	DET
ejde-502	44	25	unique	unique	ADJ
ejde-502	44	26	solution	solution	NOUN
ejde-502	44	27	to	to	ADP
ejde-502	44	28	the	the	DET
ejde-502	44	29	cauchy	cauchy	ADJ
ejde-502	44	30	problem	problem	NOUN
ejde-502	44	31	for	for	ADP
ejde-502	44	32	any	any	DET
ejde-502	44	33	initial	initial	ADJ
ejde-502	44	34	condition	condition	NOUN
ejde-502	44	35	u0	u0	ADJ
ejde-502	44	36	except	except	SCONJ
ejde-502	44	37	for	for	ADP
ejde-502	44	38	the	the	DET
ejde-502	44	39	trivial	trivial	ADJ
ejde-502	44	40	one	one	NUM
ejde-502	44	41	u0	u0	ADJ
ejde-502	44	42	≡	≡	PROPN
ejde-502	44	43	0	0	PROPN
ejde-502	44	44	,	,	PUNCT
ejde-502	44	45	where	where	SCONJ
ejde-502	44	46	two	two	NUM
ejde-502	44	47	different	different	ADJ
ejde-502	44	48	solutions	solution	NOUN
ejde-502	44	49	are	be	AUX
ejde-502	44	50	constructed	construct	VERB
ejde-502	44	51	.	.	PUNCT
ejde-502	45	1	considering	consider	VERB
ejde-502	45	2	weighted	weight	VERB
ejde-502	45	3	reaction	reaction	NOUN
ejde-502	45	4	terms	term	NOUN
ejde-502	45	5	came	come	VERB
ejde-502	45	6	as	as	ADP
ejde-502	45	7	a	a	DET
ejde-502	45	8	natural	natural	ADJ
ejde-502	45	9	extension	extension	NOUN
ejde-502	45	10	of	of	ADP
ejde-502	45	11	the	the	DET
ejde-502	45	12	already	already	ADV
ejde-502	45	13	well	well	ADV
ejde-502	45	14	developed	develop	VERB
ejde-502	45	15	knowledge	knowledge	NOUN
ejde-502	45	16	on	on	ADP
ejde-502	45	17	the	the	DET
ejde-502	45	18	“	"	PUNCT
ejde-502	45	19	classical	classical	ADJ
ejde-502	45	20	”	"	PUNCT
ejde-502	45	21	reaction	reaction	NOUN
ejde-502	45	22	-	-	PUNCT
ejde-502	45	23	diffusion	diffusion	NOUN
ejde-502	45	24	equations	equation	NOUN
ejde-502	45	25	with	with	ADP
ejde-502	45	26	reaction	reaction	NOUN
ejde-502	45	27	of	of	ADP
ejde-502	45	28	the	the	DET
ejde-502	45	29	form	form	NOUN
ejde-502	45	30	up	up	ADP
ejde-502	45	31	or	or	CCONJ
ejde-502	45	32	more	more	ADJ
ejde-502	45	33	general	general	ADJ
ejde-502	45	34	functions	function	NOUN
ejde-502	45	35	resembling	resemble	VERB
ejde-502	45	36	it	it	PRON
ejde-502	45	37	(	(	PUNCT
ejde-502	45	38	see	see	VERB
ejde-502	45	39	[	[	X
ejde-502	45	40	32	32	NUM
ejde-502	45	41	,	,	PUNCT
ejde-502	45	42	34	34	NUM
ejde-502	45	43	]	]	PUNCT
ejde-502	45	44	as	as	ADP
ejde-502	45	45	important	important	ADJ
ejde-502	45	46	monographs	monograph	NOUN
ejde-502	45	47	on	on	ADP
ejde-502	45	48	this	this	DET
ejde-502	45	49	subject	subject	NOUN
ejde-502	45	50	)	)	PUNCT
ejde-502	45	51	.	.	PUNCT
ejde-502	46	1	many	many	ADJ
ejde-502	46	2	results	result	NOUN
ejde-502	46	3	were	be	AUX
ejde-502	46	4	achieved	achieve	VERB
ejde-502	46	5	for	for	ADP
ejde-502	46	6	the	the	DET
ejde-502	46	7	semilinear	semilinear	NOUN
ejde-502	46	8	case	case	NOUN
ejde-502	46	9	ejde-2023/72	ejde-2023/72	VERB
ejde-502	46	10	reaction	reaction	NOUN
ejde-502	46	11	-	-	PUNCT
ejde-502	46	12	diffusion	diffusion	NOUN
ejde-502	46	13	with	with	ADP
ejde-502	46	14	weighted	weight	VERB
ejde-502	46	15	strong	strong	ADJ
ejde-502	46	16	reaction	reaction	NOUN
ejde-502	46	17	3	3	NUM
ejde-502	46	18	m	m	NOUN
ejde-502	46	19	=	=	SYM
ejde-502	46	20	1	1	NUM
ejde-502	46	21	and	and	CCONJ
ejde-502	46	22	unbounded	unbounded	ADJ
ejde-502	46	23	weights	weight	NOUN
ejde-502	46	24	of	of	ADP
ejde-502	46	25	the	the	DET
ejde-502	46	26	form	form	NOUN
ejde-502	46	27	|x|σup	|x|σup	PUNCT
ejde-502	46	28	(	(	PUNCT
ejde-502	46	29	and	and	CCONJ
ejde-502	46	30	sometimes	sometimes	ADV
ejde-502	46	31	even	even	ADV
ejde-502	46	32	more	more	ADV
ejde-502	46	33	general	general	ADJ
ejde-502	46	34	weights	weight	NOUN
ejde-502	46	35	v	v	X
ejde-502	46	36	(	(	PUNCT
ejde-502	46	37	x	x	NOUN
ejde-502	46	38	)	)	PUNCT
ejde-502	46	39	instead	instead	ADV
ejde-502	46	40	of	of	ADP
ejde-502	46	41	pure	pure	ADJ
ejde-502	46	42	powers	power	NOUN
ejde-502	46	43	)	)	PUNCT
ejde-502	46	44	,	,	PUNCT
ejde-502	46	45	always	always	ADV
ejde-502	46	46	with	with	ADP
ejde-502	46	47	p	p	PROPN
ejde-502	46	48	>	>	X
ejde-502	46	49	1	1	NUM
ejde-502	46	50	.	.	PUNCT
ejde-502	46	51	fujita	fujita	NOUN
ejde-502	46	52	-	-	PUNCT
ejde-502	46	53	type	type	NOUN
ejde-502	46	54	exponents	exponent	NOUN
ejde-502	46	55	and	and	CCONJ
ejde-502	46	56	conditions	condition	NOUN
ejde-502	46	57	on	on	ADP
ejde-502	46	58	the	the	DET
ejde-502	46	59	data	datum	NOUN
ejde-502	46	60	for	for	ADP
ejde-502	46	61	finite	finite	ADJ
ejde-502	46	62	time	time	NOUN
ejde-502	46	63	blow	blow	NOUN
ejde-502	46	64	-	-	PUNCT
ejde-502	46	65	up	up	NOUN
ejde-502	46	66	to	to	PART
ejde-502	46	67	occur	occur	VERB
ejde-502	46	68	were	be	AUX
ejde-502	46	69	studied	study	VERB
ejde-502	46	70	in	in	ADP
ejde-502	46	71	celebrated	celebrate	VERB
ejde-502	46	72	papers	paper	NOUN
ejde-502	46	73	by	by	ADP
ejde-502	46	74	baras	bara	NOUN
ejde-502	46	75	and	and	CCONJ
ejde-502	46	76	kersner	kersner	NOUN
ejde-502	46	77	,	,	PUNCT
ejde-502	46	78	bandle	bandle	NOUN
ejde-502	46	79	and	and	CCONJ
ejde-502	46	80	levine	levine	PROPN
ejde-502	46	81	,	,	PUNCT
ejde-502	46	82	pinsky	pinsky	PROPN
ejde-502	46	83	et	et	PROPN
ejde-502	46	84	al	al	PROPN
ejde-502	46	85	.	.	PROPN
ejde-502	46	86	,	,	PUNCT
ejde-502	46	87	see	see	VERB
ejde-502	46	88	for	for	ADP
ejde-502	46	89	example	example	NOUN
ejde-502	47	1	[	[	X
ejde-502	47	2	6	6	NUM
ejde-502	47	3	,	,	PUNCT
ejde-502	47	4	5	5	NUM
ejde-502	47	5	,	,	PUNCT
ejde-502	47	6	29	29	NUM
ejde-502	47	7	,	,	PUNCT
ejde-502	47	8	30	30	NUM
ejde-502	47	9	]	]	PUNCT
ejde-502	47	10	.	.	PUNCT
ejde-502	48	1	more	more	ADV
ejde-502	48	2	recently	recently	ADV
ejde-502	48	3	,	,	PUNCT
ejde-502	48	4	still	still	ADV
ejde-502	48	5	with	with	ADP
ejde-502	48	6	m	m	PROPN
ejde-502	48	7	=	=	SYM
ejde-502	48	8	1	1	NUM
ejde-502	48	9	,	,	PUNCT
ejde-502	48	10	an	an	DET
ejde-502	48	11	interesting	interesting	ADJ
ejde-502	48	12	question	question	NOUN
ejde-502	48	13	was	be	AUX
ejde-502	48	14	addressed	address	VERB
ejde-502	48	15	:	:	PUNCT
ejde-502	48	16	considering	consider	VERB
ejde-502	48	17	the	the	DET
ejde-502	48	18	equation	equation	NOUN
ejde-502	48	19	ut	ut	PROPN
ejde-502	48	20	=	=	PROPN
ejde-502	48	21	∆u+	∆u+	PROPN
ejde-502	48	22	|x|σup	|x|σup	NUM
ejde-502	48	23	,	,	PUNCT
ejde-502	48	24	(	(	PUNCT
ejde-502	48	25	1.6	1.6	NUM
ejde-502	48	26	)	)	PUNCT
ejde-502	48	27	it	it	PRON
ejde-502	48	28	is	be	AUX
ejde-502	48	29	natural	natural	ADJ
ejde-502	48	30	to	to	PART
ejde-502	48	31	ask	ask	VERB
ejde-502	48	32	ourselves	ourselves	PRON
ejde-502	48	33	whether	whether	SCONJ
ejde-502	48	34	x	x	AUX
ejde-502	48	35	=	=	SYM
ejde-502	48	36	0	0	NUM
ejde-502	48	37	(	(	PUNCT
ejde-502	48	38	or	or	CCONJ
ejde-502	48	39	more	more	ADV
ejde-502	48	40	generally	generally	ADV
ejde-502	48	41	,	,	PUNCT
ejde-502	48	42	any	any	DET
ejde-502	48	43	zero	zero	NUM
ejde-502	48	44	of	of	ADP
ejde-502	48	45	a	a	DET
ejde-502	48	46	powerlike	powerlike	ADJ
ejde-502	48	47	weight	weight	NOUN
ejde-502	48	48	v	v	NOUN
ejde-502	48	49	(	(	PUNCT
ejde-502	48	50	x	x	NOUN
ejde-502	48	51	)	)	PUNCT
ejde-502	48	52	)	)	PUNCT
ejde-502	48	53	can	can	AUX
ejde-502	48	54	be	be	AUX
ejde-502	48	55	a	a	DET
ejde-502	48	56	blow	blow	NOUN
ejde-502	48	57	-	-	PUNCT
ejde-502	48	58	up	up	ADP
ejde-502	48	59	point	point	NOUN
ejde-502	48	60	.	.	PUNCT
ejde-502	49	1	examples	example	NOUN
ejde-502	49	2	of	of	ADP
ejde-502	49	3	both	both	CCONJ
ejde-502	49	4	possible	possible	ADJ
ejde-502	49	5	situations	situation	NOUN
ejde-502	49	6	(	(	PUNCT
ejde-502	49	7	when	when	SCONJ
ejde-502	49	8	x	x	SYM
ejde-502	49	9	=	=	SYM
ejde-502	49	10	0	0	NUM
ejde-502	49	11	is	be	AUX
ejde-502	49	12	a	a	DET
ejde-502	49	13	blow	blow	NOUN
ejde-502	49	14	-	-	PUNCT
ejde-502	49	15	up	up	ADP
ejde-502	49	16	point	point	NOUN
ejde-502	49	17	and	and	CCONJ
ejde-502	49	18	when	when	SCONJ
ejde-502	49	19	it	it	PRON
ejde-502	49	20	is	be	AUX
ejde-502	49	21	not	not	PART
ejde-502	49	22	)	)	PUNCT
ejde-502	49	23	were	be	AUX
ejde-502	49	24	constructed	construct	VERB
ejde-502	49	25	(	(	PUNCT
ejde-502	49	26	mostly	mostly	ADV
ejde-502	49	27	for	for	ADP
ejde-502	49	28	cauchy	cauchy	PROPN
ejde-502	49	29	-	-	PUNCT
ejde-502	49	30	dirichlet	dirichlet	PROPN
ejde-502	49	31	problems	problem	NOUN
ejde-502	49	32	posed	pose	VERB
ejde-502	49	33	in	in	ADP
ejde-502	49	34	bounded	bounded	ADJ
ejde-502	49	35	domains	domain	NOUN
ejde-502	49	36	)	)	PUNCT
ejde-502	49	37	in	in	ADP
ejde-502	49	38	the	the	DET
ejde-502	49	39	series	series	NOUN
ejde-502	49	40	of	of	ADP
ejde-502	49	41	recent	recent	ADJ
ejde-502	49	42	papers	paper	NOUN
ejde-502	49	43	by	by	ADP
ejde-502	49	44	guo	guo	PROPN
ejde-502	49	45	and	and	CCONJ
ejde-502	49	46	collaborators	collaborator	NOUN
ejde-502	49	47	[	[	X
ejde-502	49	48	11	11	NUM
ejde-502	49	49	,	,	PUNCT
ejde-502	49	50	12	12	NUM
ejde-502	49	51	,	,	PUNCT
ejde-502	49	52	13	13	NUM
ejde-502	49	53	,	,	PUNCT
ejde-502	49	54	14	14	NUM
ejde-502	49	55	]	]	PUNCT
ejde-502	49	56	.	.	PUNCT
ejde-502	50	1	finer	fine	ADJ
ejde-502	50	2	analysis	analysis	NOUN
ejde-502	50	3	on	on	ADP
ejde-502	50	4	how	how	SCONJ
ejde-502	50	5	blow	blow	VERB
ejde-502	50	6	-	-	PUNCT
ejde-502	50	7	up	up	ADP
ejde-502	50	8	occurs	occur	NOUN
ejde-502	50	9	,	,	PUNCT
ejde-502	50	10	with	with	ADP
ejde-502	50	11	rates	rate	NOUN
ejde-502	50	12	and	and	CCONJ
ejde-502	50	13	local	local	ADJ
ejde-502	50	14	asymptotic	asymptotic	ADJ
ejde-502	50	15	behavior	behavior	NOUN
ejde-502	50	16	in	in	ADP
ejde-502	50	17	self	self	NOUN
ejde-502	50	18	-	-	PUNCT
ejde-502	50	19	similar	similar	ADJ
ejde-502	50	20	forms	form	NOUN
ejde-502	50	21	near	near	ADP
ejde-502	50	22	the	the	DET
ejde-502	50	23	blow	blow	NOUN
ejde-502	50	24	-	-	PUNCT
ejde-502	50	25	up	up	ADP
ejde-502	50	26	time	time	NOUN
ejde-502	50	27	and	and	CCONJ
ejde-502	50	28	points	point	NOUN
ejde-502	50	29	were	be	AUX
ejde-502	50	30	performed	perform	VERB
ejde-502	50	31	by	by	ADP
ejde-502	50	32	filippas	filippa	NOUN
ejde-502	50	33	and	and	CCONJ
ejde-502	50	34	tertikas	tertikas	ADV
ejde-502	50	35	[	[	X
ejde-502	50	36	10	10	NUM
ejde-502	50	37	]	]	PUNCT
ejde-502	50	38	and	and	CCONJ
ejde-502	50	39	in	in	ADP
ejde-502	50	40	the	the	DET
ejde-502	50	41	very	very	ADV
ejde-502	50	42	recent	recent	ADJ
ejde-502	50	43	work	work	NOUN
ejde-502	50	44	by	by	ADP
ejde-502	50	45	mukai	mukai	PROPN
ejde-502	50	46	and	and	CCONJ
ejde-502	50	47	seki	seki	PROPN
ejde-502	51	1	[	[	X
ejde-502	51	2	24	24	NUM
ejde-502	51	3	]	]	PUNCT
ejde-502	51	4	for	for	ADP
ejde-502	51	5	different	different	ADJ
ejde-502	51	6	ranges	range	NOUN
ejde-502	51	7	of	of	ADP
ejde-502	51	8	the	the	DET
ejde-502	51	9	exponent	exponent	NOUN
ejde-502	51	10	p	p	X
ejde-502	51	11	>	>	X
ejde-502	51	12	1	1	NUM
ejde-502	51	13	.	.	PUNCT
ejde-502	51	14	because	because	SCONJ
ejde-502	51	15	of	of	ADP
ejde-502	51	16	their	their	PRON
ejde-502	51	17	further	further	ADJ
ejde-502	51	18	complexity	complexity	NOUN
ejde-502	51	19	and	and	CCONJ
ejde-502	51	20	the	the	DET
ejde-502	51	21	fact	fact	NOUN
ejde-502	51	22	that	that	SCONJ
ejde-502	51	23	even	even	ADV
ejde-502	51	24	for	for	ADP
ejde-502	51	25	the	the	DET
ejde-502	51	26	non	non	ADJ
ejde-502	51	27	-	-	ADJ
ejde-502	51	28	weighted	weighted	ADJ
ejde-502	51	29	case	case	NOUN
ejde-502	52	1	σ	σ	X
ejde-502	52	2	=	=	SYM
ejde-502	52	3	0	0	NUM
ejde-502	52	4	there	there	PRON
ejde-502	52	5	are	be	VERB
ejde-502	52	6	some	some	DET
ejde-502	52	7	difficult	difficult	ADJ
ejde-502	52	8	open	open	ADJ
ejde-502	52	9	problems	problem	NOUN
ejde-502	52	10	(	(	PUNCT
ejde-502	52	11	see	see	VERB
ejde-502	52	12	for	for	ADP
ejde-502	52	13	example	example	NOUN
ejde-502	53	1	[	[	X
ejde-502	53	2	34	34	NUM
ejde-502	53	3	,	,	PUNCT
ejde-502	53	4	chapter	chapter	NOUN
ejde-502	53	5	4	4	NUM
ejde-502	53	6	]	]	PUNCT
ejde-502	53	7	)	)	PUNCT
ejde-502	53	8	,	,	PUNCT
ejde-502	53	9	equations	equation	NOUN
ejde-502	53	10	such	such	ADJ
ejde-502	53	11	as	as	ADP
ejde-502	53	12	(	(	PUNCT
ejde-502	53	13	1.1	1.1	NUM
ejde-502	53	14	)	)	PUNCT
ejde-502	53	15	or	or	CCONJ
ejde-502	53	16	its	its	PRON
ejde-502	53	17	close	close	ADJ
ejde-502	53	18	relative	relative	ADJ
ejde-502	53	19	ut	ut	PROPN
ejde-502	53	20	=	=	PROPN
ejde-502	53	21	∆um	∆um	PROPN
ejde-502	53	22	+	+	CCONJ
ejde-502	53	23	|x|σup	|x|σup	PROPN
ejde-502	53	24	,	,	PUNCT
ejde-502	53	25	(	(	PUNCT
ejde-502	53	26	1.7	1.7	NUM
ejde-502	53	27	)	)	PUNCT
ejde-502	53	28	with	with	ADP
ejde-502	53	29	m	m	PROPN
ejde-502	53	30	>	>	X
ejde-502	53	31	1	1	NUM
ejde-502	53	32	have	have	AUX
ejde-502	53	33	been	be	AUX
ejde-502	53	34	less	less	ADV
ejde-502	53	35	considered	consider	VERB
ejde-502	53	36	in	in	ADP
ejde-502	53	37	literature	literature	NOUN
ejde-502	53	38	.	.	PUNCT
ejde-502	54	1	apart	apart	ADV
ejde-502	54	2	from	from	ADP
ejde-502	54	3	the	the	DET
ejde-502	54	4	quoted	quote	VERB
ejde-502	54	5	paper	paper	NOUN
ejde-502	54	6	[	[	X
ejde-502	54	7	1	1	NUM
ejde-502	54	8	]	]	PUNCT
ejde-502	54	9	,	,	PUNCT
ejde-502	54	10	the	the	DET
ejde-502	54	11	fujita	fujita	NOUN
ejde-502	54	12	-	-	PUNCT
ejde-502	54	13	type	type	NOUN
ejde-502	54	14	exponent	exponent	NOUN
ejde-502	54	15	and	and	CCONJ
ejde-502	54	16	rather	rather	ADV
ejde-502	54	17	sharp	sharp	ADJ
ejde-502	54	18	conditions	condition	NOUN
ejde-502	54	19	on	on	ADP
ejde-502	54	20	the	the	DET
ejde-502	54	21	initial	initial	ADJ
ejde-502	54	22	data	datum	NOUN
ejde-502	54	23	for	for	ADP
ejde-502	54	24	the	the	DET
ejde-502	54	25	finite	finite	ADJ
ejde-502	54	26	time	time	NOUN
ejde-502	54	27	blow	blow	NOUN
ejde-502	54	28	-	-	PUNCT
ejde-502	54	29	up	up	NOUN
ejde-502	54	30	to	to	PART
ejde-502	54	31	hold	hold	VERB
ejde-502	54	32	have	have	AUX
ejde-502	54	33	been	be	AUX
ejde-502	54	34	obtained	obtain	VERB
ejde-502	54	35	by	by	ADP
ejde-502	54	36	qi	qi	PROPN
ejde-502	54	37	[	[	X
ejde-502	54	38	31	31	NUM
ejde-502	54	39	]	]	PUNCT
ejde-502	54	40	and	and	CCONJ
ejde-502	54	41	then	then	ADV
ejde-502	54	42	suzuki	suzuki	PROPN
ejde-502	54	43	[	[	X
ejde-502	54	44	35	35	NUM
ejde-502	54	45	]	]	PUNCT
ejde-502	54	46	for	for	ADP
ejde-502	54	47	the	the	DET
ejde-502	54	48	case	case	NOUN
ejde-502	54	49	p	p	X
ejde-502	54	50	>	>	X
ejde-502	54	51	m	m	VERB
ejde-502	54	52	>	>	X
ejde-502	54	53	1	1	NUM
ejde-502	54	54	(	(	PUNCT
ejde-502	54	55	including	include	VERB
ejde-502	54	56	also	also	ADV
ejde-502	54	57	a	a	DET
ejde-502	54	58	part	part	NOUN
ejde-502	54	59	of	of	ADP
ejde-502	54	60	the	the	DET
ejde-502	54	61	fast	fast	ADJ
ejde-502	54	62	diffusion	diffusion	NOUN
ejde-502	54	63	range	range	NOUN
ejde-502	54	64	m	m	VERB
ejde-502	54	65	<	<	X
ejde-502	54	66	1	1	NUM
ejde-502	54	67	in	in	ADP
ejde-502	54	68	[	[	X
ejde-502	54	69	31	31	NUM
ejde-502	54	70	]	]	PUNCT
ejde-502	54	71	)	)	PUNCT
ejde-502	54	72	.	.	PUNCT
ejde-502	55	1	we	we	PRON
ejde-502	55	2	recommend	recommend	VERB
ejde-502	55	3	suzuki	suzuki	PROPN
ejde-502	55	4	’s	’s	PART
ejde-502	55	5	paper	paper	NOUN
ejde-502	55	6	to	to	ADP
ejde-502	55	7	the	the	DET
ejde-502	55	8	reader	reader	NOUN
ejde-502	55	9	as	as	ADP
ejde-502	55	10	a	a	DET
ejde-502	55	11	well	well	ADV
ejde-502	55	12	-	-	PUNCT
ejde-502	55	13	written	write	VERB
ejde-502	55	14	basic	basic	ADJ
ejde-502	55	15	work	work	NOUN
ejde-502	55	16	on	on	ADP
ejde-502	55	17	the	the	DET
ejde-502	55	18	qualitative	qualitative	ADJ
ejde-502	55	19	theory	theory	NOUN
ejde-502	55	20	for	for	ADP
ejde-502	55	21	these	these	DET
ejde-502	55	22	equations	equation	NOUN
ejde-502	55	23	,	,	PUNCT
ejde-502	55	24	while	while	SCONJ
ejde-502	55	25	blow	blow	VERB
ejde-502	55	26	-	-	PUNCT
ejde-502	55	27	up	up	ADP
ejde-502	55	28	rates	rate	NOUN
ejde-502	55	29	as	as	ADP
ejde-502	55	30	t→	t→	X
ejde-502	55	31	t	t	PROPN
ejde-502	55	32	also	also	ADV
ejde-502	55	33	for	for	ADP
ejde-502	55	34	p	p	PROPN
ejde-502	55	35	>	>	X
ejde-502	55	36	m	m	VERB
ejde-502	55	37	are	be	AUX
ejde-502	55	38	proved	prove	VERB
ejde-502	55	39	in	in	ADP
ejde-502	55	40	[	[	X
ejde-502	55	41	2	2	NUM
ejde-502	55	42	]	]	PUNCT
ejde-502	55	43	.	.	PUNCT
ejde-502	56	1	in	in	ADP
ejde-502	56	2	recent	recent	ADJ
ejde-502	56	3	years	year	NOUN
ejde-502	56	4	,	,	PUNCT
ejde-502	56	5	the	the	DET
ejde-502	56	6	authors	author	NOUN
ejde-502	56	7	of	of	ADP
ejde-502	56	8	the	the	DET
ejde-502	56	9	present	present	ADJ
ejde-502	56	10	work	work	NOUN
ejde-502	56	11	started	start	VERB
ejde-502	56	12	a	a	DET
ejde-502	56	13	larger	large	ADJ
ejde-502	56	14	project	project	NOUN
ejde-502	56	15	of	of	ADP
ejde-502	56	16	understanding	understand	VERB
ejde-502	56	17	the	the	DET
ejde-502	56	18	patterns	pattern	NOUN
ejde-502	56	19	(	(	PUNCT
ejde-502	56	20	in	in	ADP
ejde-502	56	21	self	self	NOUN
ejde-502	56	22	-	-	PUNCT
ejde-502	56	23	similar	similar	ADJ
ejde-502	56	24	form	form	NOUN
ejde-502	56	25	)	)	PUNCT
ejde-502	56	26	that	that	SCONJ
ejde-502	56	27	solutions	solution	NOUN
ejde-502	56	28	may	may	AUX
ejde-502	56	29	take	take	VERB
ejde-502	56	30	either	either	ADV
ejde-502	56	31	in	in	ADP
ejde-502	56	32	the	the	DET
ejde-502	56	33	case	case	NOUN
ejde-502	56	34	of	of	ADP
ejde-502	56	35	global	global	ADJ
ejde-502	56	36	solutions	solution	NOUN
ejde-502	56	37	,	,	PUNCT
ejde-502	56	38	or	or	CCONJ
ejde-502	56	39	close	close	ADV
ejde-502	56	40	to	to	ADP
ejde-502	56	41	the	the	DET
ejde-502	56	42	blow	blow	NOUN
ejde-502	56	43	-	-	PUNCT
ejde-502	56	44	up	up	ADP
ejde-502	56	45	time	time	NOUN
ejde-502	56	46	if	if	SCONJ
ejde-502	56	47	this	this	PRON
ejde-502	56	48	occurs	occur	VERB
ejde-502	56	49	,	,	PUNCT
ejde-502	56	50	for	for	ADP
ejde-502	56	51	(	(	PUNCT
ejde-502	56	52	1.7	1.7	NUM
ejde-502	56	53	)	)	PUNCT
ejde-502	56	54	.	.	PUNCT
ejde-502	57	1	this	this	PRON
ejde-502	57	2	is	be	AUX
ejde-502	57	3	an	an	DET
ejde-502	57	4	important	important	ADJ
ejde-502	57	5	part	part	NOUN
ejde-502	57	6	of	of	ADP
ejde-502	57	7	the	the	DET
ejde-502	57	8	study	study	NOUN
ejde-502	57	9	,	,	PUNCT
ejde-502	57	10	since	since	SCONJ
ejde-502	57	11	it	it	PRON
ejde-502	57	12	is	be	AUX
ejde-502	57	13	well	well	ADV
ejde-502	57	14	-	-	PUNCT
ejde-502	57	15	known	know	VERB
ejde-502	57	16	that	that	SCONJ
ejde-502	57	17	such	such	ADJ
ejde-502	57	18	patterns	pattern	NOUN
ejde-502	57	19	are	be	AUX
ejde-502	57	20	a	a	DET
ejde-502	57	21	prototype	prototype	NOUN
ejde-502	57	22	of	of	ADP
ejde-502	57	23	the	the	DET
ejde-502	57	24	general	general	ADJ
ejde-502	57	25	behavior	behavior	NOUN
ejde-502	57	26	of	of	ADP
ejde-502	57	27	the	the	DET
ejde-502	57	28	equation	equation	NOUN
ejde-502	57	29	,	,	PUNCT
ejde-502	57	30	in	in	ADP
ejde-502	57	31	the	the	DET
ejde-502	57	32	form	form	NOUN
ejde-502	57	33	of	of	ADP
ejde-502	57	34	asymptotic	asymptotic	ADJ
ejde-502	57	35	profiles	profile	NOUN
ejde-502	57	36	as	as	ADP
ejde-502	57	37	t	t	PROPN
ejde-502	57	38	→	→	SYM
ejde-502	57	39	∞	∞	PROPN
ejde-502	57	40	or	or	CCONJ
ejde-502	57	41	t	t	PROPN
ejde-502	57	42	→	→	SYM
ejde-502	57	43	t	t	PROPN
ejde-502	57	44	and	and	CCONJ
ejde-502	57	45	also	also	ADV
ejde-502	57	46	bring	bring	VERB
ejde-502	57	47	a	a	DET
ejde-502	57	48	deeper	deep	ADJ
ejde-502	57	49	understanding	understanding	NOUN
ejde-502	57	50	on	on	ADP
ejde-502	57	51	the	the	DET
ejde-502	57	52	blow	blow	NOUN
ejde-502	57	53	-	-	PUNCT
ejde-502	57	54	up	up	ADP
ejde-502	57	55	sets	set	NOUN
ejde-502	57	56	and	and	CCONJ
ejde-502	57	57	rates	rate	NOUN
ejde-502	57	58	.	.	PUNCT
ejde-502	58	1	a	a	DET
ejde-502	58	2	series	series	NOUN
ejde-502	58	3	of	of	ADP
ejde-502	58	4	papers	paper	NOUN
ejde-502	58	5	[	[	X
ejde-502	58	6	17	17	NUM
ejde-502	58	7	,	,	PUNCT
ejde-502	58	8	20	20	NUM
ejde-502	58	9	,	,	PUNCT
ejde-502	58	10	22	22	NUM
ejde-502	58	11	,	,	PUNCT
ejde-502	58	12	15	15	NUM
ejde-502	58	13	]	]	PUNCT
ejde-502	58	14	addressed	address	VERB
ejde-502	58	15	the	the	DET
ejde-502	58	16	question	question	NOUN
ejde-502	58	17	of	of	ADP
ejde-502	58	18	the	the	DET
ejde-502	58	19	blow	blow	NOUN
ejde-502	58	20	-	-	PUNCT
ejde-502	58	21	up	up	ADP
ejde-502	58	22	profiles	profile	NOUN
ejde-502	58	23	to	to	ADP
ejde-502	58	24	(	(	PUNCT
ejde-502	58	25	1.7	1.7	NUM
ejde-502	58	26	)	)	PUNCT
ejde-502	58	27	for	for	ADP
ejde-502	58	28	m	m	PROPN
ejde-502	58	29	>	>	X
ejde-502	58	30	1	1	NUM
ejde-502	58	31	and	and	CCONJ
ejde-502	58	32	1	1	NUM
ejde-502	58	33	≤	≤	NOUN
ejde-502	58	34	p	p	X
ejde-502	58	35	≤	≤	NUM
ejde-502	58	36	m	m	NOUN
ejde-502	58	37	,	,	PUNCT
ejde-502	58	38	where	where	SCONJ
ejde-502	58	39	interesting	interesting	ADJ
ejde-502	58	40	and	and	CCONJ
ejde-502	58	41	rather	rather	ADV
ejde-502	58	42	unexpected	unexpected	ADJ
ejde-502	58	43	behaviors	behavior	NOUN
ejde-502	58	44	were	be	AUX
ejde-502	58	45	established	establish	VERB
ejde-502	58	46	.	.	PUNCT
ejde-502	59	1	in	in	ADP
ejde-502	59	2	all	all	DET
ejde-502	59	3	these	these	DET
ejde-502	59	4	cases	case	NOUN
ejde-502	59	5	,	,	PUNCT
ejde-502	59	6	solutions	solution	NOUN
ejde-502	59	7	blow	blow	VERB
ejde-502	59	8	up	up	ADP
ejde-502	59	9	in	in	ADP
ejde-502	59	10	finite	finite	ADJ
ejde-502	59	11	time	time	NOUN
ejde-502	59	12	when	when	SCONJ
ejde-502	59	13	σ	σ	X
ejde-502	59	14	>	>	X
ejde-502	59	15	0	0	NUM
ejde-502	59	16	,	,	PUNCT
ejde-502	59	17	but	but	CCONJ
ejde-502	59	18	their	their	PRON
ejde-502	59	19	specific	specific	ADJ
ejde-502	59	20	blow	blow	NOUN
ejde-502	59	21	-	-	PUNCT
ejde-502	59	22	up	up	ADP
ejde-502	59	23	behavior	behavior	NOUN
ejde-502	59	24	it	it	PRON
ejde-502	59	25	is	be	AUX
ejde-502	59	26	shown	show	VERB
ejde-502	59	27	to	to	PART
ejde-502	59	28	depend	depend	VERB
ejde-502	59	29	strongly	strongly	ADV
ejde-502	59	30	on	on	ADP
ejde-502	59	31	the	the	DET
ejde-502	59	32	magnitude	magnitude	NOUN
ejde-502	59	33	of	of	ADP
ejde-502	59	34	σ	σ	PROPN
ejde-502	59	35	.	.	PUNCT
ejde-502	60	1	going	go	VERB
ejde-502	60	2	back	back	ADV
ejde-502	60	3	to	to	ADP
ejde-502	60	4	the	the	DET
ejde-502	60	5	case	case	NOUN
ejde-502	60	6	of	of	ADP
ejde-502	60	7	interest	interest	NOUN
ejde-502	60	8	for	for	ADP
ejde-502	60	9	us	we	PRON
ejde-502	60	10	,	,	PUNCT
ejde-502	60	11	p	p	PROPN
ejde-502	60	12	∈	∈	PROPN
ejde-502	60	13	(	(	PUNCT
ejde-502	60	14	0	0	NUM
ejde-502	60	15	,	,	PUNCT
ejde-502	60	16	1	1	NUM
ejde-502	60	17	)	)	PUNCT
ejde-502	60	18	,	,	PUNCT
ejde-502	60	19	we	we	PRON
ejde-502	60	20	have	have	AUX
ejde-502	60	21	proved	prove	VERB
ejde-502	60	22	that	that	SCONJ
ejde-502	60	23	blow	blow	VERB
ejde-502	60	24	-	-	PUNCT
ejde-502	60	25	up	up	ADP
ejde-502	60	26	profiles	profile	NOUN
ejde-502	60	27	exist	exist	VERB
ejde-502	60	28	if	if	SCONJ
ejde-502	60	29	the	the	DET
ejde-502	60	30	following	follow	VERB
ejde-502	60	31	condition	condition	NOUN
ejde-502	60	32	is	be	AUX
ejde-502	60	33	fulfilled	fulfil	VERB
ejde-502	60	34	:	:	PUNCT
ejde-502	60	35	l	l	NOUN
ejde-502	60	36	:	:	PUNCT
ejde-502	61	1	=	=	SYM
ejde-502	61	2	σ(m−	σ(m−	PROPN
ejde-502	61	3	1	1	X
ejde-502	61	4	)	)	PUNCT
ejde-502	61	5	+	+	CCONJ
ejde-502	61	6	2(p−	2(p−	NUM
ejde-502	61	7	1	1	NUM
ejde-502	61	8	)	)	PUNCT
ejde-502	61	9	>	>	X
ejde-502	61	10	0	0	X
ejde-502	61	11	.	.	PUNCT
ejde-502	62	1	(	(	PUNCT
ejde-502	62	2	1.8	1.8	NUM
ejde-502	62	3	)	)	PUNCT
ejde-502	62	4	we	we	PRON
ejde-502	62	5	classified	classify	VERB
ejde-502	62	6	such	such	ADJ
ejde-502	62	7	blow	blow	NOUN
ejde-502	62	8	-	-	PUNCT
ejde-502	62	9	up	up	ADP
ejde-502	62	10	profiles	profile	NOUN
ejde-502	62	11	in	in	ADP
ejde-502	62	12	[	[	X
ejde-502	62	13	16	16	NUM
ejde-502	62	14	]	]	PUNCT
ejde-502	62	15	in	in	ADP
ejde-502	62	16	general	general	ADJ
ejde-502	62	17	dimensions	dimension	NOUN
ejde-502	62	18	n	n	CCONJ
ejde-502	62	19	≥	≥	NOUN
ejde-502	62	20	2	2	NUM
ejde-502	62	21	,	,	PUNCT
ejde-502	62	22	following	follow	VERB
ejde-502	62	23	previous	previous	ADJ
ejde-502	62	24	results	result	NOUN
ejde-502	62	25	restricted	restrict	VERB
ejde-502	62	26	to	to	PART
ejde-502	62	27	dimension	dimension	VERB
ejde-502	62	28	n	n	NOUN
ejde-502	62	29	=	=	SYM
ejde-502	62	30	1	1	NUM
ejde-502	62	31	in	in	ADP
ejde-502	62	32	[	[	X
ejde-502	62	33	19	19	NUM
ejde-502	62	34	,	,	PUNCT
ejde-502	62	35	21	21	NUM
ejde-502	62	36	]	]	PUNCT
ejde-502	62	37	,	,	PUNCT
ejde-502	62	38	obtaining	obtain	VERB
ejde-502	62	39	again	again	ADV
ejde-502	62	40	that	that	SCONJ
ejde-502	62	41	the	the	DET
ejde-502	62	42	sign	sign	NOUN
ejde-502	62	43	of	of	ADP
ejde-502	62	44	m+	m+	NOUN
ejde-502	62	45	p−	p−	NOUN
ejde-502	62	46	2	2	NUM
ejde-502	62	47	is	be	AUX
ejde-502	62	48	fundamental	fundamental	ADJ
ejde-502	62	49	for	for	ADP
ejde-502	62	50	their	their	PRON
ejde-502	62	51	existence	existence	NOUN
ejde-502	62	52	and	and	CCONJ
ejde-502	62	53	behavior	behavior	NOUN
ejde-502	62	54	.	.	PUNCT
ejde-502	63	1	•	•	ADV
ejde-502	63	2	when	when	SCONJ
ejde-502	63	3	m	m	VERB
ejde-502	63	4	+	+	NOUN
ejde-502	64	1	p	p	X
ejde-502	64	2	−	−	PROPN
ejde-502	64	3	2	2	NUM
ejde-502	64	4	>	>	SYM
ejde-502	64	5	0	0	NUM
ejde-502	64	6	,	,	PUNCT
ejde-502	64	7	all	all	DET
ejde-502	64	8	the	the	DET
ejde-502	64	9	blow	blow	NOUN
ejde-502	64	10	-	-	PUNCT
ejde-502	64	11	up	up	ADP
ejde-502	64	12	self	self	NOUN
ejde-502	64	13	-	-	PUNCT
ejde-502	64	14	similar	similar	ADJ
ejde-502	64	15	profiles	profile	NOUN
ejde-502	64	16	present	present	VERB
ejde-502	64	17	an	an	DET
ejde-502	64	18	interface	interface	NOUN
ejde-502	64	19	(	(	PUNCT
ejde-502	64	20	that	that	PRON
ejde-502	64	21	is	is	ADV
ejde-502	64	22	,	,	PUNCT
ejde-502	64	23	they	they	PRON
ejde-502	64	24	are	be	AUX
ejde-502	64	25	compactly	compactly	ADV
ejde-502	64	26	supported	support	VERB
ejde-502	64	27	)	)	PUNCT
ejde-502	64	28	and	and	CCONJ
ejde-502	64	29	there	there	PRON
ejde-502	64	30	are	be	VERB
ejde-502	64	31	two	two	NUM
ejde-502	64	32	different	different	ADJ
ejde-502	64	33	types	type	NOUN
ejde-502	64	34	of	of	ADP
ejde-502	64	35	possible	possible	ADJ
ejde-502	64	36	interface	interface	NOUN
ejde-502	64	37	behaviors	behavior	NOUN
ejde-502	64	38	.	.	PUNCT
ejde-502	65	1	this	this	PRON
ejde-502	65	2	is	be	AUX
ejde-502	65	3	both	both	PRON
ejde-502	65	4	a	a	DET
ejde-502	65	5	manifestation	manifestation	NOUN
ejde-502	65	6	of	of	ADP
ejde-502	65	7	non	non	ADJ
ejde-502	65	8	-	-	ADJ
ejde-502	65	9	uniqueness	uniqueness	ADJ
ejde-502	65	10	(	(	PUNCT
ejde-502	65	11	expected	expect	VERB
ejde-502	65	12	for	for	ADP
ejde-502	65	13	4	4	NUM
ejde-502	65	14	r.	r.	PROPN
ejde-502	65	15	g.	g.	PROPN
ejde-502	65	16	iagar	iagar	PROPN
ejde-502	65	17	,	,	PUNCT
ejde-502	65	18	a.	a.	NOUN
ejde-502	65	19	i.	i.	PROPN
ejde-502	65	20	muñoz	muñoz	PROPN
ejde-502	65	21	,	,	PUNCT
ejde-502	65	22	a.	a.	NOUN
ejde-502	65	23	sánchez	sánchez	PROPN
ejde-502	65	24	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	65	25	compactly	compactly	ADV
ejde-502	65	26	supported	support	VERB
ejde-502	65	27	data	datum	NOUN
ejde-502	65	28	)	)	PUNCT
ejde-502	65	29	and	and	CCONJ
ejde-502	65	30	a	a	DET
ejde-502	65	31	suggestion	suggestion	NOUN
ejde-502	65	32	of	of	ADP
ejde-502	65	33	possible	possible	ADJ
ejde-502	65	34	non	non	ADJ
ejde-502	65	35	-	-	NOUN
ejde-502	65	36	existence	existence	NOUN
ejde-502	65	37	of	of	ADP
ejde-502	65	38	solutions	solution	NOUN
ejde-502	65	39	when	when	SCONJ
ejde-502	65	40	u0(x	u0(x	PRON
ejde-502	65	41	)	)	PUNCT
ejde-502	65	42	>	>	X
ejde-502	65	43	0	0	NUM
ejde-502	65	44	,	,	PUNCT
ejde-502	65	45	as	as	SCONJ
ejde-502	65	46	there	there	PRON
ejde-502	65	47	is	be	VERB
ejde-502	65	48	no	no	DET
ejde-502	65	49	pattern	pattern	NOUN
ejde-502	65	50	they	they	PRON
ejde-502	65	51	can	can	AUX
ejde-502	65	52	approach	approach	VERB
ejde-502	65	53	when	when	SCONJ
ejde-502	65	54	t→	t→	X
ejde-502	65	55	t	t	X
ejde-502	65	56	.	.	PUNCT
ejde-502	66	1	•whenm+p	•whenm+p	NOUN
ejde-502	66	2	=	=	SYM
ejde-502	66	3	2	2	NUM
ejde-502	66	4	,	,	PUNCT
ejde-502	66	5	self	self	NOUN
ejde-502	66	6	-	-	PUNCT
ejde-502	66	7	similar	similar	ADJ
ejde-502	66	8	blow	blow	NOUN
ejde-502	66	9	-	-	PUNCT
ejde-502	66	10	up	up	ADP
ejde-502	66	11	patterns	pattern	NOUN
ejde-502	66	12	present	present	VERB
ejde-502	66	13	a	a	DET
ejde-502	66	14	rather	rather	ADV
ejde-502	66	15	similar	similar	ADJ
ejde-502	66	16	panorama	panorama	NOUN
ejde-502	66	17	,	,	PUNCT
ejde-502	66	18	but	but	CCONJ
ejde-502	66	19	with	with	ADP
ejde-502	66	20	a	a	DET
ejde-502	66	21	single	single	ADJ
ejde-502	66	22	type	type	NOUN
ejde-502	66	23	of	of	ADP
ejde-502	66	24	interface	interface	NOUN
ejde-502	66	25	behavior	behavior	NOUN
ejde-502	66	26	.	.	PUNCT
ejde-502	67	1	•when	•when	PROPN
ejde-502	67	2	m+p	m+p	NOUN
ejde-502	67	3	<	<	X
ejde-502	67	4	2	2	NUM
ejde-502	67	5	,	,	PUNCT
ejde-502	67	6	a	a	DET
ejde-502	67	7	rather	rather	ADV
ejde-502	67	8	striking	striking	ADJ
ejde-502	67	9	non	non	ADJ
ejde-502	67	10	-	-	NOUN
ejde-502	67	11	existence	existence	NOUN
ejde-502	67	12	of	of	ADP
ejde-502	67	13	any	any	DET
ejde-502	67	14	kind	kind	NOUN
ejde-502	67	15	of	of	ADP
ejde-502	67	16	blow	blow	NOUN
ejde-502	67	17	-	-	PUNCT
ejde-502	67	18	up	up	ADP
ejde-502	67	19	profiles	profile	NOUN
ejde-502	67	20	(	(	PUNCT
ejde-502	67	21	either	either	CCONJ
ejde-502	67	22	with	with	ADP
ejde-502	67	23	interfaces	interface	NOUN
ejde-502	67	24	or	or	CCONJ
ejde-502	67	25	not	not	PART
ejde-502	67	26	)	)	PUNCT
ejde-502	67	27	occurs	occur	VERB
ejde-502	67	28	[	[	X
ejde-502	67	29	19	19	NUM
ejde-502	67	30	]	]	PUNCT
ejde-502	67	31	.	.	PUNCT
ejde-502	68	1	such	such	DET
ejde-502	68	2	an	an	DET
ejde-502	68	3	outcome	outcome	NOUN
ejde-502	68	4	gives	give	VERB
ejde-502	68	5	the	the	DET
ejde-502	68	6	intuition	intuition	NOUN
ejde-502	68	7	of	of	ADP
ejde-502	68	8	an	an	DET
ejde-502	68	9	infinite	infinite	ADJ
ejde-502	68	10	speed	speed	NOUN
ejde-502	68	11	of	of	ADP
ejde-502	68	12	propagation	propagation	NOUN
ejde-502	68	13	and	and	CCONJ
ejde-502	68	14	a	a	DET
ejde-502	68	15	complete	complete	ADJ
ejde-502	68	16	non	non	NOUN
ejde-502	68	17	-	-	NOUN
ejde-502	68	18	existence	existence	NOUN
ejde-502	68	19	of	of	ADP
ejde-502	68	20	non	non	ADJ
ejde-502	68	21	-	-	ADJ
ejde-502	68	22	trivial	trivial	ADJ
ejde-502	68	23	solutions	solution	NOUN
ejde-502	68	24	.	.	PUNCT
ejde-502	69	1	in	in	ADP
ejde-502	69	2	this	this	DET
ejde-502	69	3	paper	paper	NOUN
ejde-502	69	4	we	we	PRON
ejde-502	69	5	thus	thus	ADV
ejde-502	69	6	begin	begin	VERB
ejde-502	69	7	the	the	DET
ejde-502	69	8	qualitative	qualitative	ADJ
ejde-502	69	9	study	study	NOUN
ejde-502	69	10	of	of	ADP
ejde-502	69	11	the	the	DET
ejde-502	69	12	cauchy	cauchy	ADJ
ejde-502	69	13	problem	problem	NOUN
ejde-502	69	14	for	for	ADP
ejde-502	69	15	(	(	PUNCT
ejde-502	69	16	1.1	1.1	NUM
ejde-502	69	17	)	)	PUNCT
ejde-502	69	18	posed	pose	VERB
ejde-502	69	19	in	in	ADP
ejde-502	69	20	rn	rn	PROPN
ejde-502	69	21	.	.	PUNCT
ejde-502	70	1	we	we	PRON
ejde-502	70	2	thus	thus	ADV
ejde-502	70	3	address	address	VERB
ejde-502	70	4	the	the	DET
ejde-502	70	5	quite	quite	ADV
ejde-502	70	6	interesting	interesting	ADJ
ejde-502	70	7	(	(	PUNCT
ejde-502	70	8	in	in	ADP
ejde-502	70	9	view	view	NOUN
ejde-502	70	10	of	of	ADP
ejde-502	70	11	precedents	precedent	NOUN
ejde-502	70	12	such	such	ADJ
ejde-502	70	13	as	as	ADP
ejde-502	70	14	[	[	X
ejde-502	70	15	26	26	NUM
ejde-502	70	16	,	,	PUNCT
ejde-502	70	17	28	28	NUM
ejde-502	70	18	]	]	PUNCT
ejde-502	70	19	)	)	PUNCT
ejde-502	70	20	question	question	NOUN
ejde-502	70	21	of	of	ADP
ejde-502	70	22	speed	speed	NOUN
ejde-502	70	23	of	of	ADP
ejde-502	70	24	propagation	propagation	NOUN
ejde-502	70	25	and	and	CCONJ
ejde-502	70	26	non	non	ADJ
ejde-502	70	27	-	-	NOUN
ejde-502	70	28	uniqueness	uniqueness	ADJ
ejde-502	70	29	for	for	ADP
ejde-502	70	30	compactly	compactly	ADV
ejde-502	70	31	supported	support	VERB
ejde-502	70	32	data	datum	NOUN
ejde-502	70	33	,	,	PUNCT
ejde-502	70	34	deriving	derive	VERB
ejde-502	70	35	in	in	ADP
ejde-502	70	36	the	the	DET
ejde-502	70	37	process	process	NOUN
ejde-502	70	38	some	some	DET
ejde-502	70	39	new	new	ADJ
ejde-502	70	40	aronson	aronson	PROPN
ejde-502	70	41	-	-	PUNCT
ejde-502	70	42	bénilan	bénilan	PROPN
ejde-502	70	43	estimates	estimate	NOUN
ejde-502	70	44	for	for	ADP
ejde-502	70	45	the	the	DET
ejde-502	70	46	solutions	solution	NOUN
ejde-502	70	47	to	to	ADP
ejde-502	70	48	(	(	PUNCT
ejde-502	70	49	1.1	1.1	NUM
ejde-502	70	50	)	)	PUNCT
ejde-502	70	51	when	when	SCONJ
ejde-502	70	52	m	m	VERB
ejde-502	70	53	+	+	VERB
ejde-502	70	54	p	p	X
ejde-502	70	55	<	<	X
ejde-502	70	56	2	2	NUM
ejde-502	70	57	.	.	PUNCT
ejde-502	70	58	we	we	PRON
ejde-502	70	59	stress	stress	VERB
ejde-502	70	60	here	here	ADV
ejde-502	70	61	that	that	SCONJ
ejde-502	70	62	uniform	uniform	VERB
ejde-502	70	63	lower	low	ADJ
ejde-502	70	64	bounds	bound	NOUN
ejde-502	70	65	on	on	ADP
ejde-502	70	66	the	the	DET
ejde-502	70	67	weight	weight	NOUN
ejde-502	70	68	are	be	AUX
ejde-502	70	69	essential	essential	ADJ
ejde-502	70	70	in	in	ADP
ejde-502	70	71	the	the	DET
ejde-502	70	72	forthcoming	forthcoming	ADJ
ejde-502	70	73	proofs	proof	NOUN
ejde-502	70	74	,	,	PUNCT
ejde-502	70	75	thus	thus	ADV
ejde-502	70	76	technical	technical	ADJ
ejde-502	70	77	complications	complication	NOUN
ejde-502	70	78	introduced	introduce	VERB
ejde-502	70	79	by	by	ADP
ejde-502	70	80	the	the	DET
ejde-502	70	81	weight	weight	NOUN
ejde-502	70	82	in	in	ADP
ejde-502	70	83	a	a	DET
ejde-502	70	84	neighborhood	neighborhood	NOUN
ejde-502	70	85	of	of	ADP
ejde-502	70	86	x	x	X
ejde-502	70	87	=	=	SYM
ejde-502	70	88	0	0	PUNCT
ejde-502	70	89	(	(	PUNCT
ejde-502	70	90	where	where	SCONJ
ejde-502	70	91	it	it	PRON
ejde-502	70	92	is	be	AUX
ejde-502	70	93	no	no	ADV
ejde-502	70	94	longer	long	ADV
ejde-502	70	95	uniformly	uniformly	ADV
ejde-502	70	96	positive	positive	ADJ
ejde-502	70	97	)	)	PUNCT
ejde-502	70	98	appear	appear	VERB
ejde-502	70	99	when	when	SCONJ
ejde-502	70	100	trying	try	VERB
ejde-502	70	101	to	to	PART
ejde-502	70	102	adapt	adapt	VERB
ejde-502	70	103	the	the	DET
ejde-502	70	104	same	same	ADJ
ejde-502	70	105	proofs	proof	NOUN
ejde-502	70	106	to	to	ADP
ejde-502	70	107	the	the	DET
ejde-502	70	108	close	close	ADJ
ejde-502	70	109	relative	relative	NOUN
ejde-502	70	110	(	(	PUNCT
ejde-502	70	111	1.7	1.7	NUM
ejde-502	70	112	)	)	PUNCT
ejde-502	70	113	.	.	PUNCT
ejde-502	71	1	but	but	CCONJ
ejde-502	71	2	let	let	VERB
ejde-502	71	3	us	we	PRON
ejde-502	71	4	present	present	VERB
ejde-502	71	5	below	below	ADP
ejde-502	71	6	in	in	ADP
ejde-502	71	7	more	more	ADJ
ejde-502	71	8	detail	detail	NOUN
ejde-502	71	9	our	our	PRON
ejde-502	71	10	main	main	ADJ
ejde-502	71	11	results	result	NOUN
ejde-502	71	12	.	.	PUNCT
ejde-502	72	1	main	main	ADJ
ejde-502	72	2	results	result	NOUN
ejde-502	72	3	.	.	PUNCT
ejde-502	73	1	to	to	PART
ejde-502	73	2	state	state	VERB
ejde-502	73	3	our	our	PRON
ejde-502	73	4	results	result	NOUN
ejde-502	73	5	concerning	concern	VERB
ejde-502	73	6	the	the	DET
ejde-502	73	7	qualitative	qualitative	ADJ
ejde-502	73	8	theory	theory	NOUN
ejde-502	73	9	of	of	ADP
ejde-502	73	10	solutions	solution	NOUN
ejde-502	73	11	to	to	ADP
ejde-502	73	12	(	(	PUNCT
ejde-502	73	13	1.1	1.1	NUM
ejde-502	73	14	)	)	PUNCT
ejde-502	73	15	,	,	PUNCT
ejde-502	73	16	we	we	PRON
ejde-502	73	17	first	first	ADV
ejde-502	73	18	have	have	VERB
ejde-502	73	19	to	to	PART
ejde-502	73	20	introduce	introduce	VERB
ejde-502	73	21	the	the	DET
ejde-502	73	22	notion	notion	NOUN
ejde-502	73	23	of	of	ADP
ejde-502	73	24	weak	weak	ADJ
ejde-502	73	25	solution	solution	NOUN
ejde-502	73	26	that	that	PRON
ejde-502	73	27	will	will	AUX
ejde-502	73	28	be	be	AUX
ejde-502	73	29	used	use	VERB
ejde-502	73	30	throughout	throughout	ADP
ejde-502	73	31	the	the	DET
ejde-502	73	32	paper	paper	NOUN
ejde-502	73	33	.	.	PUNCT
ejde-502	74	1	let	let	VERB
ejde-502	74	2	us	we	PRON
ejde-502	74	3	denote	denote	VERB
ejde-502	74	4	by	by	ADP
ejde-502	74	5	u(t	u(t	NOUN
ejde-502	74	6	)	)	PUNCT
ejde-502	74	7	the	the	DET
ejde-502	74	8	mapping	mapping	NOUN
ejde-502	74	9	x	x	SYM
ejde-502	74	10	7→	7→	NUM
ejde-502	74	11	u(x	u(x	NOUN
ejde-502	74	12	,	,	PUNCT
ejde-502	74	13	t	t	PROPN
ejde-502	74	14	)	)	PUNCT
ejde-502	74	15	for	for	ADP
ejde-502	74	16	t	t	PROPN
ejde-502	74	17	>	>	X
ejde-502	74	18	0	0	NUM
ejde-502	74	19	fixed	fix	VERB
ejde-502	74	20	.	.	PUNCT
ejde-502	75	1	we	we	PRON
ejde-502	75	2	will	will	AUX
ejde-502	75	3	slightly	slightly	ADV
ejde-502	75	4	modify	modify	VERB
ejde-502	75	5	the	the	DET
ejde-502	75	6	functional	functional	ADJ
ejde-502	75	7	framework	framework	NOUN
ejde-502	75	8	from	from	ADP
ejde-502	75	9	andreucci	andreucci	NOUN
ejde-502	75	10	and	and	CCONJ
ejde-502	75	11	dibenedetto	dibenedetto	NOUN
ejde-502	75	12	[	[	X
ejde-502	75	13	1	1	NUM
ejde-502	75	14	]	]	PUNCT
ejde-502	75	15	by	by	ADP
ejde-502	75	16	passing	pass	VERB
ejde-502	75	17	in	in	ADP
ejde-502	75	18	our	our	PRON
ejde-502	75	19	weak	weak	ADJ
ejde-502	75	20	formulation	formulation	NOUN
ejde-502	75	21	the	the	DET
ejde-502	75	22	full	full	ADJ
ejde-502	75	23	laplacian	laplacian	NOUN
ejde-502	75	24	to	to	ADP
ejde-502	75	25	the	the	DET
ejde-502	75	26	test	test	NOUN
ejde-502	75	27	function	function	NOUN
ejde-502	75	28	.	.	PUNCT
ejde-502	76	1	definition	definition	NOUN
ejde-502	76	2	1.1	1.1	NUM
ejde-502	76	3	.	.	PUNCT
ejde-502	77	1	a	a	DET
ejde-502	77	2	non	non	ADJ
ejde-502	77	3	-	-	ADJ
ejde-502	77	4	negative	negative	ADJ
ejde-502	77	5	function	function	NOUN
ejde-502	77	6	u	u	NOUN
ejde-502	77	7	:	:	PUNCT
ejde-502	77	8	rn	rn	PROPN
ejde-502	77	9	×	×	PROPN
ejde-502	77	10	(	(	PUNCT
ejde-502	77	11	0	0	NUM
ejde-502	77	12	,	,	PUNCT
ejde-502	77	13	t	t	NOUN
ejde-502	77	14	)	)	PUNCT
ejde-502	77	15	7→	7→	PROPN
ejde-502	78	1	[	[	X
ejde-502	78	2	0,∞	0,∞	NUM
ejde-502	78	3	)	)	PUNCT
ejde-502	78	4	is	be	AUX
ejde-502	78	5	said	say	VERB
ejde-502	78	6	to	to	PART
ejde-502	78	7	be	be	AUX
ejde-502	78	8	a	a	DET
ejde-502	78	9	weak	weak	ADJ
ejde-502	78	10	solution	solution	NOUN
ejde-502	78	11	to	to	ADP
ejde-502	78	12	the	the	DET
ejde-502	78	13	cauchy	cauchy	ADJ
ejde-502	78	14	problem	problem	NOUN
ejde-502	78	15	(	(	PUNCT
ejde-502	78	16	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	78	17	)	)	PUNCT
ejde-502	78	18	with	with	ADP
ejde-502	78	19	initial	initial	ADJ
ejde-502	78	20	condition	condition	NOUN
ejde-502	78	21	u0	u0	ADJ
ejde-502	78	22	as	as	ADP
ejde-502	78	23	in	in	ADP
ejde-502	78	24	(	(	PUNCT
ejde-502	78	25	1.4	1.4	NUM
ejde-502	78	26	)	)	PUNCT
ejde-502	78	27	if	if	SCONJ
ejde-502	78	28	the	the	DET
ejde-502	78	29	following	follow	VERB
ejde-502	78	30	assumptions	assumption	NOUN
ejde-502	78	31	are	be	AUX
ejde-502	78	32	satisfied	satisfied	ADJ
ejde-502	78	33	by	by	ADP
ejde-502	78	34	u	u	NOUN
ejde-502	78	35	:	:	PUNCT
ejde-502	78	36	(	(	PUNCT
ejde-502	78	37	a	a	X
ejde-502	78	38	)	)	PUNCT
ejde-502	78	39	regularity	regularity	NOUN
ejde-502	78	40	assumption	assumption	NOUN
ejde-502	78	41	:	:	PUNCT
ejde-502	78	42	we	we	PRON
ejde-502	78	43	have	have	VERB
ejde-502	78	44	u	u	PROPN
ejde-502	78	45	∈	∈	PROPN
ejde-502	78	46	c(0	c(0	PROPN
ejde-502	78	47	,	,	PUNCT
ejde-502	78	48	t	t	PROPN
ejde-502	78	49	;	;	PUNCT
ejde-502	78	50	l1	l1	PROPN
ejde-502	78	51	loc(rn	loc(rn	NUM
ejde-502	78	52	)	)	PUNCT
ejde-502	78	53	)	)	PUNCT
ejde-502	79	1	∩	∩	NOUN
ejde-502	79	2	l∞loc(rn	l∞loc(rn	PRON
ejde-502	79	3	×	×	NOUN
ejde-502	79	4	(	(	PUNCT
ejde-502	79	5	0	0	NUM
ejde-502	79	6	,	,	PUNCT
ejde-502	79	7	t	t	NOUN
ejde-502	79	8	)	)	PUNCT
ejde-502	79	9	)	)	PUNCT
ejde-502	79	10	.	.	PUNCT
ejde-502	80	1	(	(	PUNCT
ejde-502	80	2	1.9	1.9	NUM
ejde-502	80	3	)	)	PUNCT
ejde-502	80	4	(	(	PUNCT
ejde-502	80	5	b	b	X
ejde-502	80	6	)	)	PUNCT
ejde-502	80	7	weak	weak	ADJ
ejde-502	80	8	formulation	formulation	NOUN
ejde-502	80	9	of	of	ADP
ejde-502	80	10	(	(	PUNCT
ejde-502	80	11	1.1	1.1	NUM
ejde-502	80	12	):	):	PUNCT
ejde-502	80	13	for	for	ADP
ejde-502	80	14	any	any	DET
ejde-502	80	15	test	test	NOUN
ejde-502	80	16	function	function	NOUN
ejde-502	80	17	η	η	PROPN
ejde-502	80	18	∈	∈	PROPN
ejde-502	80	19	c∞0	c∞0	PROPN
ejde-502	80	20	(	(	PUNCT
ejde-502	80	21	rn	rn	PROPN
ejde-502	80	22	×	×	PROPN
ejde-502	80	23	(	(	PUNCT
ejde-502	80	24	0	0	NUM
ejde-502	80	25	,	,	PUNCT
ejde-502	80	26	t	t	NOUN
ejde-502	80	27	)	)	PUNCT
ejde-502	80	28	)	)	PUNCT
ejde-502	80	29	and	and	CCONJ
ejde-502	80	30	for	for	ADP
ejde-502	80	31	any	any	DET
ejde-502	80	32	t	t	NOUN
ejde-502	80	33	∈	∈	PROPN
ejde-502	80	34	(	(	PUNCT
ejde-502	80	35	0	0	NUM
ejde-502	80	36	,	,	PUNCT
ejde-502	80	37	t	t	NOUN
ejde-502	80	38	)	)	PUNCT
ejde-502	80	39	we	we	PRON
ejde-502	80	40	have∫	have∫	VERB
ejde-502	80	41	rn	rn	PROPN
ejde-502	80	42	u(x	u(x	PROPN
ejde-502	80	43	,	,	PUNCT
ejde-502	81	1	t)η(x	t)η(x	PROPN
ejde-502	81	2	,	,	PUNCT
ejde-502	81	3	t	t	PROPN
ejde-502	81	4	)	)	PUNCT
ejde-502	81	5	dx+	dx+	NOUN
ejde-502	82	1	∫	∫	PROPN
ejde-502	82	2	t	t	PROPN
ejde-502	82	3	0	0	NUM
ejde-502	82	4	∫	∫	PROPN
ejde-502	82	5	rn	rn	PROPN
ejde-502	82	6	(	(	PUNCT
ejde-502	82	7	−u(x	−u(x	PROPN
ejde-502	82	8	,	,	PUNCT
ejde-502	82	9	τ)ηt(x	τ)ηt(x	NUM
ejde-502	82	10	,	,	PUNCT
ejde-502	82	11	τ)−	τ)−	PROPN
ejde-502	82	12	um(x	um(x	NUM
ejde-502	82	13	,	,	PUNCT
ejde-502	82	14	τ)∆η(x	τ)∆η(x	PROPN
ejde-502	82	15	,	,	PUNCT
ejde-502	82	16	τ	τ	PROPN
ejde-502	82	17	)	)	PUNCT
ejde-502	82	18	)	)	PUNCT
ejde-502	83	1	dx	dx	PROPN
ejde-502	83	2	dτ	dτ	PROPN
ejde-502	84	1	=	=	SYM
ejde-502	84	2	∫	∫	PROPN
ejde-502	84	3	t	t	PROPN
ejde-502	84	4	0	0	NUM
ejde-502	84	5	∫	∫	PROPN
ejde-502	84	6	rn	rn	PROPN
ejde-502	84	7	(	(	PUNCT
ejde-502	84	8	1	1	NUM
ejde-502	84	9	+	+	NUM
ejde-502	84	10	|x|)σup(x	|x|)σup(x	PROPN
ejde-502	84	11	,	,	PUNCT
ejde-502	84	12	τ)η(x	τ)η(x	ADJ
ejde-502	84	13	,	,	PUNCT
ejde-502	84	14	τ)dx	τ)dx	PROPN
ejde-502	84	15	dτ	dτ	NOUN
ejde-502	84	16	.	.	PROPN
ejde-502	84	17	(	(	PUNCT
ejde-502	84	18	1.10	1.10	NUM
ejde-502	84	19	)	)	PUNCT
ejde-502	84	20	(	(	PUNCT
ejde-502	84	21	c	c	X
ejde-502	84	22	)	)	PUNCT
ejde-502	84	23	taking	take	VERB
ejde-502	84	24	the	the	DET
ejde-502	84	25	initial	initial	ADJ
ejde-502	84	26	condition	condition	NOUN
ejde-502	84	27	:	:	PUNCT
ejde-502	84	28	this	this	PRON
ejde-502	84	29	is	be	AUX
ejde-502	84	30	done	do	VERB
ejde-502	84	31	in	in	ADP
ejde-502	84	32	l1	l1	PROPN
ejde-502	84	33	sense	sense	NOUN
ejde-502	84	34	,	,	PUNCT
ejde-502	84	35	more	more	ADV
ejde-502	84	36	precisely	precisely	ADV
ejde-502	84	37	lim	lim	NOUN
ejde-502	84	38	t→0	t→0	PUNCT
ejde-502	84	39	u(t	u(t	PROPN
ejde-502	84	40	)	)	PUNCT
ejde-502	84	41	=	=	PUNCT
ejde-502	85	1	u0	u0	ADJ
ejde-502	85	2	,	,	PUNCT
ejde-502	85	3	with	with	ADP
ejde-502	85	4	convergence	convergence	NOUN
ejde-502	85	5	in	in	ADP
ejde-502	85	6	l1	l1	PROPN
ejde-502	85	7	loc(rn	loc(rn	NUM
ejde-502	85	8	)	)	PUNCT
ejde-502	85	9	(	(	PUNCT
ejde-502	85	10	1.11	1.11	NUM
ejde-502	85	11	)	)	PUNCT
ejde-502	85	12	this	this	DET
ejde-502	85	13	change	change	NOUN
ejde-502	85	14	relaxes	relax	VERB
ejde-502	85	15	the	the	DET
ejde-502	85	16	functional	functional	ADJ
ejde-502	85	17	assumptions	assumption	NOUN
ejde-502	85	18	of	of	ADP
ejde-502	85	19	regularity	regularity	NOUN
ejde-502	85	20	for	for	ADP
ejde-502	85	21	a	a	DET
ejde-502	85	22	solution	solution	NOUN
ejde-502	85	23	,	,	PUNCT
ejde-502	85	24	making	make	VERB
ejde-502	85	25	it	it	PRON
ejde-502	85	26	easier	easy	ADJ
ejde-502	85	27	to	to	PART
ejde-502	85	28	obtain	obtain	VERB
ejde-502	85	29	weak	weak	ADJ
ejde-502	85	30	solutions	solution	NOUN
ejde-502	85	31	by	by	ADP
ejde-502	85	32	limiting	limit	VERB
ejde-502	85	33	processes	process	NOUN
ejde-502	85	34	.	.	PUNCT
ejde-502	86	1	we	we	PRON
ejde-502	86	2	will	will	AUX
ejde-502	86	3	also	also	ADV
ejde-502	86	4	need	need	VERB
ejde-502	86	5	throughout	throughout	ADP
ejde-502	86	6	the	the	DET
ejde-502	86	7	paper	paper	NOUN
ejde-502	86	8	the	the	DET
ejde-502	86	9	notions	notion	NOUN
ejde-502	86	10	of	of	ADP
ejde-502	86	11	(	(	PUNCT
ejde-502	86	12	weak	weak	ADJ
ejde-502	86	13	)	)	PUNCT
ejde-502	86	14	suband	suband	NOUN
ejde-502	86	15	supersolution	supersolution	NOUN
ejde-502	86	16	to	to	ADP
ejde-502	86	17	(	(	PUNCT
ejde-502	86	18	1.1	1.1	NUM
ejde-502	86	19	)	)	PUNCT
ejde-502	86	20	.	.	PUNCT
ejde-502	87	1	we	we	PRON
ejde-502	87	2	say	say	VERB
ejde-502	87	3	that	that	SCONJ
ejde-502	87	4	u	u	PROPN
ejde-502	87	5	is	be	AUX
ejde-502	87	6	a	a	DET
ejde-502	87	7	weak	weak	ADJ
ejde-502	87	8	subsolution	subsolution	NOUN
ejde-502	87	9	(	(	PUNCT
ejde-502	87	10	respectively	respectively	ADV
ejde-502	87	11	weak	weak	ADJ
ejde-502	87	12	supersolution	supersolution	NOUN
ejde-502	87	13	)	)	PUNCT
ejde-502	87	14	to	to	ADP
ejde-502	87	15	(	(	PUNCT
ejde-502	87	16	1.1	1.1	NUM
ejde-502	87	17	)	)	PUNCT
ejde-502	87	18	if	if	SCONJ
ejde-502	87	19	condition	condition	NOUN
ejde-502	87	20	(	(	PUNCT
ejde-502	87	21	a	a	X
ejde-502	87	22	)	)	PUNCT
ejde-502	87	23	is	be	AUX
ejde-502	87	24	fulfilled	fulfil	VERB
ejde-502	87	25	and	and	CCONJ
ejde-502	87	26	condition	condition	NOUN
ejde-502	87	27	(	(	PUNCT
ejde-502	87	28	b	b	NOUN
ejde-502	87	29	)	)	PUNCT
ejde-502	87	30	is	be	AUX
ejde-502	87	31	modified	modify	VERB
ejde-502	87	32	in	in	ADP
ejde-502	87	33	the	the	DET
ejde-502	87	34	sense	sense	NOUN
ejde-502	87	35	that	that	SCONJ
ejde-502	87	36	the	the	DET
ejde-502	87	37	equal	equal	ADJ
ejde-502	87	38	sign	sign	NOUN
ejde-502	87	39	is	be	AUX
ejde-502	87	40	replaced	replace	VERB
ejde-502	87	41	by	by	ADP
ejde-502	87	42	≤	≤	NOUN
ejde-502	87	43	(	(	PUNCT
ejde-502	87	44	respectively	respectively	ADV
ejde-502	87	45	≥	≥	NUM
ejde-502	87	46	)	)	PUNCT
ejde-502	87	47	for	for	ADP
ejde-502	87	48	any	any	DET
ejde-502	87	49	test	test	NOUN
ejde-502	87	50	functions	function	NOUN
ejde-502	87	51	η	η	PROPN
ejde-502	87	52	∈	∈	PROPN
ejde-502	87	53	c∞0	c∞0	PROPN
ejde-502	87	54	(	(	PUNCT
ejde-502	87	55	rn×(0	rn×(0	PROPN
ejde-502	87	56	,	,	PUNCT
ejde-502	87	57	t	t	PROPN
ejde-502	87	58	)	)	PUNCT
ejde-502	87	59	)	)	PUNCT
ejde-502	87	60	such	such	ADJ
ejde-502	87	61	that	that	SCONJ
ejde-502	87	62	η	η	PROPN
ejde-502	87	63	≥	≥	X
ejde-502	87	64	0	0	NUM
ejde-502	87	65	and	and	CCONJ
ejde-502	87	66	for	for	ADP
ejde-502	87	67	any	any	DET
ejde-502	87	68	t	t	NOUN
ejde-502	87	69	∈	∈	PROPN
ejde-502	87	70	(	(	PUNCT
ejde-502	87	71	0	0	NUM
ejde-502	87	72	,	,	PUNCT
ejde-502	87	73	t	t	NOUN
ejde-502	87	74	)	)	PUNCT
ejde-502	87	75	.	.	PUNCT
ejde-502	88	1	moreover	moreover	ADV
ejde-502	88	2	,	,	PUNCT
ejde-502	88	3	the	the	DET
ejde-502	88	4	notions	notion	NOUN
ejde-502	88	5	of	of	ADP
ejde-502	88	6	weak	weak	ADJ
ejde-502	88	7	solution	solution	NOUN
ejde-502	88	8	,	,	PUNCT
ejde-502	88	9	subsolution	subsolution	NOUN
ejde-502	88	10	,	,	PUNCT
ejde-502	88	11	supersolution	supersolution	NOUN
ejde-502	88	12	to	to	ADP
ejde-502	88	13	(	(	PUNCT
ejde-502	88	14	1.1	1.1	NUM
ejde-502	88	15	)	)	PUNCT
ejde-502	88	16	(	(	PUNCT
ejde-502	88	17	without	without	ADP
ejde-502	88	18	the	the	DET
ejde-502	88	19	initial	initial	ADJ
ejde-502	88	20	condition	condition	NOUN
ejde-502	88	21	)	)	PUNCT
ejde-502	88	22	can	can	AUX
ejde-502	88	23	be	be	AUX
ejde-502	88	24	defined	define	VERB
ejde-502	88	25	in	in	ADP
ejde-502	88	26	an	an	DET
ejde-502	88	27	obvious	obvious	ADJ
ejde-502	88	28	way	way	NOUN
ejde-502	88	29	on	on	ADP
ejde-502	88	30	time	time	NOUN
ejde-502	88	31	intervals	interval	NOUN
ejde-502	88	32	[	[	X
ejde-502	88	33	t1	t1	NOUN
ejde-502	88	34	,	,	PUNCT
ejde-502	88	35	t2	t2	PROPN
ejde-502	88	36	]	]	X
ejde-502	88	37	⊂	⊂	X
ejde-502	88	38	(	(	PUNCT
ejde-502	88	39	0	0	NUM
ejde-502	88	40	,	,	PUNCT
ejde-502	88	41	t	t	PROPN
ejde-502	88	42	)	)	PUNCT
ejde-502	88	43	instead	instead	ADV
ejde-502	88	44	of	of	ADP
ejde-502	88	45	(	(	PUNCT
ejde-502	88	46	0	0	NUM
ejde-502	88	47	,	,	PUNCT
ejde-502	88	48	t	t	NOUN
ejde-502	88	49	)	)	PUNCT
ejde-502	88	50	by	by	ADP
ejde-502	88	51	just	just	ADV
ejde-502	88	52	removing	remove	VERB
ejde-502	88	53	assumption	assumption	NOUN
ejde-502	88	54	(	(	PUNCT
ejde-502	88	55	c	c	NOUN
ejde-502	88	56	)	)	PUNCT
ejde-502	88	57	.	.	PUNCT
ejde-502	89	1	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	89	2	reaction	reaction	NOUN
ejde-502	89	3	-	-	PUNCT
ejde-502	89	4	diffusion	diffusion	NOUN
ejde-502	89	5	with	with	ADP
ejde-502	89	6	weighted	weight	VERB
ejde-502	89	7	strong	strong	ADJ
ejde-502	89	8	reaction	reaction	NOUN
ejde-502	89	9	5	5	NUM
ejde-502	89	10	once	once	ADV
ejde-502	89	11	defined	define	VERB
ejde-502	89	12	the	the	DET
ejde-502	89	13	notion	notion	NOUN
ejde-502	89	14	of	of	ADP
ejde-502	89	15	weak	weak	ADJ
ejde-502	89	16	solution	solution	NOUN
ejde-502	89	17	,	,	PUNCT
ejde-502	89	18	we	we	PRON
ejde-502	89	19	are	be	AUX
ejde-502	89	20	in	in	ADP
ejde-502	89	21	a	a	DET
ejde-502	89	22	position	position	NOUN
ejde-502	89	23	to	to	PART
ejde-502	89	24	give	give	VERB
ejde-502	89	25	below	below	ADP
ejde-502	89	26	the	the	DET
ejde-502	89	27	main	main	ADJ
ejde-502	89	28	theorems	theorem	NOUN
ejde-502	89	29	of	of	ADP
ejde-502	89	30	this	this	DET
ejde-502	89	31	work	work	NOUN
ejde-502	89	32	.	.	PUNCT
ejde-502	90	1	as	as	SCONJ
ejde-502	90	2	explained	explain	VERB
ejde-502	90	3	above	above	ADV
ejde-502	90	4	,	,	PUNCT
ejde-502	90	5	it	it	PRON
ejde-502	90	6	is	be	AUX
ejde-502	90	7	expected	expect	VERB
ejde-502	90	8	from	from	ADP
ejde-502	90	9	the	the	DET
ejde-502	90	10	results	result	NOUN
ejde-502	90	11	in	in	ADP
ejde-502	90	12	papers	paper	NOUN
ejde-502	90	13	such	such	ADJ
ejde-502	90	14	as	as	ADP
ejde-502	90	15	[	[	X
ejde-502	90	16	26	26	NUM
ejde-502	90	17	,	,	PUNCT
ejde-502	90	18	27	27	NUM
ejde-502	90	19	,	,	PUNCT
ejde-502	90	20	28	28	NUM
ejde-502	90	21	]	]	PUNCT
ejde-502	90	22	for	for	ADP
ejde-502	90	23	equations	equation	NOUN
ejde-502	90	24	with	with	ADP
ejde-502	90	25	non	non	ADJ
ejde-502	90	26	-	-	ADJ
ejde-502	90	27	weighted	weighted	ADJ
ejde-502	90	28	reaction	reaction	NOUN
ejde-502	90	29	that	that	SCONJ
ejde-502	90	30	,	,	PUNCT
ejde-502	90	31	in	in	ADP
ejde-502	90	32	our	our	PRON
ejde-502	90	33	range	range	NOUN
ejde-502	90	34	of	of	ADP
ejde-502	90	35	exponents	exponent	NOUN
ejde-502	90	36	,	,	PUNCT
ejde-502	90	37	ill	ill	ADJ
ejde-502	90	38	-	-	PUNCT
ejde-502	90	39	posedness	posedness	NOUN
ejde-502	90	40	of	of	ADP
ejde-502	90	41	the	the	DET
ejde-502	90	42	cauchy	cauchy	ADJ
ejde-502	90	43	problem	problem	NOUN
ejde-502	90	44	will	will	AUX
ejde-502	90	45	still	still	ADV
ejde-502	90	46	hold	hold	VERB
ejde-502	90	47	and	and	CCONJ
ejde-502	90	48	both	both	DET
ejde-502	90	49	existence	existence	NOUN
ejde-502	90	50	and	and	CCONJ
ejde-502	90	51	uniqueness	uniqueness	NOUN
ejde-502	90	52	of	of	ADP
ejde-502	90	53	solutions	solution	NOUN
ejde-502	90	54	are	be	AUX
ejde-502	90	55	an	an	DET
ejde-502	90	56	issue	issue	NOUN
ejde-502	90	57	.	.	PUNCT
ejde-502	91	1	as	as	SCONJ
ejde-502	91	2	we	we	PRON
ejde-502	91	3	shall	shall	AUX
ejde-502	91	4	see	see	VERB
ejde-502	91	5	below	below	ADV
ejde-502	91	6	,	,	PUNCT
ejde-502	91	7	if	if	SCONJ
ejde-502	91	8	we	we	PRON
ejde-502	91	9	consider	consider	VERB
ejde-502	91	10	initial	initial	ADJ
ejde-502	91	11	conditions	condition	NOUN
ejde-502	91	12	u0	u0	ADJ
ejde-502	91	13	as	as	ADP
ejde-502	91	14	in	in	ADP
ejde-502	91	15	(	(	PUNCT
ejde-502	91	16	1.4	1.4	NUM
ejde-502	91	17	)	)	PUNCT
ejde-502	91	18	,	,	PUNCT
ejde-502	91	19	these	these	DET
ejde-502	91	20	basic	basic	ADJ
ejde-502	91	21	properties	property	NOUN
ejde-502	91	22	of	of	ADP
ejde-502	91	23	existence	existence	NOUN
ejde-502	91	24	and	and	CCONJ
ejde-502	91	25	uniqueness	uniqueness	NOUN
ejde-502	91	26	of	of	ADP
ejde-502	91	27	weak	weak	ADJ
ejde-502	91	28	solutions	solution	NOUN
ejde-502	91	29	are	be	AUX
ejde-502	91	30	strongly	strongly	ADV
ejde-502	91	31	related	relate	VERB
ejde-502	91	32	to	to	ADP
ejde-502	91	33	the	the	DET
ejde-502	91	34	finite	finite	NOUN
ejde-502	91	35	or	or	CCONJ
ejde-502	91	36	infinite	infinite	ADJ
ejde-502	91	37	speed	speed	NOUN
ejde-502	91	38	of	of	ADP
ejde-502	91	39	propagation	propagation	NOUN
ejde-502	91	40	of	of	ADP
ejde-502	91	41	their	their	PRON
ejde-502	91	42	edge	edge	NOUN
ejde-502	91	43	of	of	ADP
ejde-502	91	44	the	the	DET
ejde-502	91	45	support	support	NOUN
ejde-502	91	46	.	.	PUNCT
ejde-502	92	1	throughout	throughout	ADP
ejde-502	92	2	the	the	DET
ejde-502	92	3	paper	paper	NOUN
ejde-502	92	4	,	,	PUNCT
ejde-502	92	5	we	we	PRON
ejde-502	92	6	will	will	AUX
ejde-502	92	7	denote	denote	VERB
ejde-502	92	8	by	by	ADP
ejde-502	92	9	c0(rn	c0(rn	PROPN
ejde-502	92	10	)	)	PUNCT
ejde-502	92	11	the	the	DET
ejde-502	92	12	space	space	NOUN
ejde-502	92	13	of	of	ADP
ejde-502	92	14	continuous	continuous	ADJ
ejde-502	92	15	,	,	PUNCT
ejde-502	92	16	compactly	compactly	ADV
ejde-502	92	17	supported	support	VERB
ejde-502	92	18	functions	function	NOUN
ejde-502	92	19	on	on	ADP
ejde-502	92	20	rn	rn	PROPN
ejde-502	92	21	.	.	PUNCT
ejde-502	93	1	range	range	NOUN
ejde-502	93	2	m	m	PROPN
ejde-502	93	3	+	+	NOUN
ejde-502	93	4	p	p	X
ejde-502	93	5	≥	≥	NOUN
ejde-502	93	6	2	2	NUM
ejde-502	93	7	.	.	PUNCT
ejde-502	93	8	let	let	VERB
ejde-502	93	9	us	we	PRON
ejde-502	93	10	first	first	ADJ
ejde-502	93	11	focus	focus	VERB
ejde-502	93	12	on	on	ADP
ejde-502	93	13	the	the	DET
ejde-502	93	14	range	range	NOUN
ejde-502	93	15	of	of	ADP
ejde-502	93	16	exponents	exponent	NOUN
ejde-502	93	17	for	for	ADP
ejde-502	93	18	which	which	PRON
ejde-502	93	19	m+p	m+p	NUM
ejde-502	93	20	≥	≥	NUM
ejde-502	93	21	2	2	NUM
ejde-502	93	22	and	and	CCONJ
ejde-502	93	23	σ	σ	NUM
ejde-502	93	24	>	>	X
ejde-502	93	25	0	0	X
ejde-502	93	26	.	.	PUNCT
ejde-502	94	1	in	in	ADP
ejde-502	94	2	this	this	DET
ejde-502	94	3	case	case	NOUN
ejde-502	94	4	,	,	PUNCT
ejde-502	94	5	we	we	PRON
ejde-502	94	6	shall	shall	AUX
ejde-502	94	7	prove	prove	VERB
ejde-502	94	8	that	that	SCONJ
ejde-502	94	9	,	,	PUNCT
ejde-502	94	10	at	at	ADP
ejde-502	94	11	least	least	ADJ
ejde-502	94	12	for	for	ADP
ejde-502	94	13	some	some	DET
ejde-502	94	14	interval	interval	NOUN
ejde-502	94	15	of	of	ADP
ejde-502	94	16	time	time	NOUN
ejde-502	94	17	t	t	PROPN
ejde-502	94	18	∈	∈	PROPN
ejde-502	94	19	(	(	PUNCT
ejde-502	94	20	0	0	NUM
ejde-502	94	21	,	,	PUNCT
ejde-502	94	22	t	t	NOUN
ejde-502	94	23	)	)	PUNCT
ejde-502	94	24	,	,	PUNCT
ejde-502	94	25	there	there	PRON
ejde-502	94	26	exist	exist	VERB
ejde-502	94	27	infinitely	infinitely	ADV
ejde-502	94	28	many	many	ADJ
ejde-502	94	29	weak	weak	ADJ
ejde-502	94	30	solutions	solution	NOUN
ejde-502	94	31	to	to	ADP
ejde-502	94	32	(	(	PUNCT
ejde-502	94	33	1.1	1.1	NUM
ejde-502	94	34	)	)	PUNCT
ejde-502	94	35	.	.	PUNCT
ejde-502	95	1	we	we	PRON
ejde-502	95	2	begin	begin	VERB
ejde-502	95	3	with	with	ADP
ejde-502	95	4	the	the	DET
ejde-502	95	5	existence	existence	NOUN
ejde-502	95	6	of	of	ADP
ejde-502	95	7	at	at	ADV
ejde-502	95	8	least	least	ADV
ejde-502	95	9	one	one	NUM
ejde-502	95	10	local	local	ADJ
ejde-502	95	11	(	(	PUNCT
ejde-502	95	12	in	in	ADP
ejde-502	95	13	time	time	NOUN
ejde-502	95	14	)	)	PUNCT
ejde-502	95	15	weak	weak	ADJ
ejde-502	95	16	solution	solution	NOUN
ejde-502	95	17	,	,	PUNCT
ejde-502	95	18	as	as	SCONJ
ejde-502	95	19	stated	state	VERB
ejde-502	95	20	in	in	ADP
ejde-502	95	21	the	the	DET
ejde-502	95	22	following	follow	VERB
ejde-502	95	23	result	result	NOUN
ejde-502	95	24	.	.	PUNCT
ejde-502	96	1	theorem	theorem	VERB
ejde-502	96	2	1.2	1.2	NUM
ejde-502	96	3	(	(	PUNCT
ejde-502	96	4	existence	existence	NOUN
ejde-502	96	5	of	of	ADP
ejde-502	96	6	compactly	compactly	ADV
ejde-502	96	7	supported	support	VERB
ejde-502	96	8	solutions	solution	NOUN
ejde-502	96	9	)	)	PUNCT
ejde-502	96	10	.	.	PUNCT
ejde-502	97	1	in	in	ADP
ejde-502	97	2	our	our	PRON
ejde-502	97	3	framework	framework	NOUN
ejde-502	97	4	and	and	CCONJ
ejde-502	97	5	notation	notation	NOUN
ejde-502	97	6	,	,	PUNCT
ejde-502	97	7	assume	assume	VERB
ejde-502	97	8	that	that	SCONJ
ejde-502	97	9	m+	m+	PRON
ejde-502	97	10	p	p	NOUN
ejde-502	97	11	≥	≥	NUM
ejde-502	97	12	2	2	NUM
ejde-502	97	13	and	and	CCONJ
ejde-502	97	14	let	let	VERB
ejde-502	97	15	u0	u0	ADJ
ejde-502	97	16	be	be	AUX
ejde-502	97	17	an	an	DET
ejde-502	97	18	initial	initial	ADJ
ejde-502	97	19	condition	condition	NOUN
ejde-502	97	20	as	as	ADP
ejde-502	97	21	in	in	ADP
ejde-502	97	22	(	(	PUNCT
ejde-502	97	23	1.4	1.4	NUM
ejde-502	97	24	)	)	PUNCT
ejde-502	97	25	satisfying	satisfy	VERB
ejde-502	97	26	moreover	moreover	ADV
ejde-502	97	27	that	that	SCONJ
ejde-502	97	28	u0	u0	PROPN
ejde-502	97	29	∈	∈	PROPN
ejde-502	97	30	c0(rn	c0(rn	PROPN
ejde-502	97	31	)	)	PUNCT
ejde-502	97	32	.	.	PUNCT
ejde-502	98	1	then	then	ADV
ejde-502	98	2	there	there	PRON
ejde-502	98	3	exists	exist	VERB
ejde-502	98	4	t	t	PROPN
ejde-502	98	5	>	>	X
ejde-502	98	6	0	0	PUNCT
ejde-502	99	1	and	and	CCONJ
ejde-502	99	2	there	there	PRON
ejde-502	99	3	exists	exist	VERB
ejde-502	99	4	at	at	ADP
ejde-502	99	5	least	least	ADJ
ejde-502	99	6	a	a	DET
ejde-502	99	7	weak	weak	ADJ
ejde-502	99	8	solution	solution	NOUN
ejde-502	99	9	u	u	NOUN
ejde-502	99	10	to	to	ADP
ejde-502	99	11	the	the	DET
ejde-502	99	12	cauchy	cauchy	ADJ
ejde-502	99	13	problem	problem	NOUN
ejde-502	99	14	(	(	PUNCT
ejde-502	99	15	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	99	16	)	)	PUNCT
ejde-502	99	17	for	for	ADP
ejde-502	99	18	t	t	PROPN
ejde-502	99	19	∈	∈	PROPN
ejde-502	99	20	(	(	PUNCT
ejde-502	99	21	0	0	NUM
ejde-502	99	22	,	,	PUNCT
ejde-502	99	23	t	t	PROPN
ejde-502	99	24	)	)	PUNCT
ejde-502	99	25	which	which	PRON
ejde-502	99	26	remains	remain	VERB
ejde-502	99	27	continuous	continuous	ADJ
ejde-502	99	28	and	and	CCONJ
ejde-502	99	29	compactly	compactly	ADV
ejde-502	99	30	supported	support	VERB
ejde-502	99	31	:	:	PUNCT
ejde-502	99	32	u(t	u(t	NOUN
ejde-502	99	33	)	)	PUNCT
ejde-502	99	34	∈	∈	PROPN
ejde-502	99	35	c0(rn	c0(rn	PROPN
ejde-502	99	36	)	)	PUNCT
ejde-502	99	37	for	for	ADP
ejde-502	99	38	any	any	DET
ejde-502	99	39	t	t	NOUN
ejde-502	99	40	∈	∈	PROPN
ejde-502	99	41	(	(	PUNCT
ejde-502	99	42	0	0	NUM
ejde-502	99	43	,	,	PUNCT
ejde-502	99	44	t	t	NOUN
ejde-502	99	45	)	)	PUNCT
ejde-502	99	46	.	.	PUNCT
ejde-502	100	1	the	the	DET
ejde-502	100	2	proof	proof	NOUN
ejde-502	100	3	relies	rely	VERB
ejde-502	100	4	on	on	ADP
ejde-502	100	5	the	the	DET
ejde-502	100	6	construction	construction	NOUN
ejde-502	100	7	of	of	ADP
ejde-502	100	8	so	so	ADV
ejde-502	100	9	-	-	PUNCT
ejde-502	100	10	called	call	VERB
ejde-502	100	11	minimal	minimal	ADJ
ejde-502	100	12	solutions	solution	NOUN
ejde-502	100	13	to	to	ADP
ejde-502	100	14	the	the	DET
ejde-502	100	15	cauchy	cauchy	ADJ
ejde-502	100	16	problem	problem	NOUN
ejde-502	100	17	(	(	PUNCT
ejde-502	100	18	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	100	19	)	)	PUNCT
ejde-502	100	20	,	,	PUNCT
ejde-502	100	21	which	which	PRON
ejde-502	100	22	are	be	AUX
ejde-502	100	23	obtained	obtain	VERB
ejde-502	100	24	through	through	ADP
ejde-502	100	25	a	a	DET
ejde-502	100	26	limit	limit	NOUN
ejde-502	100	27	process	process	NOUN
ejde-502	100	28	from	from	ADP
ejde-502	100	29	a	a	DET
ejde-502	100	30	family	family	NOUN
ejde-502	100	31	of	of	ADP
ejde-502	100	32	approximating	approximate	VERB
ejde-502	100	33	(	(	PUNCT
ejde-502	100	34	regularized	regularize	VERB
ejde-502	100	35	)	)	PUNCT
ejde-502	100	36	cauchy	cauchy	NOUN
ejde-502	100	37	problems	problem	NOUN
ejde-502	100	38	.	.	PUNCT
ejde-502	101	1	it	it	PRON
ejde-502	101	2	will	will	AUX
ejde-502	101	3	be	be	AUX
ejde-502	101	4	shown	show	VERB
ejde-502	101	5	that	that	SCONJ
ejde-502	101	6	such	such	DET
ejde-502	101	7	a	a	DET
ejde-502	101	8	minimal	minimal	ADJ
ejde-502	101	9	solution	solution	NOUN
ejde-502	101	10	exists	exist	VERB
ejde-502	101	11	for	for	ADP
ejde-502	101	12	any	any	DET
ejde-502	101	13	compactly	compactly	ADV
ejde-502	101	14	supported	support	VERB
ejde-502	101	15	condition	condition	NOUN
ejde-502	101	16	u0	u0	ADJ
ejde-502	101	17	and	and	CCONJ
ejde-502	101	18	stays	stays	AUX
ejde-502	101	19	compactly	compactly	ADV
ejde-502	101	20	supported	support	VERB
ejde-502	101	21	for	for	ADP
ejde-502	101	22	t	t	PROPN
ejde-502	101	23	∈	∈	PROPN
ejde-502	101	24	(	(	PUNCT
ejde-502	101	25	0	0	NUM
ejde-502	101	26	,	,	PUNCT
ejde-502	101	27	t	t	PROPN
ejde-502	101	28	)	)	PUNCT
ejde-502	101	29	,	,	PUNCT
ejde-502	101	30	a	a	DET
ejde-502	101	31	property	property	NOUN
ejde-502	101	32	known	know	VERB
ejde-502	101	33	as	as	ADP
ejde-502	101	34	finite	finite	ADJ
ejde-502	101	35	speed	speed	NOUN
ejde-502	101	36	of	of	ADP
ejde-502	101	37	propagation	propagation	NOUN
ejde-502	101	38	.	.	PUNCT
ejde-502	102	1	the	the	DET
ejde-502	102	2	(	(	PUNCT
ejde-502	102	3	local	local	ADJ
ejde-502	102	4	in	in	ADP
ejde-502	102	5	time	time	NOUN
ejde-502	102	6	)	)	PUNCT
ejde-502	102	7	finite	finite	VERB
ejde-502	102	8	speed	speed	NOUN
ejde-502	102	9	of	of	ADP
ejde-502	102	10	propagation	propagation	NOUN
ejde-502	102	11	will	will	AUX
ejde-502	102	12	be	be	AUX
ejde-502	102	13	proved	prove	VERB
ejde-502	102	14	with	with	ADP
ejde-502	102	15	the	the	DET
ejde-502	102	16	aid	aid	NOUN
ejde-502	102	17	of	of	ADP
ejde-502	102	18	comparison	comparison	NOUN
ejde-502	102	19	with	with	ADP
ejde-502	102	20	solutions	solution	NOUN
ejde-502	102	21	and	and	CCONJ
ejde-502	102	22	supersolutions	supersolution	NOUN
ejde-502	102	23	in	in	ADP
ejde-502	102	24	self	self	NOUN
ejde-502	102	25	-	-	PUNCT
ejde-502	102	26	similar	similar	ADJ
ejde-502	102	27	form	form	NOUN
ejde-502	102	28	introduced	introduce	VERB
ejde-502	102	29	in	in	ADP
ejde-502	102	30	the	the	DET
ejde-502	102	31	recent	recent	ADJ
ejde-502	102	32	works	work	NOUN
ejde-502	102	33	[	[	X
ejde-502	102	34	19	19	NUM
ejde-502	102	35	,	,	PUNCT
ejde-502	102	36	21	21	NUM
ejde-502	102	37	,	,	PUNCT
ejde-502	102	38	16	16	NUM
ejde-502	102	39	]	]	PUNCT
ejde-502	102	40	.	.	PUNCT
ejde-502	103	1	the	the	DET
ejde-502	103	2	statement	statement	NOUN
ejde-502	103	3	of	of	ADP
ejde-502	103	4	theorem	theorem	ADJ
ejde-502	103	5	1.2	1.2	NUM
ejde-502	103	6	and	and	CCONJ
ejde-502	103	7	precedents	precedent	NOUN
ejde-502	103	8	in	in	ADP
ejde-502	103	9	the	the	DET
ejde-502	103	10	non	non	ADJ
ejde-502	103	11	-	-	ADJ
ejde-502	103	12	weighted	weighted	ADJ
ejde-502	103	13	case	case	NOUN
ejde-502	103	14	[	[	X
ejde-502	103	15	28	28	NUM
ejde-502	103	16	]	]	PUNCT
ejde-502	103	17	give	give	VERB
ejde-502	103	18	the	the	DET
ejde-502	103	19	idea	idea	NOUN
ejde-502	103	20	that	that	SCONJ
ejde-502	103	21	uniqueness	uniqueness	NOUN
ejde-502	103	22	does	do	AUX
ejde-502	103	23	not	not	PART
ejde-502	103	24	hold	hold	VERB
ejde-502	103	25	.	.	PUNCT
ejde-502	104	1	in	in	ADP
ejde-502	104	2	fact	fact	NOUN
ejde-502	104	3	,	,	PUNCT
ejde-502	104	4	we	we	PRON
ejde-502	104	5	infer	infer	VERB
ejde-502	104	6	from	from	ADP
ejde-502	104	7	theorem	theorem	ADJ
ejde-502	104	8	1.2	1.2	NUM
ejde-502	104	9	that	that	SCONJ
ejde-502	104	10	at	at	ADP
ejde-502	104	11	intuitive	intuitive	ADJ
ejde-502	104	12	level	level	NOUN
ejde-502	104	13	the	the	DET
ejde-502	104	14	weight	weight	NOUN
ejde-502	104	15	v	v	NOUN
ejde-502	104	16	(	(	PUNCT
ejde-502	104	17	x	x	NOUN
ejde-502	104	18	)	)	PUNCT
ejde-502	104	19	=	=	SYM
ejde-502	104	20	(	(	PUNCT
ejde-502	104	21	1	1	NUM
ejde-502	104	22	+	+	CCONJ
ejde-502	104	23	|x|)σ	|x|)σ	PROPN
ejde-502	104	24	is	be	AUX
ejde-502	104	25	equivalent	equivalent	ADJ
ejde-502	104	26	to	to	ADP
ejde-502	104	27	a	a	DET
ejde-502	104	28	constant	constant	NOUN
ejde-502	104	29	for	for	ADP
ejde-502	104	30	times	time	NOUN
ejde-502	104	31	t	t	PROPN
ejde-502	104	32	∈	∈	PROPN
ejde-502	104	33	(	(	PUNCT
ejde-502	104	34	0	0	NUM
ejde-502	104	35	,	,	PUNCT
ejde-502	104	36	t	t	NOUN
ejde-502	104	37	)	)	PUNCT
ejde-502	104	38	(	(	PUNCT
ejde-502	104	39	that	that	PRON
ejde-502	104	40	is	be	AUX
ejde-502	104	41	,	,	PUNCT
ejde-502	104	42	while	while	SCONJ
ejde-502	104	43	the	the	DET
ejde-502	104	44	support	support	NOUN
ejde-502	104	45	of	of	ADP
ejde-502	104	46	u(t	u(t	NOUN
ejde-502	104	47	)	)	PUNCT
ejde-502	104	48	remains	remain	VERB
ejde-502	104	49	finite	finite	ADJ
ejde-502	104	50	)	)	PUNCT
ejde-502	104	51	,	,	PUNCT
ejde-502	104	52	thus	thus	ADV
ejde-502	104	53	a	a	DET
ejde-502	104	54	similar	similar	ADJ
ejde-502	104	55	property	property	NOUN
ejde-502	104	56	to	to	ADP
ejde-502	104	57	the	the	DET
ejde-502	104	58	non	non	ADJ
ejde-502	104	59	-	-	ADJ
ejde-502	104	60	weighted	weighted	ADJ
ejde-502	104	61	case	case	NOUN
ejde-502	104	62	concerning	concern	VERB
ejde-502	104	63	non	non	NOUN
ejde-502	104	64	-	-	NOUN
ejde-502	104	65	uniqueness	uniqueness	NOUN
ejde-502	104	66	of	of	ADP
ejde-502	104	67	solutions	solution	NOUN
ejde-502	104	68	is	be	AUX
ejde-502	104	69	expected	expect	VERB
ejde-502	104	70	.	.	PUNCT
ejde-502	105	1	the	the	DET
ejde-502	105	2	next	next	ADJ
ejde-502	105	3	result	result	NOUN
ejde-502	105	4	characterizes	characterize	VERB
ejde-502	105	5	the	the	DET
ejde-502	105	6	extent	extent	NOUN
ejde-502	105	7	of	of	ADP
ejde-502	105	8	this	this	DET
ejde-502	105	9	non	non	ADJ
ejde-502	105	10	-	-	NOUN
ejde-502	105	11	uniqueness	uniqueness	ADJ
ejde-502	105	12	,	,	PUNCT
ejde-502	105	13	by	by	ADP
ejde-502	105	14	showing	show	VERB
ejde-502	105	15	that	that	SCONJ
ejde-502	105	16	we	we	PRON
ejde-502	105	17	can	can	AUX
ejde-502	105	18	prescribe	prescribe	VERB
ejde-502	105	19	in	in	ADP
ejde-502	105	20	infinitely	infinitely	ADV
ejde-502	105	21	many	many	ADJ
ejde-502	105	22	ways	way	NOUN
ejde-502	105	23	the	the	DET
ejde-502	105	24	evolution	evolution	NOUN
ejde-502	105	25	of	of	ADP
ejde-502	105	26	the	the	DET
ejde-502	105	27	interface	interface	NOUN
ejde-502	105	28	of	of	ADP
ejde-502	105	29	a	a	DET
ejde-502	105	30	solution	solution	NOUN
ejde-502	105	31	stemming	stem	VERB
ejde-502	105	32	from	from	ADP
ejde-502	105	33	the	the	DET
ejde-502	105	34	same	same	ADJ
ejde-502	105	35	initial	initial	ADJ
ejde-502	105	36	condition	condition	NOUN
ejde-502	105	37	.	.	PUNCT
ejde-502	106	1	theorem	theorem	VERB
ejde-502	106	2	1.3	1.3	NUM
ejde-502	106	3	(	(	PUNCT
ejde-502	106	4	non	non	ADJ
ejde-502	106	5	-	-	NOUN
ejde-502	106	6	uniqueness	uniqueness	NOUN
ejde-502	106	7	of	of	ADP
ejde-502	106	8	compactly	compactly	ADV
ejde-502	106	9	supported	support	VERB
ejde-502	106	10	solutions	solution	NOUN
ejde-502	106	11	)	)	PUNCT
ejde-502	106	12	.	.	PUNCT
ejde-502	107	1	in	in	ADP
ejde-502	107	2	our	our	PRON
ejde-502	107	3	framework	framework	NOUN
ejde-502	107	4	and	and	CCONJ
ejde-502	107	5	notation	notation	NOUN
ejde-502	107	6	,	,	PUNCT
ejde-502	107	7	assume	assume	VERB
ejde-502	107	8	that	that	SCONJ
ejde-502	107	9	m+	m+	PRON
ejde-502	107	10	p	p	NOUN
ejde-502	107	11	≥	≥	NUM
ejde-502	107	12	2	2	NUM
ejde-502	107	13	and	and	CCONJ
ejde-502	107	14	let	let	VERB
ejde-502	107	15	u0	u0	ADJ
ejde-502	107	16	be	be	AUX
ejde-502	107	17	an	an	DET
ejde-502	107	18	initial	initial	ADJ
ejde-502	107	19	condition	condition	NOUN
ejde-502	107	20	as	as	ADP
ejde-502	107	21	in	in	ADP
ejde-502	107	22	(	(	PUNCT
ejde-502	107	23	1.4	1.4	NUM
ejde-502	107	24	)	)	PUNCT
ejde-502	107	25	satisfying	satisfy	VERB
ejde-502	107	26	moreover	moreover	ADV
ejde-502	107	27	that	that	SCONJ
ejde-502	107	28	u0	u0	PROPN
ejde-502	107	29	∈	∈	PROPN
ejde-502	107	30	c0(r	c0(r	PROPN
ejde-502	107	31	)	)	PUNCT
ejde-502	107	32	.	.	PUNCT
ejde-502	108	1	then	then	ADV
ejde-502	108	2	there	there	PRON
ejde-502	108	3	exists	exist	VERB
ejde-502	108	4	t	t	PROPN
ejde-502	108	5	>	>	X
ejde-502	108	6	0	0	PUNCT
ejde-502	109	1	and	and	CCONJ
ejde-502	109	2	infinitely	infinitely	ADV
ejde-502	109	3	many	many	ADJ
ejde-502	109	4	weak	weak	ADJ
ejde-502	109	5	solutions	solution	NOUN
ejde-502	109	6	to	to	ADP
ejde-502	109	7	the	the	DET
ejde-502	109	8	cauchy	cauchy	ADJ
ejde-502	109	9	problem	problem	NOUN
ejde-502	109	10	(	(	PUNCT
ejde-502	109	11	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	109	12	)	)	PUNCT
ejde-502	109	13	for	for	ADP
ejde-502	109	14	t	t	PROPN
ejde-502	109	15	∈	∈	PROPN
ejde-502	109	16	(	(	PUNCT
ejde-502	109	17	0	0	NUM
ejde-502	109	18	,	,	PUNCT
ejde-502	109	19	t	t	NOUN
ejde-502	109	20	)	)	PUNCT
ejde-502	109	21	.	.	PUNCT
ejde-502	110	1	the	the	DET
ejde-502	110	2	same	same	ADJ
ejde-502	110	3	existence	existence	NOUN
ejde-502	110	4	of	of	ADP
ejde-502	110	5	infinitely	infinitely	ADV
ejde-502	110	6	many	many	ADJ
ejde-502	110	7	weak	weak	ADJ
ejde-502	110	8	solutions	solution	NOUN
ejde-502	110	9	holds	hold	VERB
ejde-502	110	10	true	true	ADJ
ejde-502	110	11	for	for	ADP
ejde-502	110	12	radially	radially	ADV
ejde-502	110	13	symmetric	symmetric	ADJ
ejde-502	110	14	initial	initial	ADJ
ejde-502	110	15	conditions	condition	NOUN
ejde-502	110	16	u0	u0	VERB
ejde-502	110	17	∈	∈	PROPN
ejde-502	110	18	c0(rn	c0(rn	PROPN
ejde-502	110	19	)	)	PUNCT
ejde-502	111	1	satisfying	satisfy	VERB
ejde-502	111	2	(	(	PUNCT
ejde-502	111	3	1.4	1.4	NUM
ejde-502	111	4	)	)	PUNCT
ejde-502	111	5	.	.	PUNCT
ejde-502	112	1	this	this	DET
ejde-502	112	2	result	result	NOUN
ejde-502	112	3	is	be	AUX
ejde-502	112	4	rather	rather	ADV
ejde-502	112	5	similar	similar	ADJ
ejde-502	112	6	to	to	ADP
ejde-502	112	7	the	the	DET
ejde-502	112	8	non	non	ADJ
ejde-502	112	9	-	-	ADJ
ejde-502	112	10	weighted	weighted	ADJ
ejde-502	112	11	case	case	NOUN
ejde-502	112	12	σ	σ	X
ejde-502	112	13	=	=	SYM
ejde-502	112	14	0	0	NUM
ejde-502	112	15	,	,	PUNCT
ejde-502	112	16	although	although	SCONJ
ejde-502	112	17	the	the	DET
ejde-502	112	18	proof	proof	NOUN
ejde-502	112	19	will	will	AUX
ejde-502	112	20	be	be	AUX
ejde-502	112	21	technically	technically	ADV
ejde-502	112	22	more	more	ADV
ejde-502	112	23	involved	involved	ADJ
ejde-502	112	24	:	:	PUNCT
ejde-502	112	25	it	it	PRON
ejde-502	112	26	is	be	AUX
ejde-502	112	27	not	not	PART
ejde-502	112	28	clear	clear	ADJ
ejde-502	112	29	whether	whether	SCONJ
ejde-502	112	30	there	there	PRON
ejde-502	112	31	exists	exist	VERB
ejde-502	112	32	when	when	SCONJ
ejde-502	112	33	σ	σ	X
ejde-502	112	34	>	>	X
ejde-502	112	35	0	0	PUNCT
ejde-502	112	36	a	a	DET
ejde-502	112	37	maximal	maximal	ADJ
ejde-502	112	38	solution	solution	NOUN
ejde-502	112	39	,	,	PUNCT
ejde-502	112	40	in	in	ADP
ejde-502	112	41	order	order	NOUN
ejde-502	112	42	to	to	PART
ejde-502	112	43	get	get	VERB
ejde-502	112	44	almost	almost	ADV
ejde-502	112	45	for	for	ADP
ejde-502	112	46	free	free	ADJ
ejde-502	112	47	a	a	DET
ejde-502	112	48	different	different	ADJ
ejde-502	112	49	,	,	PUNCT
ejde-502	112	50	second	second	ADJ
ejde-502	112	51	solution	solution	NOUN
ejde-502	112	52	(	(	PUNCT
ejde-502	112	53	as	as	SCONJ
ejde-502	112	54	it	it	PRON
ejde-502	112	55	holds	hold	VERB
ejde-502	112	56	true	true	ADJ
ejde-502	112	57	for	for	ADP
ejde-502	112	58	σ	σ	PROPN
ejde-502	112	59	=	=	SYM
ejde-502	112	60	0	0	NUM
ejde-502	112	61	,	,	PUNCT
ejde-502	112	62	see	see	VERB
ejde-502	112	63	[	[	X
ejde-502	112	64	26	26	NUM
ejde-502	112	65	,	,	PUNCT
ejde-502	112	66	27	27	NUM
ejde-502	112	67	]	]	NUM
ejde-502	112	68	)	)	PUNCT
ejde-502	112	69	,	,	PUNCT
ejde-502	112	70	thus	thus	ADV
ejde-502	112	71	we	we	PRON
ejde-502	112	72	have	have	VERB
ejde-502	112	73	to	to	PART
ejde-502	112	74	use	use	VERB
ejde-502	112	75	a	a	DET
ejde-502	112	76	different	different	ADJ
ejde-502	112	77	approach	approach	NOUN
ejde-502	112	78	.	.	PUNCT
ejde-502	113	1	the	the	DET
ejde-502	113	2	statement	statement	NOUN
ejde-502	113	3	of	of	ADP
ejde-502	113	4	theorem	theorem	ADJ
ejde-502	113	5	1.3	1.3	NUM
ejde-502	113	6	will	will	AUX
ejde-502	113	7	be	be	AUX
ejde-502	113	8	thus	thus	ADV
ejde-502	113	9	enforced	enforce	VERB
ejde-502	113	10	and	and	CCONJ
ejde-502	113	11	made	make	VERB
ejde-502	113	12	more	more	ADV
ejde-502	113	13	precise	precise	ADJ
ejde-502	113	14	in	in	ADP
ejde-502	113	15	section	section	NOUN
ejde-502	113	16	3	3	NUM
ejde-502	113	17	,	,	PUNCT
ejde-502	113	18	where	where	SCONJ
ejde-502	113	19	we	we	PRON
ejde-502	113	20	show	show	VERB
ejde-502	113	21	that	that	SCONJ
ejde-502	113	22	the	the	DET
ejde-502	113	23	existence	existence	NOUN
ejde-502	113	24	of	of	ADP
ejde-502	113	25	infinitely	infinitely	ADV
ejde-502	113	26	many	many	ADJ
ejde-502	113	27	solutions	solution	NOUN
ejde-502	113	28	is	be	AUX
ejde-502	113	29	linked	link	VERB
ejde-502	113	30	to	to	ADP
ejde-502	113	31	a	a	DET
ejde-502	113	32	6	6	NUM
ejde-502	113	33	r.	r.	PROPN
ejde-502	113	34	g.	g.	PROPN
ejde-502	113	35	iagar	iagar	PROPN
ejde-502	113	36	,	,	PUNCT
ejde-502	113	37	a.	a.	NOUN
ejde-502	113	38	i.	i.	PROPN
ejde-502	113	39	muñoz	muñoz	PROPN
ejde-502	113	40	,	,	PUNCT
ejde-502	113	41	a.	a.	NOUN
ejde-502	113	42	sánchez	sánchez	PROPN
ejde-502	113	43	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	113	44	prescribed	prescribe	VERB
ejde-502	113	45	evolution	evolution	NOUN
ejde-502	113	46	of	of	ADP
ejde-502	113	47	their	their	PRON
ejde-502	113	48	interface	interface	NOUN
ejde-502	113	49	in	in	ADP
ejde-502	113	50	time	time	NOUN
ejde-502	113	51	,	,	PUNCT
ejde-502	113	52	adapting	adapt	VERB
ejde-502	113	53	but	but	CCONJ
ejde-502	113	54	also	also	ADV
ejde-502	113	55	slightly	slightly	ADV
ejde-502	113	56	improving	improve	VERB
ejde-502	113	57	techniques	technique	NOUN
ejde-502	113	58	from	from	ADP
ejde-502	113	59	[	[	X
ejde-502	113	60	28	28	NUM
ejde-502	113	61	]	]	PUNCT
ejde-502	113	62	.	.	PUNCT
ejde-502	114	1	range	range	NOUN
ejde-502	114	2	m	m	VERB
ejde-502	115	1	+	+	X
ejde-502	115	2	p	p	X
ejde-502	115	3	<	<	X
ejde-502	115	4	2	2	NUM
ejde-502	115	5	.	.	PUNCT
ejde-502	115	6	in	in	ADP
ejde-502	115	7	this	this	DET
ejde-502	115	8	complementary	complementary	ADJ
ejde-502	115	9	range	range	NOUN
ejde-502	115	10	,	,	PUNCT
ejde-502	115	11	our	our	PRON
ejde-502	115	12	main	main	ADJ
ejde-502	115	13	goal	goal	NOUN
ejde-502	115	14	is	be	AUX
ejde-502	115	15	to	to	PART
ejde-502	115	16	prove	prove	VERB
ejde-502	115	17	that	that	SCONJ
ejde-502	115	18	solutions	solution	NOUN
ejde-502	115	19	to	to	ADP
ejde-502	115	20	the	the	DET
ejde-502	115	21	cauchy	cauchy	ADJ
ejde-502	115	22	problem	problem	NOUN
ejde-502	115	23	for	for	ADP
ejde-502	115	24	(	(	PUNCT
ejde-502	115	25	1.1	1.1	NUM
ejde-502	115	26	)	)	PUNCT
ejde-502	115	27	with	with	ADP
ejde-502	115	28	initial	initial	ADJ
ejde-502	115	29	conditions	condition	NOUN
ejde-502	115	30	as	as	ADP
ejde-502	115	31	in	in	ADP
ejde-502	115	32	(	(	PUNCT
ejde-502	115	33	1.4	1.4	NUM
ejde-502	115	34	)	)	PUNCT
ejde-502	115	35	have	have	VERB
ejde-502	115	36	infinite	infinite	ADJ
ejde-502	115	37	speed	speed	NOUN
ejde-502	115	38	of	of	ADP
ejde-502	115	39	propagation	propagation	NOUN
ejde-502	115	40	,	,	PUNCT
ejde-502	115	41	that	that	ADV
ejde-502	115	42	is	is	ADV
ejde-502	115	43	,	,	PUNCT
ejde-502	115	44	they	they	PRON
ejde-502	115	45	become	become	VERB
ejde-502	115	46	immediately	immediately	ADV
ejde-502	115	47	positive	positive	ADJ
ejde-502	115	48	at	at	ADP
ejde-502	115	49	any	any	DET
ejde-502	115	50	point	point	NOUN
ejde-502	115	51	x	x	X
ejde-502	115	52	∈	∈	PROPN
ejde-502	115	53	rn	rn	PROPN
ejde-502	115	54	.	.	PUNCT
ejde-502	116	1	this	this	DET
ejde-502	116	2	fact	fact	NOUN
ejde-502	116	3	extends	extend	VERB
ejde-502	116	4	to	to	ADP
ejde-502	116	5	the	the	DET
ejde-502	116	6	non	non	ADJ
ejde-502	116	7	-	-	ADJ
ejde-502	116	8	homogeneous	homogeneous	ADJ
ejde-502	116	9	range	range	NOUN
ejde-502	116	10	σ	σ	X
ejde-502	116	11	>	>	X
ejde-502	116	12	0	0	PUNCT
ejde-502	117	1	a	a	DET
ejde-502	117	2	similar	similar	ADJ
ejde-502	117	3	result	result	NOUN
ejde-502	117	4	holding	hold	VERB
ejde-502	117	5	true	true	ADJ
ejde-502	117	6	for	for	ADP
ejde-502	117	7	σ	σ	PROPN
ejde-502	117	8	=	=	SYM
ejde-502	117	9	0	0	NUM
ejde-502	117	10	,	,	PUNCT
ejde-502	117	11	established	establish	VERB
ejde-502	117	12	in	in	ADP
ejde-502	117	13	[	[	X
ejde-502	117	14	26	26	NUM
ejde-502	117	15	]	]	PUNCT
ejde-502	117	16	,	,	PUNCT
ejde-502	117	17	but	but	CCONJ
ejde-502	117	18	we	we	PRON
ejde-502	117	19	use	use	VERB
ejde-502	117	20	a	a	DET
ejde-502	117	21	different	different	ADJ
ejde-502	117	22	approach	approach	NOUN
ejde-502	117	23	which	which	PRON
ejde-502	117	24	gives	give	VERB
ejde-502	117	25	,	,	PUNCT
ejde-502	117	26	along	along	ADP
ejde-502	117	27	the	the	DET
ejde-502	117	28	way	way	NOUN
ejde-502	117	29	,	,	PUNCT
ejde-502	117	30	a	a	DET
ejde-502	117	31	result	result	NOUN
ejde-502	117	32	of	of	ADP
ejde-502	117	33	independent	independent	ADJ
ejde-502	117	34	interest	interest	NOUN
ejde-502	117	35	in	in	ADP
ejde-502	117	36	the	the	DET
ejde-502	117	37	study	study	NOUN
ejde-502	117	38	of	of	ADP
ejde-502	117	39	the	the	DET
ejde-502	117	40	homogeneous	homogeneous	ADJ
ejde-502	117	41	equation	equation	NOUN
ejde-502	117	42	(	(	PUNCT
ejde-502	117	43	1.5	1.5	NUM
ejde-502	117	44	)	)	PUNCT
ejde-502	117	45	.	.	PUNCT
ejde-502	118	1	we	we	PRON
ejde-502	118	2	begin	begin	VERB
ejde-502	118	3	by	by	ADP
ejde-502	118	4	establishing	establish	VERB
ejde-502	118	5	the	the	DET
ejde-502	118	6	following	follow	VERB
ejde-502	118	7	aronsonbénilan	aronsonbénilan	PROPN
ejde-502	118	8	estimates	estimate	NOUN
ejde-502	118	9	(	(	PUNCT
ejde-502	118	10	whose	whose	DET
ejde-502	118	11	name	name	NOUN
ejde-502	118	12	stems	stem	VERB
ejde-502	118	13	from	from	ADP
ejde-502	118	14	the	the	DET
ejde-502	118	15	celebrated	celebrated	ADJ
ejde-502	118	16	short	short	ADJ
ejde-502	118	17	note	note	NOUN
ejde-502	119	1	[	[	X
ejde-502	119	2	3	3	X
ejde-502	119	3	]	]	PUNCT
ejde-502	119	4	on	on	ADP
ejde-502	119	5	solutions	solution	NOUN
ejde-502	119	6	to	to	ADP
ejde-502	119	7	the	the	DET
ejde-502	119	8	porous	porous	ADJ
ejde-502	119	9	medium	medium	ADJ
ejde-502	119	10	equation	equation	NOUN
ejde-502	119	11	)	)	PUNCT
ejde-502	119	12	for	for	ADP
ejde-502	119	13	solutions	solution	NOUN
ejde-502	119	14	to	to	ADP
ejde-502	119	15	the	the	DET
ejde-502	119	16	homogeneous	homogeneous	ADJ
ejde-502	119	17	case	case	NOUN
ejde-502	119	18	σ	σ	X
ejde-502	119	19	=	=	SYM
ejde-502	119	20	0	0	X
ejde-502	119	21	.	.	PUNCT
ejde-502	120	1	we	we	PRON
ejde-502	120	2	thus	thus	ADV
ejde-502	120	3	introduce	introduce	VERB
ejde-502	120	4	the	the	DET
ejde-502	120	5	pressure	pressure	NOUN
ejde-502	120	6	function	function	NOUN
ejde-502	120	7	v	v	ADP
ejde-502	120	8	=	=	NOUN
ejde-502	120	9	m	m	VERB
ejde-502	120	10	m−	m−	PROPN
ejde-502	120	11	1	1	NUM
ejde-502	120	12	um−1	um−1	PROPN
ejde-502	120	13	.	.	PUNCT
ejde-502	121	1	theorem	theorem	VERB
ejde-502	121	2	1.4	1.4	NUM
ejde-502	121	3	(	(	PUNCT
ejde-502	121	4	aronson	aronson	PROPN
ejde-502	121	5	-	-	PUNCT
ejde-502	121	6	bénilan	bénilan	PROPN
ejde-502	121	7	estimates	estimate	VERB
ejde-502	121	8	when	when	SCONJ
ejde-502	121	9	m+	m+	NOUN
ejde-502	121	10	p	p	X
ejde-502	121	11	<	<	X
ejde-502	121	12	2	2	NUM
ejde-502	121	13	and	and	CCONJ
ejde-502	121	14	σ	σ	NUM
ejde-502	121	15	=	=	SYM
ejde-502	121	16	0	0	NUM
ejde-502	121	17	)	)	PUNCT
ejde-502	121	18	.	.	PUNCT
ejde-502	122	1	assume	assume	VERB
ejde-502	122	2	that	that	DET
ejde-502	122	3	m+	m+	PRON
ejde-502	122	4	p	p	X
ejde-502	122	5	<	<	X
ejde-502	122	6	2	2	NUM
ejde-502	122	7	.	.	PUNCT
ejde-502	123	1	let	let	VERB
ejde-502	123	2	u	u	PRON
ejde-502	123	3	be	be	AUX
ejde-502	123	4	a	a	DET
ejde-502	123	5	weak	weak	ADJ
ejde-502	123	6	solution	solution	NOUN
ejde-502	123	7	to	to	ADP
ejde-502	123	8	(	(	PUNCT
ejde-502	123	9	1.5	1.5	NUM
ejde-502	123	10	)	)	PUNCT
ejde-502	123	11	in	in	ADP
ejde-502	123	12	rn	rn	PROPN
ejde-502	123	13	×	×	PROPN
ejde-502	123	14	(	(	PUNCT
ejde-502	123	15	0	0	NUM
ejde-502	123	16	,	,	PUNCT
ejde-502	123	17	t	t	PROPN
ejde-502	123	18	)	)	PUNCT
ejde-502	123	19	with	with	ADP
ejde-502	123	20	a	a	DET
ejde-502	123	21	continuous	continuous	ADJ
ejde-502	123	22	initial	initial	ADJ
ejde-502	123	23	condition	condition	NOUN
ejde-502	123	24	u0(x	u0(x	NUM
ejde-502	123	25	)	)	PUNCT
ejde-502	123	26	=	=	PUNCT
ejde-502	123	27	u(x	u(x	NOUN
ejde-502	123	28	,	,	PUNCT
ejde-502	123	29	0	0	NUM
ejde-502	123	30	)	)	PUNCT
ejde-502	124	1	satisfying	satisfying	NOUN
ejde-502	124	2	(	(	PUNCT
ejde-502	124	3	1.4	1.4	NUM
ejde-502	124	4	)	)	PUNCT
ejde-502	124	5	and	and	CCONJ
ejde-502	124	6	let	let	VERB
ejde-502	124	7	v	v	PART
ejde-502	124	8	be	be	AUX
ejde-502	124	9	the	the	DET
ejde-502	124	10	pressure	pressure	NOUN
ejde-502	124	11	variable	variable	NOUN
ejde-502	124	12	introduced	introduce	VERB
ejde-502	124	13	above	above	ADV
ejde-502	124	14	.	.	PUNCT
ejde-502	125	1	then	then	ADV
ejde-502	125	2	the	the	DET
ejde-502	125	3	inequality	inequality	NOUN
ejde-502	125	4	∆v	∆v	PROPN
ejde-502	125	5	≥	≥	NOUN
ejde-502	125	6	−k	−k	PROPN
ejde-502	125	7	t	t	PROPN
ejde-502	125	8	,	,	PUNCT
ejde-502	125	9	k	k	PROPN
ejde-502	125	10	=	=	PUNCT
ejde-502	125	11	n	n	CCONJ
ejde-502	125	12	n(m−	n(m−	PROPN
ejde-502	125	13	1	1	NUM
ejde-502	125	14	)	)	PUNCT
ejde-502	125	15	+	+	NUM
ejde-502	125	16	2	2	NUM
ejde-502	125	17	,	,	PUNCT
ejde-502	125	18	(	(	PUNCT
ejde-502	125	19	1.12	1.12	NUM
ejde-502	125	20	)	)	PUNCT
ejde-502	125	21	holds	hold	VERB
ejde-502	125	22	in	in	ADP
ejde-502	125	23	the	the	DET
ejde-502	125	24	sense	sense	NOUN
ejde-502	125	25	of	of	ADP
ejde-502	125	26	distributions	distribution	NOUN
ejde-502	125	27	in	in	ADP
ejde-502	125	28	rn	rn	PROPN
ejde-502	125	29	,	,	PUNCT
ejde-502	125	30	that	that	PRON
ejde-502	125	31	means,∫	means,∫	NOUN
ejde-502	125	32	t	t	PROPN
ejde-502	125	33	0	0	NUM
ejde-502	125	34	∫	∫	PROPN
ejde-502	125	35	rn	rn	PROPN
ejde-502	125	36	(	(	PUNCT
ejde-502	125	37	v(x	v(x	PROPN
ejde-502	125	38	,	,	PUNCT
ejde-502	125	39	t)∆ϕ(x	t)∆ϕ(x	PROPN
ejde-502	125	40	,	,	PUNCT
ejde-502	125	41	t	t	PROPN
ejde-502	125	42	)	)	PUNCT
ejde-502	126	1	+	+	CCONJ
ejde-502	126	2	k	k	PROPN
ejde-502	126	3	t	t	PROPN
ejde-502	126	4	ϕ(x	ϕ(x	PROPN
ejde-502	126	5	,	,	PUNCT
ejde-502	126	6	t	t	PROPN
ejde-502	126	7	)	)	PUNCT
ejde-502	126	8	)	)	PUNCT
ejde-502	127	1	dx	dx	PROPN
ejde-502	127	2	dt	dt	X
ejde-502	127	3	≥	≥	PROPN
ejde-502	127	4	0	0	NUM
ejde-502	127	5	,	,	PUNCT
ejde-502	127	6	(	(	PUNCT
ejde-502	127	7	1.13	1.13	NUM
ejde-502	127	8	)	)	PUNCT
ejde-502	127	9	for	for	ADP
ejde-502	127	10	any	any	DET
ejde-502	127	11	test	test	NOUN
ejde-502	127	12	function	function	NOUN
ejde-502	127	13	ϕ	ϕ	PROPN
ejde-502	127	14	∈	∈	PROPN
ejde-502	127	15	c∞0	c∞0	X
ejde-502	127	16	(	(	PUNCT
ejde-502	127	17	rn	rn	PROPN
ejde-502	127	18	×	×	PROPN
ejde-502	127	19	(	(	PUNCT
ejde-502	127	20	0	0	NUM
ejde-502	127	21	,	,	PUNCT
ejde-502	127	22	t	t	NOUN
ejde-502	127	23	)	)	PUNCT
ejde-502	127	24	)	)	PUNCT
ejde-502	128	1	such	such	ADJ
ejde-502	128	2	that	that	SCONJ
ejde-502	128	3	ϕ	ϕ	PROPN
ejde-502	128	4	≥	≥	NOUN
ejde-502	128	5	0	0	NUM
ejde-502	128	6	in	in	ADP
ejde-502	128	7	rn	rn	PROPN
ejde-502	128	8	×	×	PROPN
ejde-502	128	9	(	(	PUNCT
ejde-502	128	10	0	0	NUM
ejde-502	128	11	,	,	PUNCT
ejde-502	128	12	t	t	NOUN
ejde-502	128	13	)	)	PUNCT
ejde-502	128	14	.	.	PUNCT
ejde-502	128	15	remark	remark	PROPN
ejde-502	128	16	.	.	PUNCT
ejde-502	129	1	theorem	theorem	VERB
ejde-502	129	2	1.4	1.4	NUM
ejde-502	129	3	is	be	AUX
ejde-502	129	4	expected	expect	VERB
ejde-502	129	5	to	to	PART
ejde-502	129	6	hold	hold	VERB
ejde-502	129	7	also	also	ADV
ejde-502	129	8	for	for	ADP
ejde-502	129	9	any	any	DET
ejde-502	129	10	σ	σ	PROPN
ejde-502	129	11	>	>	X
ejde-502	129	12	0	0	NUM
ejde-502	129	13	,	,	PUNCT
ejde-502	129	14	provided	provide	VERB
ejde-502	129	15	n	n	PRON
ejde-502	129	16	≥	≥	NUM
ejde-502	129	17	2	2	NUM
ejde-502	129	18	.	.	PUNCT
ejde-502	130	1	we	we	PRON
ejde-502	130	2	give	give	VERB
ejde-502	130	3	a	a	DET
ejde-502	130	4	formal	formal	ADJ
ejde-502	130	5	proof	proof	NOUN
ejde-502	130	6	of	of	ADP
ejde-502	130	7	this	this	DET
ejde-502	130	8	fact	fact	NOUN
ejde-502	130	9	at	at	ADP
ejde-502	130	10	the	the	DET
ejde-502	130	11	end	end	NOUN
ejde-502	130	12	of	of	ADP
ejde-502	130	13	section	section	NOUN
ejde-502	130	14	4	4	NUM
ejde-502	130	15	.	.	PUNCT
ejde-502	131	1	however	however	ADV
ejde-502	131	2	,	,	PUNCT
ejde-502	131	3	transforming	transform	VERB
ejde-502	131	4	it	it	PRON
ejde-502	131	5	into	into	ADP
ejde-502	131	6	a	a	DET
ejde-502	131	7	rigorous	rigorous	ADJ
ejde-502	131	8	proof	proof	NOUN
ejde-502	131	9	is	be	AUX
ejde-502	131	10	impossible	impossible	ADJ
ejde-502	131	11	by	by	ADP
ejde-502	131	12	now	now	ADV
ejde-502	131	13	for	for	ADP
ejde-502	131	14	σ	σ	PROPN
ejde-502	131	15	>	>	X
ejde-502	131	16	0	0	NUM
ejde-502	131	17	,	,	PUNCT
ejde-502	131	18	as	as	SCONJ
ejde-502	131	19	we	we	PRON
ejde-502	131	20	are	be	AUX
ejde-502	131	21	lacking	lack	VERB
ejde-502	131	22	a	a	DET
ejde-502	131	23	wellposedness	wellposedness	ADJ
ejde-502	131	24	result	result	NOUN
ejde-502	131	25	for	for	ADP
ejde-502	131	26	the	the	DET
ejde-502	131	27	cauchy	cauchy	ADJ
ejde-502	131	28	problem	problem	NOUN
ejde-502	131	29	(	(	PUNCT
ejde-502	131	30	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	131	31	)	)	PUNCT
ejde-502	131	32	in	in	ADP
ejde-502	131	33	the	the	DET
ejde-502	131	34	range	range	NOUN
ejde-502	131	35	m	m	VERB
ejde-502	131	36	+	+	X
ejde-502	131	37	p	p	X
ejde-502	131	38	<	<	X
ejde-502	131	39	2	2	NUM
ejde-502	131	40	.	.	PUNCT
ejde-502	132	1	this	this	PRON
ejde-502	132	2	is	be	AUX
ejde-502	132	3	why	why	SCONJ
ejde-502	132	4	,	,	PUNCT
ejde-502	132	5	we	we	PRON
ejde-502	132	6	introduce	introduce	VERB
ejde-502	132	7	this	this	DET
ejde-502	132	8	formal	formal	ADJ
ejde-502	132	9	proof	proof	NOUN
ejde-502	132	10	as	as	ADP
ejde-502	132	11	a	a	DET
ejde-502	132	12	remark	remark	NOUN
ejde-502	132	13	.	.	PUNCT
ejde-502	133	1	we	we	PRON
ejde-502	133	2	then	then	ADV
ejde-502	133	3	employ	employ	VERB
ejde-502	133	4	the	the	DET
ejde-502	133	5	aronson	aronson	PROPN
ejde-502	133	6	-	-	PUNCT
ejde-502	133	7	bénilan	bénilan	PROPN
ejde-502	133	8	estimates	estimate	NOUN
ejde-502	133	9	established	establish	VERB
ejde-502	133	10	in	in	ADP
ejde-502	133	11	theorem	theorem	ADJ
ejde-502	133	12	1.4	1.4	NUM
ejde-502	133	13	and	and	CCONJ
ejde-502	133	14	some	some	DET
ejde-502	133	15	consequences	consequence	NOUN
ejde-502	133	16	of	of	ADP
ejde-502	133	17	them	they	PRON
ejde-502	133	18	in	in	ADP
ejde-502	133	19	order	order	NOUN
ejde-502	133	20	to	to	PART
ejde-502	133	21	prove	prove	VERB
ejde-502	133	22	the	the	DET
ejde-502	133	23	infinite	infinite	ADJ
ejde-502	133	24	speed	speed	NOUN
ejde-502	133	25	of	of	ADP
ejde-502	133	26	propagation	propagation	NOUN
ejde-502	133	27	of	of	ADP
ejde-502	133	28	the	the	DET
ejde-502	133	29	supports	support	NOUN
ejde-502	133	30	of	of	ADP
ejde-502	133	31	solutions	solution	NOUN
ejde-502	133	32	to	to	ADP
ejde-502	133	33	(	(	PUNCT
ejde-502	133	34	1.1	1.1	NUM
ejde-502	133	35	)	)	PUNCT
ejde-502	133	36	when	when	SCONJ
ejde-502	133	37	m+p−2	m+p−2	PROPN
ejde-502	133	38	<	<	X
ejde-502	133	39	0	0	PROPN
ejde-502	133	40	,	,	PUNCT
ejde-502	133	41	which	which	PRON
ejde-502	133	42	is	be	AUX
ejde-502	133	43	strongly	strongly	ADV
ejde-502	133	44	contrasting	contrast	VERB
ejde-502	133	45	to	to	ADP
ejde-502	133	46	the	the	DET
ejde-502	133	47	results	result	NOUN
ejde-502	133	48	established	establish	VERB
ejde-502	133	49	in	in	ADP
ejde-502	133	50	the	the	DET
ejde-502	133	51	range	range	NOUN
ejde-502	133	52	m	m	VERB
ejde-502	133	53	+	+	NOUN
ejde-502	133	54	p	p	X
ejde-502	133	55	≥	≥	NOUN
ejde-502	133	56	2	2	NUM
ejde-502	133	57	.	.	PUNCT
ejde-502	134	1	we	we	PRON
ejde-502	134	2	include	include	VERB
ejde-502	134	3	the	the	DET
ejde-502	134	4	range	range	NOUN
ejde-502	134	5	σ	σ	NOUN
ejde-502	134	6	=	=	SYM
ejde-502	134	7	0	0	NUM
ejde-502	134	8	in	in	ADP
ejde-502	134	9	the	the	DET
ejde-502	134	10	statement	statement	NOUN
ejde-502	134	11	,	,	PUNCT
ejde-502	134	12	as	as	SCONJ
ejde-502	134	13	our	our	PRON
ejde-502	134	14	approach	approach	NOUN
ejde-502	134	15	gives	give	VERB
ejde-502	134	16	an	an	DET
ejde-502	134	17	alternative	alternative	ADJ
ejde-502	134	18	proof	proof	NOUN
ejde-502	134	19	to	to	ADP
ejde-502	134	20	the	the	DET
ejde-502	134	21	one	one	NUM
ejde-502	134	22	in	in	ADP
ejde-502	134	23	[	[	X
ejde-502	134	24	26	26	NUM
ejde-502	134	25	]	]	PUNCT
ejde-502	134	26	.	.	PUNCT
ejde-502	135	1	theorem	theorem	ADJ
ejde-502	135	2	1.5	1.5	NUM
ejde-502	135	3	(	(	PUNCT
ejde-502	135	4	infinite	infinite	ADJ
ejde-502	135	5	speed	speed	NOUN
ejde-502	135	6	of	of	ADP
ejde-502	135	7	propagation	propagation	NOUN
ejde-502	135	8	when	when	SCONJ
ejde-502	135	9	m	m	VERB
ejde-502	135	10	+	+	NOUN
ejde-502	135	11	p	p	X
ejde-502	135	12	<	<	X
ejde-502	135	13	2	2	NUM
ejde-502	135	14	)	)	PUNCT
ejde-502	135	15	.	.	PUNCT
ejde-502	136	1	if	if	SCONJ
ejde-502	136	2	the	the	DET
ejde-502	136	3	exponents	exponent	NOUN
ejde-502	136	4	m	m	VERB
ejde-502	136	5	and	and	CCONJ
ejde-502	136	6	p	p	X
ejde-502	136	7	satisfy	satisfy	NOUN
ejde-502	136	8	m	m	VERB
ejde-502	136	9	+	+	NOUN
ejde-502	136	10	p	p	X
ejde-502	136	11	<	<	X
ejde-502	136	12	2	2	NUM
ejde-502	136	13	,	,	PUNCT
ejde-502	136	14	equation	equation	NOUN
ejde-502	136	15	(	(	PUNCT
ejde-502	136	16	1.1	1.1	NUM
ejde-502	136	17	)	)	PUNCT
ejde-502	136	18	has	have	VERB
ejde-502	136	19	the	the	DET
ejde-502	136	20	property	property	NOUN
ejde-502	136	21	of	of	ADP
ejde-502	136	22	infinite	infinite	ADJ
ejde-502	136	23	speed	speed	NOUN
ejde-502	136	24	of	of	ADP
ejde-502	136	25	propagation	propagation	NOUN
ejde-502	136	26	for	for	ADP
ejde-502	136	27	any	any	DET
ejde-502	136	28	σ	σ	X
ejde-502	136	29	≥	≥	NOUN
ejde-502	136	30	0	0	NUM
ejde-502	136	31	,	,	PUNCT
ejde-502	136	32	that	that	ADV
ejde-502	136	33	is	is	ADV
ejde-502	136	34	,	,	PUNCT
ejde-502	136	35	every	every	DET
ejde-502	136	36	weak	weak	ADJ
ejde-502	136	37	solution	solution	NOUN
ejde-502	136	38	(	(	PUNCT
ejde-502	136	39	if	if	SCONJ
ejde-502	136	40	it	it	PRON
ejde-502	136	41	exists	exist	VERB
ejde-502	136	42	)	)	PUNCT
ejde-502	136	43	to	to	ADP
ejde-502	136	44	the	the	DET
ejde-502	136	45	cauchy	cauchy	ADJ
ejde-502	136	46	problem	problem	NOUN
ejde-502	136	47	associated	associate	VERB
ejde-502	136	48	to	to	ADP
ejde-502	136	49	(	(	PUNCT
ejde-502	136	50	1.1	1.1	NUM
ejde-502	136	51	)	)	PUNCT
ejde-502	136	52	with	with	ADP
ejde-502	136	53	continuous	continuous	ADJ
ejde-502	136	54	initial	initial	ADJ
ejde-502	136	55	condition	condition	NOUN
ejde-502	136	56	u0	u0	ADJ
ejde-502	136	57	as	as	ADP
ejde-502	136	58	in	in	ADP
ejde-502	136	59	(	(	PUNCT
ejde-502	136	60	1.4	1.4	NUM
ejde-502	136	61	)	)	PUNCT
ejde-502	136	62	satisfies	satisfie	NOUN
ejde-502	136	63	u(x	u(x	NOUN
ejde-502	136	64	,	,	PUNCT
ejde-502	136	65	t	t	PROPN
ejde-502	136	66	)	)	PUNCT
ejde-502	136	67	>	>	X
ejde-502	136	68	0	0	PUNCT
ejde-502	137	1	for	for	ADP
ejde-502	137	2	any	any	DET
ejde-502	137	3	x	x	SYM
ejde-502	137	4	∈	∈	PROPN
ejde-502	137	5	rn	rn	PROPN
ejde-502	137	6	and	and	CCONJ
ejde-502	137	7	t	t	PROPN
ejde-502	137	8	>	>	X
ejde-502	137	9	0	0	X
ejde-502	137	10	.	.	PUNCT
ejde-502	138	1	the	the	DET
ejde-502	138	2	rest	rest	NOUN
ejde-502	138	3	of	of	ADP
ejde-502	138	4	this	this	DET
ejde-502	138	5	article	article	NOUN
ejde-502	138	6	is	be	AUX
ejde-502	138	7	devoted	devote	VERB
ejde-502	138	8	to	to	ADP
ejde-502	138	9	the	the	DET
ejde-502	138	10	proofs	proof	NOUN
ejde-502	138	11	of	of	ADP
ejde-502	138	12	the	the	DET
ejde-502	138	13	main	main	ADJ
ejde-502	138	14	theorems	theorem	NOUN
ejde-502	138	15	,	,	PUNCT
ejde-502	138	16	following	follow	VERB
ejde-502	138	17	the	the	DET
ejde-502	138	18	outlines	outline	NOUN
ejde-502	138	19	explained	explain	VERB
ejde-502	138	20	in	in	ADP
ejde-502	138	21	the	the	DET
ejde-502	138	22	comments	comment	NOUN
ejde-502	138	23	near	near	ADP
ejde-502	138	24	their	their	PRON
ejde-502	138	25	statements	statement	NOUN
ejde-502	138	26	.	.	PUNCT
ejde-502	139	1	we	we	PRON
ejde-502	139	2	then	then	ADV
ejde-502	139	3	give	give	VERB
ejde-502	139	4	at	at	ADP
ejde-502	139	5	the	the	DET
ejde-502	139	6	end	end	NOUN
ejde-502	139	7	of	of	ADP
ejde-502	139	8	the	the	DET
ejde-502	139	9	paper	paper	NOUN
ejde-502	139	10	a	a	DET
ejde-502	139	11	list	list	NOUN
ejde-502	139	12	of	of	ADP
ejde-502	139	13	open	open	ADJ
ejde-502	139	14	problems	problem	NOUN
ejde-502	139	15	and	and	CCONJ
ejde-502	139	16	possible	possible	ADJ
ejde-502	139	17	extensions	extension	NOUN
ejde-502	139	18	of	of	ADP
ejde-502	139	19	our	our	PRON
ejde-502	139	20	study	study	NOUN
ejde-502	139	21	that	that	SCONJ
ejde-502	139	22	we	we	PRON
ejde-502	139	23	believe	believe	VERB
ejde-502	139	24	to	to	PART
ejde-502	139	25	be	be	AUX
ejde-502	139	26	interesting	interesting	ADJ
ejde-502	139	27	for	for	ADP
ejde-502	139	28	future	future	ADJ
ejde-502	139	29	developments	development	NOUN
ejde-502	139	30	.	.	PUNCT
ejde-502	140	1	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	140	2	reaction	reaction	NOUN
ejde-502	140	3	-	-	PUNCT
ejde-502	140	4	diffusion	diffusion	NOUN
ejde-502	140	5	with	with	ADP
ejde-502	140	6	weighted	weight	VERB
ejde-502	140	7	strong	strong	ADJ
ejde-502	140	8	reaction	reaction	NOUN
ejde-502	140	9	7	7	NUM
ejde-502	140	10	2	2	NUM
ejde-502	140	11	.	.	PUNCT
ejde-502	140	12	existence	existence	NOUN
ejde-502	140	13	and	and	CCONJ
ejde-502	140	14	finite	finite	VERB
ejde-502	140	15	speed	speed	NOUN
ejde-502	140	16	of	of	ADP
ejde-502	140	17	propagation	propagation	NOUN
ejde-502	140	18	when	when	SCONJ
ejde-502	140	19	m+	m+	PRON
ejde-502	140	20	p	p	NOUN
ejde-502	140	21	≥	≥	NUM
ejde-502	140	22	2	2	NUM
ejde-502	140	23	the	the	DET
ejde-502	140	24	goal	goal	NOUN
ejde-502	140	25	of	of	ADP
ejde-502	140	26	this	this	DET
ejde-502	140	27	section	section	NOUN
ejde-502	140	28	is	be	AUX
ejde-502	140	29	to	to	PART
ejde-502	140	30	prove	prove	VERB
ejde-502	140	31	theorem	theorem	VERB
ejde-502	140	32	1.2	1.2	NUM
ejde-502	140	33	in	in	ADP
ejde-502	140	34	the	the	DET
ejde-502	140	35	range	range	NOUN
ejde-502	140	36	of	of	ADP
ejde-502	140	37	parameters	parameter	NOUN
ejde-502	141	1	m	m	VERB
ejde-502	141	2	+	+	ADJ
ejde-502	141	3	p	p	X
ejde-502	141	4	≥	≥	NOUN
ejde-502	141	5	2	2	NUM
ejde-502	141	6	.	.	PUNCT
ejde-502	142	1	let	let	VERB
ejde-502	142	2	u0	u0	ADJ
ejde-502	142	3	be	be	AUX
ejde-502	142	4	a	a	DET
ejde-502	142	5	continuous	continuous	ADJ
ejde-502	142	6	initial	initial	ADJ
ejde-502	142	7	condition	condition	NOUN
ejde-502	142	8	as	as	ADP
ejde-502	142	9	in	in	ADP
ejde-502	142	10	(	(	PUNCT
ejde-502	142	11	1.4	1.4	NUM
ejde-502	142	12	)	)	PUNCT
ejde-502	142	13	.	.	PUNCT
ejde-502	143	1	the	the	DET
ejde-502	143	2	scheme	scheme	NOUN
ejde-502	143	3	of	of	ADP
ejde-502	143	4	the	the	DET
ejde-502	143	5	proof	proof	NOUN
ejde-502	143	6	is	be	AUX
ejde-502	143	7	based	base	VERB
ejde-502	143	8	on	on	ADP
ejde-502	143	9	the	the	DET
ejde-502	143	10	construction	construction	NOUN
ejde-502	143	11	of	of	ADP
ejde-502	143	12	a	a	DET
ejde-502	143	13	minimal	minimal	ADJ
ejde-502	143	14	solution	solution	NOUN
ejde-502	143	15	via	via	ADP
ejde-502	143	16	an	an	DET
ejde-502	143	17	approximation	approximation	NOUN
ejde-502	143	18	process	process	NOUN
ejde-502	143	19	and	and	CCONJ
ejde-502	143	20	showing	show	VERB
ejde-502	143	21	that	that	SCONJ
ejde-502	143	22	this	this	DET
ejde-502	143	23	minimal	minimal	ADJ
ejde-502	143	24	solution	solution	NOUN
ejde-502	143	25	is	be	AUX
ejde-502	143	26	a	a	DET
ejde-502	143	27	weak	weak	ADJ
ejde-502	143	28	solution	solution	NOUN
ejde-502	143	29	to	to	ADP
ejde-502	143	30	the	the	DET
ejde-502	143	31	cauchy	cauchy	ADJ
ejde-502	143	32	problem	problem	NOUN
ejde-502	143	33	(	(	PUNCT
ejde-502	143	34	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	143	35	)	)	PUNCT
ejde-502	143	36	with	with	ADP
ejde-502	143	37	compact	compact	ADJ
ejde-502	143	38	support	support	NOUN
ejde-502	143	39	at	at	ADP
ejde-502	143	40	any	any	DET
ejde-502	143	41	time	time	NOUN
ejde-502	143	42	t	t	X
ejde-502	143	43	∈	∈	PROPN
ejde-502	143	44	(	(	PUNCT
ejde-502	143	45	0	0	NUM
ejde-502	143	46	,	,	PUNCT
ejde-502	143	47	t	t	PROPN
ejde-502	143	48	)	)	PUNCT
ejde-502	143	49	for	for	ADP
ejde-502	143	50	some	some	DET
ejde-502	143	51	t	t	PROPN
ejde-502	143	52	>	>	X
ejde-502	143	53	0	0	X
ejde-502	143	54	.	.	PUNCT
ejde-502	144	1	proposition	proposition	NOUN
ejde-502	144	2	2.1	2.1	NUM
ejde-502	144	3	.	.	PUNCT
ejde-502	145	1	there	there	PRON
ejde-502	145	2	exists	exist	VERB
ejde-502	145	3	some	some	DET
ejde-502	145	4	t	t	PROPN
ejde-502	145	5	>	>	X
ejde-502	145	6	0	0	PUNCT
ejde-502	146	1	and	and	CCONJ
ejde-502	146	2	a	a	DET
ejde-502	146	3	continuous	continuous	ADJ
ejde-502	146	4	weak	weak	ADJ
ejde-502	146	5	solution	solution	NOUN
ejde-502	146	6	u	u	NOUN
ejde-502	146	7	defined	define	VERB
ejde-502	146	8	and	and	CCONJ
ejde-502	146	9	compactly	compactly	ADV
ejde-502	146	10	supported	support	VERB
ejde-502	146	11	for	for	ADP
ejde-502	146	12	t	t	PROPN
ejde-502	146	13	∈	∈	PROPN
ejde-502	146	14	(	(	PUNCT
ejde-502	146	15	0	0	NUM
ejde-502	146	16	,	,	PUNCT
ejde-502	146	17	t	t	NOUN
ejde-502	146	18	)	)	PUNCT
ejde-502	146	19	to	to	ADP
ejde-502	146	20	the	the	DET
ejde-502	146	21	cauchy	cauchy	ADJ
ejde-502	146	22	problem	problem	NOUN
ejde-502	146	23	(	(	PUNCT
ejde-502	146	24	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	146	25	)	)	PUNCT
ejde-502	146	26	with	with	ADP
ejde-502	146	27	u0	u0	ADJ
ejde-502	146	28	as	as	ADP
ejde-502	146	29	above	above	ADP
ejde-502	146	30	such	such	ADJ
ejde-502	146	31	that	that	PRON
ejde-502	146	32	for	for	ADP
ejde-502	146	33	any	any	DET
ejde-502	146	34	other	other	ADJ
ejde-502	146	35	weak	weak	ADJ
ejde-502	146	36	solution	solution	NOUN
ejde-502	146	37	u	u	NOUN
ejde-502	146	38	to	to	ADP
ejde-502	146	39	the	the	DET
ejde-502	146	40	cauchy	cauchy	ADJ
ejde-502	146	41	problem	problem	NOUN
ejde-502	146	42	(	(	PUNCT
ejde-502	146	43	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	146	44	)	)	PUNCT
ejde-502	146	45	(	(	PUNCT
ejde-502	146	46	if	if	SCONJ
ejde-502	146	47	it	it	PRON
ejde-502	146	48	exists	exist	VERB
ejde-502	146	49	)	)	PUNCT
ejde-502	146	50	,	,	PUNCT
ejde-502	146	51	we	we	PRON
ejde-502	146	52	have	have	VERB
ejde-502	146	53	u(x	u(x	NOUN
ejde-502	146	54	,	,	PUNCT
ejde-502	146	55	t	t	PROPN
ejde-502	146	56	)	)	PUNCT
ejde-502	146	57	≤	≤	NOUN
ejde-502	146	58	u(x	u(x	PROPN
ejde-502	146	59	,	,	PUNCT
ejde-502	146	60	t	t	PROPN
ejde-502	146	61	)	)	PUNCT
ejde-502	146	62	,	,	PUNCT
ejde-502	146	63	for	for	ADP
ejde-502	146	64	all	all	DET
ejde-502	146	65	(	(	PUNCT
ejde-502	146	66	x	x	NOUN
ejde-502	146	67	,	,	PUNCT
ejde-502	146	68	t	t	PROPN
ejde-502	146	69	)	)	PUNCT
ejde-502	146	70	∈	∈	PROPN
ejde-502	146	71	rn	rn	PROPN
ejde-502	146	72	×	×	PROPN
ejde-502	146	73	(	(	PUNCT
ejde-502	146	74	0	0	NUM
ejde-502	146	75	,	,	PUNCT
ejde-502	146	76	t	t	NOUN
ejde-502	146	77	)	)	PUNCT
ejde-502	146	78	.	.	PUNCT
ejde-502	147	1	proof	proof	NOUN
ejde-502	147	2	.	.	PUNCT
ejde-502	148	1	we	we	PRON
ejde-502	148	2	divide	divide	VERB
ejde-502	148	3	the	the	DET
ejde-502	148	4	proof	proof	NOUN
ejde-502	148	5	into	into	ADP
ejde-502	148	6	three	three	NUM
ejde-502	148	7	steps	step	NOUN
ejde-502	148	8	for	for	ADP
ejde-502	148	9	the	the	DET
ejde-502	148	10	reader	reader	NOUN
ejde-502	148	11	’s	’s	PART
ejde-502	148	12	convenience	convenience	NOUN
ejde-502	148	13	.	.	PUNCT
ejde-502	149	1	let	let	VERB
ejde-502	149	2	us	we	PRON
ejde-502	149	3	stress	stress	VERB
ejde-502	149	4	at	at	ADP
ejde-502	149	5	this	this	DET
ejde-502	149	6	point	point	NOUN
ejde-502	149	7	that	that	SCONJ
ejde-502	149	8	,	,	PUNCT
ejde-502	149	9	while	while	SCONJ
ejde-502	149	10	step	step	NOUN
ejde-502	149	11	1	1	NUM
ejde-502	149	12	of	of	ADP
ejde-502	149	13	the	the	DET
ejde-502	149	14	proof	proof	NOUN
ejde-502	149	15	is	be	AUX
ejde-502	149	16	an	an	DET
ejde-502	149	17	adaptation	adaptation	NOUN
ejde-502	149	18	of	of	ADP
ejde-502	149	19	an	an	DET
ejde-502	149	20	analogous	analogous	ADJ
ejde-502	149	21	construction	construction	NOUN
ejde-502	149	22	in	in	ADP
ejde-502	149	23	[	[	X
ejde-502	149	24	27	27	NUM
ejde-502	149	25	]	]	PUNCT
ejde-502	149	26	,	,	PUNCT
ejde-502	149	27	the	the	DET
ejde-502	149	28	idea	idea	NOUN
ejde-502	149	29	in	in	ADP
ejde-502	149	30	steps	step	NOUN
ejde-502	149	31	2	2	NUM
ejde-502	149	32	and	and	CCONJ
ejde-502	149	33	3	3	NUM
ejde-502	149	34	strongly	strongly	ADV
ejde-502	149	35	departs	depart	VERB
ejde-502	149	36	from	from	ADP
ejde-502	149	37	the	the	DET
ejde-502	149	38	one	one	NOUN
ejde-502	149	39	used	use	VERB
ejde-502	149	40	in	in	ADP
ejde-502	149	41	the	the	DET
ejde-502	149	42	previously	previously	ADV
ejde-502	149	43	mentioned	mention	VERB
ejde-502	149	44	work	work	NOUN
ejde-502	149	45	and	and	CCONJ
ejde-502	149	46	employs	employ	VERB
ejde-502	149	47	results	result	NOUN
ejde-502	149	48	on	on	ADP
ejde-502	149	49	self	self	NOUN
ejde-502	149	50	-	-	PUNCT
ejde-502	149	51	similar	similar	ADJ
ejde-502	149	52	solutions	solution	NOUN
ejde-502	149	53	to	to	ADP
ejde-502	149	54	(	(	PUNCT
ejde-502	149	55	1.7	1.7	NUM
ejde-502	149	56	)	)	PUNCT
ejde-502	149	57	published	publish	VERB
ejde-502	149	58	recently	recently	ADV
ejde-502	149	59	by	by	ADP
ejde-502	149	60	the	the	DET
ejde-502	149	61	authors	author	NOUN
ejde-502	149	62	.	.	PUNCT
ejde-502	150	1	step	step	NOUN
ejde-502	150	2	1	1	NUM
ejde-502	150	3	.	.	PUNCT
ejde-502	151	1	construction	construction	NOUN
ejde-502	151	2	of	of	ADP
ejde-502	151	3	the	the	DET
ejde-502	151	4	minimal	minimal	ADJ
ejde-502	151	5	solution	solution	NOUN
ejde-502	151	6	.	.	PUNCT
ejde-502	152	1	we	we	PRON
ejde-502	152	2	approximate	approximate	VERB
ejde-502	152	3	the	the	DET
ejde-502	152	4	cauchy	cauchy	ADJ
ejde-502	152	5	problem	problem	NOUN
ejde-502	152	6	(	(	PUNCT
ejde-502	152	7	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	152	8	)	)	PUNCT
ejde-502	152	9	by	by	ADP
ejde-502	152	10	the	the	DET
ejde-502	152	11	following	follow	VERB
ejde-502	152	12	sequence	sequence	NOUN
ejde-502	152	13	of	of	ADP
ejde-502	152	14	cauchy	cauchy	ADJ
ejde-502	152	15	problems	problem	NOUN
ejde-502	152	16	for	for	ADP
ejde-502	152	17	any	any	DET
ejde-502	152	18	positive	positive	ADJ
ejde-502	152	19	integer	integer	NOUN
ejde-502	152	20	k	k	PROPN
ejde-502	152	21	≥	≥	NUM
ejde-502	152	22	1	1	NUM
ejde-502	152	23	wt	wt	NOUN
ejde-502	152	24	=	=	PUNCT
ejde-502	152	25	∆wm	∆wm	NOUN
ejde-502	152	26	+	+	NOUN
ejde-502	152	27	min{(1	min{(1	NOUN
ejde-502	153	1	+	+	CCONJ
ejde-502	153	2	|x|)σ	|x|)σ	NOUN
ejde-502	153	3	,	,	PUNCT
ejde-502	153	4	k}fk(w	k}fk(w	NOUN
ejde-502	153	5	)	)	PUNCT
ejde-502	153	6	,	,	PUNCT
ejde-502	153	7	(	(	PUNCT
ejde-502	153	8	x	x	X
ejde-502	153	9	,	,	PUNCT
ejde-502	153	10	t	t	PROPN
ejde-502	153	11	)	)	PUNCT
ejde-502	153	12	∈	∈	PROPN
ejde-502	153	13	rn	rn	PROPN
ejde-502	153	14	×	×	PROPN
ejde-502	153	15	(	(	PUNCT
ejde-502	153	16	0,∞	0,∞	NUM
ejde-502	153	17	)	)	PUNCT
ejde-502	153	18	,	,	PUNCT
ejde-502	153	19	w(x	w(x	PROPN
ejde-502	153	20	,	,	PUNCT
ejde-502	153	21	0	0	NUM
ejde-502	153	22	)	)	PUNCT
ejde-502	153	23	=	=	SYM
ejde-502	153	24	u0(x	u0(x	NOUN
ejde-502	153	25	)	)	PUNCT
ejde-502	153	26	,	,	PUNCT
ejde-502	153	27	x	x	PROPN
ejde-502	153	28	∈	∈	PROPN
ejde-502	153	29	rn	rn	PROPN
ejde-502	153	30	,	,	PUNCT
ejde-502	153	31	(	(	PUNCT
ejde-502	153	32	2.1	2.1	NUM
ejde-502	153	33	)	)	PUNCT
ejde-502	153	34	where	where	SCONJ
ejde-502	153	35	fk(w	fk(w	VERB
ejde-502	153	36	)	)	PUNCT
ejde-502	153	37	=	=	PRON
ejde-502	153	38	{	{	PUNCT
ejde-502	153	39	(	(	PUNCT
ejde-502	153	40	1	1	NUM
ejde-502	153	41	k	k	NOUN
ejde-502	153	42	)	)	PUNCT
ejde-502	153	43	p−1w	p−1w	VERB
ejde-502	153	44	,	,	PUNCT
ejde-502	153	45	if	if	SCONJ
ejde-502	153	46	0	0	NUM
ejde-502	153	47	≤	≤	NUM
ejde-502	153	48	w	w	NOUN
ejde-502	153	49	≤	≤	NUM
ejde-502	153	50	1	1	NUM
ejde-502	153	51	k	k	NOUN
ejde-502	153	52	,	,	PUNCT
ejde-502	153	53	wp	wp	INTJ
ejde-502	153	54	,	,	PUNCT
ejde-502	153	55	if	if	SCONJ
ejde-502	153	56	w	w	PROPN
ejde-502	153	57	≥	≥	NOUN
ejde-502	153	58	1	1	NUM
ejde-502	153	59	k	k	NOUN
ejde-502	153	60	.	.	PUNCT
ejde-502	154	1	since	since	SCONJ
ejde-502	154	2	the	the	DET
ejde-502	154	3	nonlinearity	nonlinearity	NOUN
ejde-502	154	4	in	in	ADP
ejde-502	154	5	(	(	PUNCT
ejde-502	154	6	2.1	2.1	NUM
ejde-502	154	7	)	)	PUNCT
ejde-502	154	8	is	be	AUX
ejde-502	154	9	of	of	ADP
ejde-502	154	10	the	the	DET
ejde-502	154	11	form	form	NOUN
ejde-502	154	12	g(x)h(w	g(x)h(w	NOUN
ejde-502	154	13	)	)	PUNCT
ejde-502	154	14	with	with	ADP
ejde-502	154	15	g	g	PROPN
ejde-502	154	16	∈	∈	PROPN
ejde-502	154	17	l∞(rn	l∞(rn	PROPN
ejde-502	154	18	)	)	PUNCT
ejde-502	154	19	,	,	PUNCT
ejde-502	154	20	g(x	g(x	NOUN
ejde-502	154	21	)	)	PUNCT
ejde-502	154	22	≥	≥	NOUN
ejde-502	154	23	1	1	NUM
ejde-502	154	24	for	for	ADP
ejde-502	154	25	any	any	DET
ejde-502	154	26	x	x	SYM
ejde-502	154	27	∈	∈	PROPN
ejde-502	154	28	rn	rn	PROPN
ejde-502	154	29	and	and	CCONJ
ejde-502	154	30	h	h	NOUN
ejde-502	154	31	is	be	AUX
ejde-502	154	32	a	a	DET
ejde-502	154	33	lipschitz	lipschitz	NOUN
ejde-502	154	34	function	function	NOUN
ejde-502	154	35	,	,	PUNCT
ejde-502	154	36	we	we	PRON
ejde-502	154	37	infer	infer	VERB
ejde-502	154	38	by	by	ADP
ejde-502	154	39	standard	standard	ADJ
ejde-502	154	40	results	result	NOUN
ejde-502	154	41	for	for	ADP
ejde-502	154	42	quasilinear	quasilinear	PROPN
ejde-502	154	43	parabolic	parabolic	ADJ
ejde-502	154	44	equations	equation	NOUN
ejde-502	154	45	(	(	PUNCT
ejde-502	154	46	see	see	VERB
ejde-502	154	47	for	for	ADP
ejde-502	154	48	example	example	NOUN
ejde-502	154	49	[	[	X
ejde-502	154	50	9	9	NUM
ejde-502	154	51	,	,	PUNCT
ejde-502	154	52	33	33	NUM
ejde-502	154	53	]	]	PUNCT
ejde-502	154	54	)	)	PUNCT
ejde-502	154	55	that	that	SCONJ
ejde-502	154	56	the	the	DET
ejde-502	154	57	cauchy	cauchy	PROPN
ejde-502	154	58	problem	problem	NOUN
ejde-502	154	59	(	(	PUNCT
ejde-502	154	60	2.1	2.1	NUM
ejde-502	154	61	)	)	PUNCT
ejde-502	154	62	admits	admit	VERB
ejde-502	154	63	a	a	DET
ejde-502	154	64	unique	unique	ADJ
ejde-502	154	65	solution	solution	NOUN
ejde-502	154	66	wk	wk	ADP
ejde-502	154	67	defined	define	VERB
ejde-502	154	68	for	for	ADP
ejde-502	154	69	(	(	PUNCT
ejde-502	154	70	x	x	NOUN
ejde-502	154	71	,	,	PUNCT
ejde-502	154	72	t	t	PROPN
ejde-502	154	73	)	)	PUNCT
ejde-502	154	74	∈	∈	PROPN
ejde-502	154	75	rn	rn	PROPN
ejde-502	154	76	×	×	PROPN
ejde-502	154	77	(	(	PUNCT
ejde-502	154	78	0,∞	0,∞	NOUN
ejde-502	154	79	)	)	PUNCT
ejde-502	154	80	,	,	PUNCT
ejde-502	154	81	which	which	PRON
ejde-502	154	82	is	be	AUX
ejde-502	154	83	compactly	compactly	ADV
ejde-502	154	84	supported	support	VERB
ejde-502	154	85	and	and	CCONJ
ejde-502	154	86	continuous	continuous	ADJ
ejde-502	154	87	.	.	PUNCT
ejde-502	155	1	the	the	DET
ejde-502	155	2	comparison	comparison	NOUN
ejde-502	155	3	principle	principle	NOUN
ejde-502	155	4	is	be	AUX
ejde-502	155	5	in	in	ADP
ejde-502	155	6	force	force	NOUN
ejde-502	155	7	for	for	ADP
ejde-502	155	8	the	the	DET
ejde-502	155	9	cauchy	cauchy	ADJ
ejde-502	155	10	problem	problem	NOUN
ejde-502	155	11	(	(	PUNCT
ejde-502	155	12	2.1	2.1	NUM
ejde-502	155	13	)	)	PUNCT
ejde-502	155	14	and	and	CCONJ
ejde-502	155	15	wk+1	wk+1	NOUN
ejde-502	155	16	is	be	AUX
ejde-502	155	17	a	a	DET
ejde-502	155	18	supersolution	supersolution	NOUN
ejde-502	155	19	to	to	ADP
ejde-502	155	20	the	the	DET
ejde-502	155	21	problem	problem	NOUN
ejde-502	155	22	(	(	PUNCT
ejde-502	155	23	2.1	2.1	NUM
ejde-502	155	24	)	)	PUNCT
ejde-502	155	25	,	,	PUNCT
ejde-502	155	26	thus	thus	ADV
ejde-502	155	27	wk+1	wk+1	X
ejde-502	155	28	≥	≥	X
ejde-502	155	29	wk	wk	INTJ
ejde-502	155	30	for	for	ADP
ejde-502	155	31	any	any	DET
ejde-502	155	32	k	k	PROPN
ejde-502	155	33	≥	≥	NUM
ejde-502	155	34	1	1	NUM
ejde-502	155	35	.	.	PUNCT
ejde-502	156	1	this	this	PRON
ejde-502	156	2	allows	allow	VERB
ejde-502	156	3	us	we	PRON
ejde-502	156	4	to	to	PART
ejde-502	156	5	introduce	introduce	VERB
ejde-502	156	6	the	the	DET
ejde-502	156	7	pointwise	pointwise	NOUN
ejde-502	156	8	(	(	PUNCT
ejde-502	156	9	and	and	CCONJ
ejde-502	156	10	monotone	monotone	ADJ
ejde-502	156	11	increasing	increase	VERB
ejde-502	156	12	)	)	PUNCT
ejde-502	156	13	limit	limit	NOUN
ejde-502	156	14	(	(	PUNCT
ejde-502	156	15	which	which	PRON
ejde-502	156	16	might	might	AUX
ejde-502	156	17	become	become	VERB
ejde-502	156	18	infinite	infinite	ADJ
ejde-502	156	19	starting	start	VERB
ejde-502	156	20	from	from	ADP
ejde-502	156	21	some	some	DET
ejde-502	156	22	finite	finite	ADJ
ejde-502	156	23	time	time	NOUN
ejde-502	156	24	)	)	PUNCT
ejde-502	156	25	u(x	u(x	PROPN
ejde-502	156	26	,	,	PUNCT
ejde-502	156	27	t	t	NOUN
ejde-502	156	28	)	)	PUNCT
ejde-502	156	29	=	=	PROPN
ejde-502	156	30	lim	lim	PROPN
ejde-502	156	31	k→∞	k→∞	PROPN
ejde-502	156	32	wk(x	wk(x	PROPN
ejde-502	156	33	,	,	PUNCT
ejde-502	156	34	t	t	PROPN
ejde-502	156	35	)	)	PUNCT
ejde-502	156	36	<	<	X
ejde-502	156	37	∞	∞	PROPN
ejde-502	156	38	,	,	PUNCT
ejde-502	156	39	(	(	PUNCT
ejde-502	156	40	x	x	X
ejde-502	156	41	,	,	PUNCT
ejde-502	156	42	t	t	PROPN
ejde-502	156	43	)	)	PUNCT
ejde-502	156	44	∈	∈	PROPN
ejde-502	156	45	rn	rn	PROPN
ejde-502	156	46	×	×	PROPN
ejde-502	156	47	(	(	PUNCT
ejde-502	156	48	0	0	NUM
ejde-502	156	49	,	,	PUNCT
ejde-502	156	50	t∞	t∞	NUM
ejde-502	156	51	)	)	PUNCT
ejde-502	156	52	,	,	PUNCT
ejde-502	156	53	which	which	PRON
ejde-502	156	54	is	be	AUX
ejde-502	156	55	well	well	ADV
ejde-502	156	56	defined	define	VERB
ejde-502	156	57	provided	provide	VERB
ejde-502	156	58	that	that	SCONJ
ejde-502	156	59	t∞	t∞	PROPN
ejde-502	156	60	>	>	X
ejde-502	156	61	0	0	X
ejde-502	156	62	.	.	PUNCT
ejde-502	157	1	this	this	DET
ejde-502	157	2	fact	fact	NOUN
ejde-502	157	3	will	will	AUX
ejde-502	157	4	follow	follow	VERB
ejde-502	157	5	from	from	ADP
ejde-502	157	6	the	the	DET
ejde-502	157	7	construction	construction	NOUN
ejde-502	157	8	of	of	ADP
ejde-502	157	9	a	a	DET
ejde-502	157	10	“	"	PUNCT
ejde-502	157	11	universal	universal	ADJ
ejde-502	157	12	”	"	PUNCT
ejde-502	157	13	family	family	NOUN
ejde-502	157	14	of	of	ADP
ejde-502	157	15	supersolutions	supersolution	NOUN
ejde-502	157	16	in	in	ADP
ejde-502	157	17	self	self	NOUN
ejde-502	157	18	-	-	PUNCT
ejde-502	157	19	similar	similar	ADJ
ejde-502	157	20	form	form	NOUN
ejde-502	157	21	which	which	PRON
ejde-502	157	22	is	be	AUX
ejde-502	157	23	postponed	postpone	VERB
ejde-502	157	24	to	to	PART
ejde-502	157	25	step	step	VERB
ejde-502	157	26	2	2	NUM
ejde-502	157	27	(	(	PUNCT
ejde-502	157	28	in	in	ADP
ejde-502	157	29	dimension	dimension	NOUN
ejde-502	157	30	n	n	NOUN
ejde-502	157	31	=	=	SYM
ejde-502	157	32	1	1	NUM
ejde-502	157	33	)	)	PUNCT
ejde-502	157	34	and	and	CCONJ
ejde-502	157	35	step	step	NOUN
ejde-502	157	36	3	3	NUM
ejde-502	157	37	(	(	PUNCT
ejde-502	157	38	in	in	ADP
ejde-502	157	39	dimension	dimension	NOUN
ejde-502	157	40	n	n	CCONJ
ejde-502	157	41	≥	≥	NOUN
ejde-502	157	42	2	2	NUM
ejde-502	157	43	)	)	PUNCT
ejde-502	157	44	below	below	ADV
ejde-502	157	45	.	.	PUNCT
ejde-502	158	1	moreover	moreover	ADV
ejde-502	158	2	,	,	PUNCT
ejde-502	158	3	the	the	DET
ejde-502	158	4	solutions	solution	NOUN
ejde-502	158	5	wk	wk	INTJ
ejde-502	158	6	are	be	AUX
ejde-502	158	7	thus	thus	ADV
ejde-502	158	8	uniformly	uniformly	ADV
ejde-502	158	9	bounded	bound	VERB
ejde-502	158	10	on	on	ADP
ejde-502	158	11	rn×(0	rn×(0	PROPN
ejde-502	158	12	,	,	PUNCT
ejde-502	158	13	t	t	PROPN
ejde-502	158	14	)	)	PUNCT
ejde-502	158	15	for	for	ADP
ejde-502	158	16	some	some	DET
ejde-502	158	17	t	t	NOUN
ejde-502	158	18	=	=	SYM
ejde-502	158	19	t	t	PROPN
ejde-502	158	20	(	(	PUNCT
ejde-502	158	21	u0	u0	PROPN
ejde-502	158	22	)	)	PUNCT
ejde-502	158	23	>	>	X
ejde-502	158	24	0	0	PUNCT
ejde-502	159	1	depending	depend	VERB
ejde-502	159	2	on	on	ADP
ejde-502	159	3	u0	u0	ADJ
ejde-502	159	4	,	,	PUNCT
ejde-502	159	5	and	and	CCONJ
ejde-502	159	6	this	this	DET
ejde-502	159	7	uniform	uniform	ADJ
ejde-502	159	8	boundedness	boundedness	NOUN
ejde-502	159	9	,	,	PUNCT
ejde-502	159	10	together	together	ADV
ejde-502	159	11	with	with	ADP
ejde-502	159	12	classical	classical	ADJ
ejde-502	159	13	results	result	NOUN
ejde-502	159	14	in	in	ADP
ejde-502	159	15	[	[	X
ejde-502	159	16	9	9	NUM
ejde-502	159	17	,	,	PUNCT
ejde-502	159	18	33	33	NUM
ejde-502	159	19	]	]	PUNCT
ejde-502	159	20	,	,	PUNCT
ejde-502	159	21	imply	imply	VERB
ejde-502	159	22	that	that	SCONJ
ejde-502	159	23	the	the	DET
ejde-502	159	24	family	family	NOUN
ejde-502	159	25	(	(	PUNCT
ejde-502	159	26	wk)k≥1	wk)k≥1	NOUN
ejde-502	159	27	is	be	AUX
ejde-502	159	28	uniformly	uniformly	ADV
ejde-502	159	29	equicontinuous	equicontinuous	ADJ
ejde-502	159	30	in	in	ADP
ejde-502	159	31	rn	rn	PROPN
ejde-502	159	32	×	×	PROPN
ejde-502	159	33	[	[	X
ejde-502	159	34	0	0	NUM
ejde-502	159	35	,	,	PUNCT
ejde-502	159	36	t	t	X
ejde-502	159	37	]	]	PUNCT
ejde-502	159	38	,	,	PUNCT
ejde-502	159	39	hence	hence	ADV
ejde-502	159	40	there	there	PRON
ejde-502	159	41	exists	exist	VERB
ejde-502	159	42	a	a	DET
ejde-502	159	43	subsequence	subsequence	NOUN
ejde-502	159	44	(	(	PUNCT
ejde-502	159	45	relabeled	relabele	VERB
ejde-502	159	46	also	also	ADV
ejde-502	159	47	wk	wk	INTJ
ejde-502	159	48	for	for	ADP
ejde-502	159	49	simplicity	simplicity	NOUN
ejde-502	159	50	)	)	PUNCT
ejde-502	159	51	which	which	PRON
ejde-502	159	52	converges	converge	VERB
ejde-502	159	53	locally	locally	ADV
ejde-502	159	54	uniformly	uniformly	ADV
ejde-502	159	55	to	to	ADP
ejde-502	159	56	the	the	DET
ejde-502	159	57	same	same	ADJ
ejde-502	159	58	function	function	NOUN
ejde-502	159	59	u(x	u(x	NOUN
ejde-502	159	60	,	,	PUNCT
ejde-502	159	61	t	t	PROPN
ejde-502	159	62	)	)	PUNCT
ejde-502	159	63	.	.	PUNCT
ejde-502	160	1	then	then	ADV
ejde-502	160	2	the	the	DET
ejde-502	160	3	fact	fact	NOUN
ejde-502	160	4	that	that	SCONJ
ejde-502	160	5	u	u	NOUN
ejde-502	160	6	is	be	AUX
ejde-502	160	7	a	a	DET
ejde-502	160	8	continuous	continuous	ADJ
ejde-502	160	9	weak	weak	ADJ
ejde-502	160	10	solution	solution	NOUN
ejde-502	160	11	to	to	ADP
ejde-502	160	12	(	(	PUNCT
ejde-502	160	13	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	160	14	)	)	PUNCT
ejde-502	160	15	for	for	ADP
ejde-502	160	16	(	(	PUNCT
ejde-502	160	17	x	x	NOUN
ejde-502	160	18	,	,	PUNCT
ejde-502	160	19	t	t	PROPN
ejde-502	160	20	)	)	PUNCT
ejde-502	160	21	∈	∈	PROPN
ejde-502	160	22	rn×(0	rn×(0	PROPN
ejde-502	160	23	,	,	PUNCT
ejde-502	160	24	t∞	t∞	PROPN
ejde-502	160	25	)	)	PUNCT
ejde-502	160	26	follows	follow	VERB
ejde-502	160	27	now	now	ADV
ejde-502	160	28	readily	readily	ADV
ejde-502	160	29	from	from	ADP
ejde-502	160	30	the	the	DET
ejde-502	160	31	previous	previous	ADJ
ejde-502	160	32	convergences	convergence	NOUN
ejde-502	160	33	(	(	PUNCT
ejde-502	160	34	assuming	assume	VERB
ejde-502	160	35	for	for	ADP
ejde-502	160	36	now	now	ADV
ejde-502	160	37	the	the	DET
ejde-502	160	38	outcome	outcome	NOUN
ejde-502	160	39	of	of	ADP
ejde-502	160	40	steps	step	NOUN
ejde-502	160	41	2	2	NUM
ejde-502	160	42	and	and	CCONJ
ejde-502	160	43	3	3	NUM
ejde-502	160	44	below	below	ADV
ejde-502	160	45	):	):	PUNCT
ejde-502	160	46	indeed	indeed	ADV
ejde-502	160	47	,	,	PUNCT
ejde-502	160	48	lebesgue	lebesgue	PROPN
ejde-502	160	49	’s	’s	PART
ejde-502	160	50	monotone	monotone	ADJ
ejde-502	160	51	convergence	convergence	NOUN
ejde-502	160	52	8	8	NUM
ejde-502	160	53	r.	r.	PROPN
ejde-502	160	54	g.	g.	PROPN
ejde-502	160	55	iagar	iagar	PROPN
ejde-502	160	56	,	,	PUNCT
ejde-502	160	57	a.	a.	NOUN
ejde-502	160	58	i.	i.	PROPN
ejde-502	160	59	muñoz	muñoz	PROPN
ejde-502	160	60	,	,	PUNCT
ejde-502	160	61	a.	a.	NOUN
ejde-502	160	62	sánchez	sánchez	PROPN
ejde-502	160	63	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	160	64	theorem	theorem	VERB
ejde-502	160	65	ensures	ensure	VERB
ejde-502	160	66	the	the	DET
ejde-502	160	67	assumptions	assumption	NOUN
ejde-502	160	68	(	(	PUNCT
ejde-502	160	69	b	b	NOUN
ejde-502	160	70	)	)	PUNCT
ejde-502	160	71	and	and	CCONJ
ejde-502	160	72	(	(	PUNCT
ejde-502	160	73	c	c	X
ejde-502	160	74	)	)	PUNCT
ejde-502	160	75	in	in	ADP
ejde-502	160	76	definition	definition	NOUN
ejde-502	160	77	1.1	1.1	NUM
ejde-502	160	78	,	,	PUNCT
ejde-502	160	79	while	while	SCONJ
ejde-502	160	80	the	the	DET
ejde-502	160	81	uniform	uniform	NOUN
ejde-502	160	82	bound	bind	VERB
ejde-502	160	83	by	by	ADP
ejde-502	160	84	the	the	DET
ejde-502	160	85	supersolutions	supersolution	NOUN
ejde-502	160	86	constructed	construct	VERB
ejde-502	160	87	in	in	ADP
ejde-502	160	88	steps	step	NOUN
ejde-502	160	89	2	2	NUM
ejde-502	160	90	and	and	CCONJ
ejde-502	160	91	3	3	NUM
ejde-502	160	92	below	below	ADV
ejde-502	160	93	,	,	PUNCT
ejde-502	160	94	together	together	ADV
ejde-502	160	95	with	with	ADP
ejde-502	160	96	the	the	DET
ejde-502	160	97	equicontinuity	equicontinuity	NOUN
ejde-502	160	98	,	,	PUNCT
ejde-502	160	99	give	give	VERB
ejde-502	160	100	that	that	SCONJ
ejde-502	160	101	u	u	NOUN
ejde-502	160	102	satisfies	satisfy	VERB
ejde-502	160	103	the	the	DET
ejde-502	160	104	regularity	regularity	NOUN
ejde-502	160	105	assumption	assumption	NOUN
ejde-502	160	106	(	(	PUNCT
ejde-502	160	107	a	a	X
ejde-502	160	108	)	)	PUNCT
ejde-502	160	109	in	in	ADP
ejde-502	160	110	definition	definition	NOUN
ejde-502	160	111	1.1	1.1	NUM
ejde-502	160	112	.	.	PUNCT
ejde-502	161	1	moreover	moreover	ADV
ejde-502	161	2	,	,	PUNCT
ejde-502	161	3	if	if	SCONJ
ejde-502	161	4	u	u	NOUN
ejde-502	161	5	is	be	AUX
ejde-502	161	6	another	another	DET
ejde-502	161	7	weak	weak	ADJ
ejde-502	161	8	solution	solution	NOUN
ejde-502	161	9	to	to	ADP
ejde-502	161	10	the	the	DET
ejde-502	161	11	cauchy	cauchy	ADJ
ejde-502	161	12	problem	problem	NOUN
ejde-502	161	13	(	(	PUNCT
ejde-502	161	14	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	161	15	)	)	PUNCT
ejde-502	161	16	,	,	PUNCT
ejde-502	161	17	then	then	ADV
ejde-502	161	18	it	it	PRON
ejde-502	161	19	is	be	AUX
ejde-502	161	20	a	a	DET
ejde-502	161	21	supersolution	supersolution	NOUN
ejde-502	161	22	to	to	ADP
ejde-502	161	23	the	the	DET
ejde-502	161	24	problem	problem	NOUN
ejde-502	161	25	(	(	PUNCT
ejde-502	161	26	2.1	2.1	NUM
ejde-502	161	27	)	)	PUNCT
ejde-502	161	28	for	for	ADP
ejde-502	161	29	any	any	DET
ejde-502	161	30	k	k	PROPN
ejde-502	161	31	≥	≥	NUM
ejde-502	161	32	1	1	NUM
ejde-502	161	33	,	,	PUNCT
ejde-502	161	34	whence	whence	NOUN
ejde-502	161	35	u(x	u(x	NOUN
ejde-502	161	36	,	,	PUNCT
ejde-502	161	37	t	t	PROPN
ejde-502	161	38	)	)	PUNCT
ejde-502	161	39	≥	≥	NOUN
ejde-502	161	40	wk(x	wk(x	PROPN
ejde-502	161	41	,	,	PUNCT
ejde-502	161	42	t	t	PROPN
ejde-502	161	43	)	)	PUNCT
ejde-502	161	44	for	for	ADP
ejde-502	161	45	any	any	DET
ejde-502	161	46	k	k	PROPN
ejde-502	161	47	≥	≥	NUM
ejde-502	161	48	1	1	NUM
ejde-502	161	49	and	and	CCONJ
ejde-502	161	50	(	(	PUNCT
ejde-502	161	51	x	x	NOUN
ejde-502	161	52	,	,	PUNCT
ejde-502	161	53	t	t	PROPN
ejde-502	161	54	)	)	PUNCT
ejde-502	161	55	∈	∈	PROPN
ejde-502	161	56	rn	rn	PROPN
ejde-502	161	57	×	×	PROPN
ejde-502	161	58	(	(	PUNCT
ejde-502	161	59	0,∞	0,∞	NOUN
ejde-502	161	60	)	)	PUNCT
ejde-502	161	61	and	and	CCONJ
ejde-502	161	62	by	by	ADP
ejde-502	161	63	passing	pass	VERB
ejde-502	161	64	to	to	ADP
ejde-502	161	65	the	the	DET
ejde-502	161	66	limit	limit	NOUN
ejde-502	161	67	u	u	NOUN
ejde-502	161	68	≥	≥	NOUN
ejde-502	161	69	u	u	NOUN
ejde-502	161	70	in	in	ADP
ejde-502	161	71	rn	rn	PROPN
ejde-502	161	72	×	×	PROPN
ejde-502	161	73	(	(	PUNCT
ejde-502	161	74	0	0	NUM
ejde-502	161	75	,	,	PUNCT
ejde-502	161	76	t∞	t∞	NUM
ejde-502	161	77	)	)	PUNCT
ejde-502	161	78	,	,	PUNCT
ejde-502	161	79	proving	prove	VERB
ejde-502	161	80	the	the	DET
ejde-502	161	81	minimality	minimality	NOUN
ejde-502	161	82	of	of	ADP
ejde-502	161	83	u.	u.	PROPN
ejde-502	161	84	step	step	NOUN
ejde-502	161	85	2	2	NUM
ejde-502	161	86	.	.	PUNCT
ejde-502	161	87	supersolutions	supersolution	NOUN
ejde-502	161	88	in	in	ADP
ejde-502	161	89	dimensions	dimension	NOUN
ejde-502	161	90	n	n	NOUN
ejde-502	161	91	=	=	SYM
ejde-502	161	92	1	1	X
ejde-502	161	93	.	.	PUNCT
ejde-502	162	1	we	we	PRON
ejde-502	162	2	are	be	AUX
ejde-502	162	3	left	leave	VERB
ejde-502	162	4	with	with	ADP
ejde-502	162	5	the	the	DET
ejde-502	162	6	task	task	NOUN
ejde-502	162	7	of	of	ADP
ejde-502	162	8	obtaining	obtain	VERB
ejde-502	162	9	a	a	DET
ejde-502	162	10	uniform	uniform	NOUN
ejde-502	162	11	bound	bind	VERB
ejde-502	162	12	from	from	ADP
ejde-502	162	13	above	above	ADV
ejde-502	162	14	for	for	ADP
ejde-502	162	15	all	all	DET
ejde-502	162	16	solutions	solution	NOUN
ejde-502	162	17	wk	wk	INTJ
ejde-502	162	18	to	to	ADP
ejde-502	162	19	the	the	DET
ejde-502	162	20	cauchy	cauchy	ADJ
ejde-502	162	21	problems	problem	NOUN
ejde-502	162	22	(	(	PUNCT
ejde-502	162	23	2.1	2.1	NUM
ejde-502	162	24	)	)	PUNCT
ejde-502	162	25	,	,	PUNCT
ejde-502	163	1	k	k	PROPN
ejde-502	163	2	≥	≥	NUM
ejde-502	163	3	1	1	NUM
ejde-502	163	4	,	,	PUNCT
ejde-502	163	5	at	at	ADP
ejde-502	163	6	least	least	ADJ
ejde-502	163	7	up	up	ADP
ejde-502	163	8	to	to	ADP
ejde-502	163	9	some	some	DET
ejde-502	163	10	(	(	PUNCT
ejde-502	163	11	short	short	ADJ
ejde-502	163	12	)	)	PUNCT
ejde-502	163	13	finite	finite	ADJ
ejde-502	163	14	time	time	NOUN
ejde-502	163	15	.	.	PUNCT
ejde-502	164	1	this	this	PRON
ejde-502	164	2	follows	follow	VERB
ejde-502	164	3	by	by	ADP
ejde-502	164	4	comparison	comparison	NOUN
ejde-502	164	5	with	with	ADP
ejde-502	164	6	suitable	suitable	ADJ
ejde-502	164	7	supersolutions	supersolution	NOUN
ejde-502	164	8	in	in	ADP
ejde-502	164	9	self	self	NOUN
ejde-502	164	10	-	-	PUNCT
ejde-502	164	11	similar	similar	ADJ
ejde-502	164	12	form	form	NOUN
ejde-502	164	13	constructed	construct	VERB
ejde-502	164	14	in	in	ADP
ejde-502	164	15	recent	recent	ADJ
ejde-502	164	16	works	work	NOUN
ejde-502	164	17	by	by	ADP
ejde-502	164	18	the	the	DET
ejde-502	164	19	authors	author	NOUN
ejde-502	164	20	such	such	ADJ
ejde-502	164	21	as	as	ADP
ejde-502	164	22	[	[	X
ejde-502	164	23	16	16	NUM
ejde-502	164	24	]	]	PUNCT
ejde-502	164	25	for	for	ADP
ejde-502	164	26	n	n	PROPN
ejde-502	164	27	≥	≥	NUM
ejde-502	164	28	2	2	NUM
ejde-502	164	29	and	and	CCONJ
ejde-502	164	30	m	m	PROPN
ejde-502	164	31	+	+	NOUN
ejde-502	164	32	p	p	X
ejde-502	164	33	≥	≥	NOUN
ejde-502	164	34	2	2	NUM
ejde-502	164	35	,	,	PUNCT
ejde-502	164	36	respectively	respectively	ADV
ejde-502	164	37	[	[	X
ejde-502	164	38	19	19	NUM
ejde-502	164	39	,	,	PUNCT
ejde-502	164	40	21	21	NUM
ejde-502	164	41	]	]	PUNCT
ejde-502	164	42	in	in	ADP
ejde-502	164	43	dimension	dimension	NOUN
ejde-502	164	44	n	n	NOUN
ejde-502	164	45	=	=	SYM
ejde-502	164	46	1	1	NUM
ejde-502	164	47	and	and	CCONJ
ejde-502	164	48	either	either	DET
ejde-502	164	49	m+	m+	NOUN
ejde-502	164	50	p	p	NOUN
ejde-502	164	51	>	>	X
ejde-502	164	52	2	2	NUM
ejde-502	164	53	or	or	CCONJ
ejde-502	164	54	m+	m+	NUM
ejde-502	164	55	p	p	NOUN
ejde-502	164	56	=	=	NOUN
ejde-502	164	57	2	2	X
ejde-502	164	58	.	.	PUNCT
ejde-502	165	1	if	if	SCONJ
ejde-502	165	2	we	we	PRON
ejde-502	165	3	restrict	restrict	VERB
ejde-502	165	4	ourselves	ourselves	PRON
ejde-502	165	5	only	only	ADV
ejde-502	165	6	to	to	PART
ejde-502	165	7	dimension	dimension	VERB
ejde-502	165	8	n	n	NOUN
ejde-502	165	9	=	=	SYM
ejde-502	165	10	1	1	NUM
ejde-502	165	11	for	for	ADP
ejde-502	165	12	this	this	DET
ejde-502	165	13	step	step	NOUN
ejde-502	165	14	,	,	PUNCT
ejde-502	165	15	it	it	PRON
ejde-502	165	16	is	be	AUX
ejde-502	165	17	shown	show	VERB
ejde-502	165	18	in	in	ADP
ejde-502	165	19	the	the	DET
ejde-502	165	20	previously	previously	ADV
ejde-502	165	21	quoted	quote	VERB
ejde-502	165	22	works	work	NOUN
ejde-502	165	23	that	that	PRON
ejde-502	165	24	,	,	PUNCT
ejde-502	165	25	in	in	ADP
ejde-502	165	26	our	our	PRON
ejde-502	165	27	range	range	NOUN
ejde-502	165	28	of	of	ADP
ejde-502	165	29	exponents	exponent	NOUN
ejde-502	165	30	together	together	ADV
ejde-502	165	31	with	with	ADP
ejde-502	165	32	the	the	DET
ejde-502	165	33	extra	extra	ADJ
ejde-502	165	34	condition	condition	NOUN
ejde-502	165	35	σ	σ	X
ejde-502	165	36	>	>	X
ejde-502	165	37	2(1−p)/(m−1	2(1−p)/(m−1	NUM
ejde-502	165	38	)	)	PUNCT
ejde-502	165	39	,	,	PUNCT
ejde-502	165	40	there	there	PRON
ejde-502	165	41	are	be	VERB
ejde-502	165	42	radially	radially	ADV
ejde-502	165	43	symmetric	symmetric	ADJ
ejde-502	165	44	blow	blow	NOUN
ejde-502	165	45	-	-	PUNCT
ejde-502	165	46	up	up	ADP
ejde-502	165	47	self	self	NOUN
ejde-502	165	48	-	-	PUNCT
ejde-502	165	49	similar	similar	ADJ
ejde-502	165	50	supersolutions	supersolution	NOUN
ejde-502	165	51	to	to	ADP
ejde-502	165	52	(	(	PUNCT
ejde-502	165	53	1.7	1.7	NUM
ejde-502	165	54	)	)	PUNCT
ejde-502	165	55	in	in	ADP
ejde-502	165	56	the	the	DET
ejde-502	165	57	form	form	NOUN
ejde-502	165	58	u(x	u(x	NOUN
ejde-502	165	59	,	,	PUNCT
ejde-502	165	60	t	t	NOUN
ejde-502	165	61	)	)	PUNCT
ejde-502	165	62	=	=	PUNCT
ejde-502	166	1	(	(	PUNCT
ejde-502	166	2	t	t	PROPN
ejde-502	166	3	−	−	PROPN
ejde-502	166	4	t)−αf(|x|(t	t)−αf(|x|(t	PROPN
ejde-502	166	5	−	−	PROPN
ejde-502	166	6	t)β	t)β	NOUN
ejde-502	166	7	)	)	PUNCT
ejde-502	166	8	,	,	PUNCT
ejde-502	167	1	α	α	X
ejde-502	167	2	=	=	PUNCT
ejde-502	167	3	σ	σ	PROPN
ejde-502	168	1	+	+	NUM
ejde-502	168	2	2	2	NUM
ejde-502	168	3	l	l	NOUN
ejde-502	168	4	,	,	PUNCT
ejde-502	168	5	β	β	X
ejde-502	168	6	=	=	PUNCT
ejde-502	168	7	m−	m−	PROPN
ejde-502	168	8	p	p	PROPN
ejde-502	168	9	l	l	NOUN
ejde-502	168	10	,	,	PUNCT
ejde-502	168	11	(	(	PUNCT
ejde-502	168	12	2.2	2.2	NUM
ejde-502	168	13	)	)	PUNCT
ejde-502	168	14	where	where	SCONJ
ejde-502	168	15	l	l	NOUN
ejde-502	168	16	>	>	X
ejde-502	168	17	0	0	NUM
ejde-502	168	18	has	have	AUX
ejde-502	168	19	been	be	AUX
ejde-502	168	20	defined	define	VERB
ejde-502	168	21	in	in	ADP
ejde-502	168	22	(	(	PUNCT
ejde-502	168	23	1.8	1.8	NUM
ejde-502	168	24	)	)	PUNCT
ejde-502	168	25	,	,	PUNCT
ejde-502	168	26	such	such	ADJ
ejde-502	168	27	that	that	SCONJ
ejde-502	168	28	their	their	PRON
ejde-502	168	29	self	self	NOUN
ejde-502	168	30	-	-	PUNCT
ejde-502	168	31	similar	similar	ADJ
ejde-502	168	32	profiles	profile	NOUN
ejde-502	168	33	f	f	X
ejde-502	168	34	solve	solve	VERB
ejde-502	168	35	the	the	DET
ejde-502	168	36	differential	differential	ADJ
ejde-502	168	37	equation	equation	NOUN
ejde-502	168	38	(	(	PUNCT
ejde-502	168	39	fm)′′(ξ)−	fm)′′(ξ)−	PROPN
ejde-502	168	40	αf(ξ	αf(ξ	NUM
ejde-502	168	41	)	)	PUNCT
ejde-502	168	42	+	+	CCONJ
ejde-502	168	43	βξf	βξf	PRON
ejde-502	168	44	′(ξ	′(ξ	NOUN
ejde-502	168	45	)	)	PUNCT
ejde-502	168	46	+	+	CCONJ
ejde-502	168	47	ξσf(ξ)p	ξσf(ξ)p	VERB
ejde-502	168	48	=	=	SYM
ejde-502	168	49	0	0	NUM
ejde-502	168	50	,	,	PUNCT
ejde-502	168	51	ξ	ξ	X
ejde-502	168	52	=	=	SYM
ejde-502	168	53	|x|σ(t	|x|σ(t	PROPN
ejde-502	168	54	−	−	PROPN
ejde-502	168	55	t)β	t)β	NOUN
ejde-502	168	56	.	.	PUNCT
ejde-502	169	1	(	(	PUNCT
ejde-502	169	2	2.3	2.3	NUM
ejde-502	169	3	)	)	PUNCT
ejde-502	169	4	moreover	moreover	ADV
ejde-502	169	5	,	,	PUNCT
ejde-502	169	6	it	it	PRON
ejde-502	169	7	is	be	AUX
ejde-502	169	8	established	establish	VERB
ejde-502	169	9	in	in	ADP
ejde-502	169	10	[	[	X
ejde-502	169	11	19	19	NUM
ejde-502	169	12	,	,	PUNCT
ejde-502	169	13	proposition	proposition	NOUN
ejde-502	169	14	4.1	4.1	NUM
ejde-502	169	15	]	]	PUNCT
ejde-502	169	16	if	if	SCONJ
ejde-502	169	17	m+	m+	NOUN
ejde-502	169	18	p	p	X
ejde-502	169	19	>	>	X
ejde-502	169	20	2	2	NUM
ejde-502	169	21	and	and	CCONJ
ejde-502	169	22	in	in	ADP
ejde-502	169	23	[	[	X
ejde-502	169	24	21	21	NUM
ejde-502	169	25	,	,	PUNCT
ejde-502	169	26	proposition	proposition	NOUN
ejde-502	169	27	3.1	3.1	NUM
ejde-502	169	28	]	]	PUNCT
ejde-502	169	29	if	if	SCONJ
ejde-502	169	30	m	m	VERB
ejde-502	169	31	+	+	NOUN
ejde-502	169	32	p	p	X
ejde-502	169	33	=	=	ADJ
ejde-502	169	34	2	2	NUM
ejde-502	169	35	that	that	PRON
ejde-502	169	36	the	the	DET
ejde-502	169	37	self	self	NOUN
ejde-502	169	38	-	-	PUNCT
ejde-502	169	39	similar	similar	ADJ
ejde-502	169	40	profiles	profile	NOUN
ejde-502	169	41	f	f	X
ejde-502	169	42	of	of	ADP
ejde-502	169	43	the	the	DET
ejde-502	169	44	supersolutions	supersolution	NOUN
ejde-502	169	45	in	in	ADP
ejde-502	169	46	the	the	DET
ejde-502	169	47	form	form	NOUN
ejde-502	169	48	(	(	PUNCT
ejde-502	169	49	2.2	2.2	NUM
ejde-502	169	50	)	)	PUNCT
ejde-502	169	51	fulfill	fulfill	VERB
ejde-502	169	52	the	the	DET
ejde-502	169	53	following	follow	VERB
ejde-502	169	54	two	two	NUM
ejde-502	169	55	additional	additional	ADJ
ejde-502	169	56	properties	property	NOUN
ejde-502	169	57	:	:	PUNCT
ejde-502	169	58	•	•	NUM
ejde-502	169	59	f	f	NOUN
ejde-502	169	60	is	be	AUX
ejde-502	169	61	strictly	strictly	ADV
ejde-502	169	62	decreasing	decrease	VERB
ejde-502	169	63	until	until	ADP
ejde-502	169	64	reaching	reach	VERB
ejde-502	169	65	the	the	DET
ejde-502	169	66	zero	zero	NUM
ejde-502	169	67	level	level	NOUN
ejde-502	169	68	:	:	PUNCT
ejde-502	169	69	f(0	f(0	NOUN
ejde-502	169	70	)	)	PUNCT
ejde-502	169	71	=	=	SYM
ejde-502	169	72	a	a	DET
ejde-502	169	73	>	>	X
ejde-502	169	74	0	0	NUM
ejde-502	169	75	,	,	PUNCT
ejde-502	169	76	f	f	PROPN
ejde-502	169	77	′(ξ	′(ξ	PROPN
ejde-502	169	78	)	)	PUNCT
ejde-502	169	79	<	<	X
ejde-502	169	80	0	0	PUNCT
ejde-502	169	81	at	at	ADP
ejde-502	169	82	points	point	NOUN
ejde-502	169	83	ξ	ξ	PROPN
ejde-502	169	84	≥	≥	NOUN
ejde-502	169	85	0	0	NUM
ejde-502	169	86	where	where	SCONJ
ejde-502	169	87	f(ξ	f(ξ	NOUN
ejde-502	169	88	)	)	PUNCT
ejde-502	169	89	>	>	X
ejde-502	170	1	0	0	X
ejde-502	170	2	.	.	X
ejde-502	170	3	•	•	NUM
ejde-502	170	4	f	f	PROPN
ejde-502	170	5	presents	present	VERB
ejde-502	170	6	an	an	DET
ejde-502	170	7	interface	interface	NOUN
ejde-502	170	8	at	at	ADP
ejde-502	170	9	some	some	DET
ejde-502	170	10	finite	finite	ADJ
ejde-502	170	11	point	point	NOUN
ejde-502	170	12	ξ0	ξ0	PROPN
ejde-502	170	13	∈	∈	PROPN
ejde-502	170	14	(	(	PUNCT
ejde-502	170	15	0,∞	0,∞	NOUN
ejde-502	170	16	)	)	PUNCT
ejde-502	170	17	,	,	PUNCT
ejde-502	170	18	that	that	ADV
ejde-502	170	19	is	is	ADV
ejde-502	170	20	,	,	PUNCT
ejde-502	170	21	f(ξ0	f(ξ0	ADJ
ejde-502	170	22	)	)	PUNCT
ejde-502	170	23	=	=	SYM
ejde-502	170	24	0	0	NUM
ejde-502	170	25	,	,	PUNCT
ejde-502	170	26	f(ξ	f(ξ	NOUN
ejde-502	170	27	)	)	PUNCT
ejde-502	170	28	>	>	X
ejde-502	170	29	0	0	NUM
ejde-502	171	1	for	for	ADP
ejde-502	171	2	any	any	DET
ejde-502	171	3	ξ	ξ	PROPN
ejde-502	171	4	∈	∈	PROPN
ejde-502	171	5	(	(	PUNCT
ejde-502	171	6	0	0	NUM
ejde-502	171	7	,	,	PUNCT
ejde-502	171	8	ξ0	ξ0	NOUN
ejde-502	171	9	)	)	PUNCT
ejde-502	171	10	and	and	CCONJ
ejde-502	171	11	(	(	PUNCT
ejde-502	171	12	fm)′(ξ0	fm)′(ξ0	NOUN
ejde-502	171	13	)	)	PUNCT
ejde-502	171	14	=	=	SYM
ejde-502	172	1	0	0	X
ejde-502	172	2	.	.	PUNCT
ejde-502	173	1	here	here	ADV
ejde-502	173	2	the	the	DET
ejde-502	173	3	blow	blow	NOUN
ejde-502	173	4	-	-	PUNCT
ejde-502	173	5	up	up	ADP
ejde-502	173	6	time	time	NOUN
ejde-502	173	7	t	t	PROPN
ejde-502	173	8	>	>	X
ejde-502	173	9	0	0	PUNCT
ejde-502	173	10	is	be	AUX
ejde-502	173	11	a	a	DET
ejde-502	173	12	free	free	ADJ
ejde-502	173	13	parameter	parameter	NOUN
ejde-502	173	14	and	and	CCONJ
ejde-502	173	15	the	the	DET
ejde-502	173	16	functions	function	NOUN
ejde-502	173	17	defined	define	VERB
ejde-502	173	18	in	in	ADP
ejde-502	173	19	(	(	PUNCT
ejde-502	173	20	2.2	2.2	NUM
ejde-502	173	21	)	)	PUNCT
ejde-502	173	22	are	be	AUX
ejde-502	173	23	actually	actually	ADV
ejde-502	173	24	weak	weak	ADJ
ejde-502	173	25	solutions	solution	NOUN
ejde-502	173	26	to	to	ADP
ejde-502	173	27	(	(	PUNCT
ejde-502	173	28	1.7	1.7	NUM
ejde-502	173	29	)	)	PUNCT
ejde-502	173	30	except	except	SCONJ
ejde-502	173	31	at	at	ADP
ejde-502	173	32	the	the	DET
ejde-502	173	33	point	point	NOUN
ejde-502	173	34	x	x	PUNCT
ejde-502	173	35	=	=	SYM
ejde-502	173	36	0	0	NUM
ejde-502	173	37	where	where	SCONJ
ejde-502	173	38	the	the	DET
ejde-502	173	39	condition	condition	NOUN
ejde-502	173	40	(	(	PUNCT
ejde-502	173	41	fm)′(0	fm)′(0	PROPN
ejde-502	173	42	)	)	PUNCT
ejde-502	173	43	=	=	SYM
ejde-502	173	44	0	0	NUM
ejde-502	173	45	is	be	AUX
ejde-502	173	46	not	not	PART
ejde-502	173	47	fulfilled	fulfil	VERB
ejde-502	173	48	in	in	ADP
ejde-502	173	49	order	order	NOUN
ejde-502	173	50	to	to	PART
ejde-502	173	51	be	be	AUX
ejde-502	173	52	a	a	DET
ejde-502	173	53	weak	weak	ADJ
ejde-502	173	54	solution	solution	NOUN
ejde-502	173	55	.	.	PUNCT
ejde-502	174	1	we	we	PRON
ejde-502	174	2	adapt	adapt	VERB
ejde-502	174	3	these	these	DET
ejde-502	174	4	supersolutions	supersolution	NOUN
ejde-502	174	5	to	to	ADP
ejde-502	174	6	our	our	PRON
ejde-502	174	7	equation	equation	NOUN
ejde-502	174	8	(	(	PUNCT
ejde-502	174	9	1.1	1.1	NUM
ejde-502	174	10	)	)	PUNCT
ejde-502	174	11	by	by	ADP
ejde-502	174	12	defining	define	VERB
ejde-502	174	13	z(x	z(x	PROPN
ejde-502	174	14	,	,	PUNCT
ejde-502	174	15	t	t	PROPN
ejde-502	174	16	)	)	PUNCT
ejde-502	174	17	=	=	PUNCT
ejde-502	175	1	(	(	PUNCT
ejde-502	175	2	t	t	PROPN
ejde-502	175	3	−	−	PROPN
ejde-502	175	4	t)−αf((1	t)−αf((1	PROPN
ejde-502	175	5	+	+	CCONJ
ejde-502	175	6	|x|)(t	|x|)(t	ADJ
ejde-502	175	7	−	−	NOUN
ejde-502	175	8	t)β	t)β	NOUN
ejde-502	175	9	)	)	PUNCT
ejde-502	175	10	,	,	PUNCT
ejde-502	175	11	(	(	PUNCT
ejde-502	175	12	2.4	2.4	NUM
ejde-502	175	13	)	)	PUNCT
ejde-502	175	14	with	with	ADP
ejde-502	175	15	α	α	PROPN
ejde-502	175	16	,	,	PUNCT
ejde-502	175	17	β	β	X
ejde-502	175	18	and	and	CCONJ
ejde-502	175	19	f	f	PROPN
ejde-502	175	20	as	as	ADP
ejde-502	175	21	in	in	ADP
ejde-502	175	22	(	(	PUNCT
ejde-502	175	23	2.2	2.2	NUM
ejde-502	175	24	)	)	PUNCT
ejde-502	175	25	.	.	PUNCT
ejde-502	176	1	since	since	SCONJ
ejde-502	176	2	f	f	PROPN
ejde-502	176	3	is	be	AUX
ejde-502	176	4	a	a	DET
ejde-502	176	5	supersolution	supersolution	NOUN
ejde-502	176	6	to	to	ADP
ejde-502	176	7	the	the	DET
ejde-502	176	8	differential	differential	ADJ
ejde-502	176	9	equation	equation	NOUN
ejde-502	176	10	(	(	PUNCT
ejde-502	176	11	2.3	2.3	NUM
ejde-502	176	12	)	)	PUNCT
ejde-502	176	13	,	,	PUNCT
ejde-502	176	14	it	it	PRON
ejde-502	176	15	is	be	AUX
ejde-502	176	16	straightforward	straightforward	ADJ
ejde-502	176	17	to	to	PART
ejde-502	176	18	check	check	VERB
ejde-502	176	19	that	that	SCONJ
ejde-502	176	20	z	z	NOUN
ejde-502	176	21	is	be	AUX
ejde-502	176	22	a	a	DET
ejde-502	176	23	supersolution	supersolution	NOUN
ejde-502	176	24	to	to	ADP
ejde-502	176	25	(	(	PUNCT
ejde-502	176	26	1.1	1.1	NUM
ejde-502	176	27	)	)	PUNCT
ejde-502	176	28	.	.	PUNCT
ejde-502	177	1	the	the	DET
ejde-502	177	2	amplitude	amplitude	NOUN
ejde-502	177	3	s(t	s(t	PROPN
ejde-502	177	4	)	)	PUNCT
ejde-502	177	5	of	of	ADP
ejde-502	177	6	the	the	DET
ejde-502	177	7	support	support	NOUN
ejde-502	177	8	of	of	ADP
ejde-502	177	9	z(t	z(t	NOUN
ejde-502	177	10	)	)	PUNCT
ejde-502	177	11	at	at	ADP
ejde-502	177	12	some	some	DET
ejde-502	177	13	t	t	NOUN
ejde-502	177	14	∈	∈	PROPN
ejde-502	177	15	(	(	PUNCT
ejde-502	177	16	0	0	NUM
ejde-502	177	17	,	,	PUNCT
ejde-502	177	18	t	t	PROPN
ejde-502	177	19	)	)	PUNCT
ejde-502	177	20	is	be	AUX
ejde-502	177	21	given	give	VERB
ejde-502	177	22	by	by	ADP
ejde-502	177	23	ξ0	ξ0	PROPN
ejde-502	177	24	=	=	SYM
ejde-502	177	25	(	(	PUNCT
ejde-502	177	26	1	1	NUM
ejde-502	177	27	+	+	CCONJ
ejde-502	177	28	s(t))(t	s(t))(t	ADJ
ejde-502	177	29	−	−	NOUN
ejde-502	177	30	t)β	t)β	NOUN
ejde-502	177	31	or	or	CCONJ
ejde-502	177	32	equivalently	equivalently	ADV
ejde-502	177	33	s(t	s(t	PROPN
ejde-502	177	34	)	)	PUNCT
ejde-502	177	35	=	=	PUNCT
ejde-502	178	1	(	(	PUNCT
ejde-502	178	2	t	t	PROPN
ejde-502	178	3	−	−	PROPN
ejde-502	178	4	t)−βξ0	t)−βξ0	NOUN
ejde-502	178	5	−	−	PROPN
ejde-502	178	6	1→∞	1→∞	NUM
ejde-502	178	7	,	,	PUNCT
ejde-502	178	8	as	as	ADP
ejde-502	178	9	t→	t→	PRON
ejde-502	178	10	t.	t.	X
ejde-502	178	11	moreover	moreover	ADV
ejde-502	178	12	,	,	PUNCT
ejde-502	178	13	the	the	DET
ejde-502	178	14	above	above	ADJ
ejde-502	178	15	supersolutions	supersolution	NOUN
ejde-502	178	16	have	have	AUX
ejde-502	178	17	been	be	AUX
ejde-502	178	18	established	establish	VERB
ejde-502	178	19	in	in	ADP
ejde-502	178	20	[	[	X
ejde-502	178	21	19	19	NUM
ejde-502	178	22	,	,	PUNCT
ejde-502	178	23	21	21	NUM
ejde-502	178	24	]	]	PUNCT
ejde-502	178	25	only	only	ADV
ejde-502	178	26	under	under	ADP
ejde-502	178	27	the	the	DET
ejde-502	178	28	condition	condition	NOUN
ejde-502	178	29	σ	σ	X
ejde-502	178	30	>	>	X
ejde-502	178	31	2(1	2(1	NUM
ejde-502	179	1	−	−	PUNCT
ejde-502	179	2	p)/(m	p)/(m	INTJ
ejde-502	179	3	−	−	NOUN
ejde-502	179	4	1	1	NUM
ejde-502	179	5	)	)	PUNCT
ejde-502	179	6	.	.	PUNCT
ejde-502	180	1	however	however	ADV
ejde-502	180	2	,	,	PUNCT
ejde-502	180	3	if	if	SCONJ
ejde-502	180	4	0	0	NUM
ejde-502	180	5	<	<	X
ejde-502	180	6	σ	σ	X
ejde-502	180	7	≤	≤	NOUN
ejde-502	180	8	2(1	2(1	NUM
ejde-502	180	9	−	−	NOUN
ejde-502	181	1	p)/(m	p)/(m	DET
ejde-502	181	2	−	−	NOUN
ejde-502	181	3	1	1	NUM
ejde-502	181	4	)	)	PUNCT
ejde-502	181	5	,	,	PUNCT
ejde-502	181	6	it	it	PRON
ejde-502	181	7	is	be	AUX
ejde-502	181	8	obvious	obvious	ADJ
ejde-502	181	9	from	from	ADP
ejde-502	181	10	the	the	DET
ejde-502	181	11	fact	fact	NOUN
ejde-502	181	12	that	that	SCONJ
ejde-502	181	13	1	1	X
ejde-502	181	14	+	+	NUM
ejde-502	181	15	|x|	|x|	PROPN
ejde-502	181	16	≥	≥	NUM
ejde-502	181	17	1	1	NUM
ejde-502	181	18	,	,	PUNCT
ejde-502	181	19	that	that	SCONJ
ejde-502	181	20	(	(	PUNCT
ejde-502	181	21	1	1	NUM
ejde-502	181	22	+	+	CCONJ
ejde-502	181	23	|x|)σ	|x|)σ	PROPN
ejde-502	181	24	≤	≤	X
ejde-502	181	25	(	(	PUNCT
ejde-502	181	26	1	1	NUM
ejde-502	181	27	+	+	CCONJ
ejde-502	181	28	|x|)σ1	|x|)σ1	PUNCT
ejde-502	181	29	for	for	ADP
ejde-502	181	30	any	any	DET
ejde-502	181	31	σ1	σ1	PROPN
ejde-502	181	32	>	>	X
ejde-502	181	33	2(1	2(1	NUM
ejde-502	181	34	−	−	NOUN
ejde-502	182	1	p)/(m	p)/(m	INTJ
ejde-502	182	2	−	−	NOUN
ejde-502	182	3	1	1	NUM
ejde-502	182	4	)	)	PUNCT
ejde-502	182	5	and	and	CCONJ
ejde-502	182	6	for	for	ADP
ejde-502	182	7	any	any	DET
ejde-502	182	8	x	x	SYM
ejde-502	182	9	∈	∈	PROPN
ejde-502	182	10	r.	r.	NOUN
ejde-502	182	11	it	it	PRON
ejde-502	182	12	thus	thus	ADV
ejde-502	182	13	follows	follow	VERB
ejde-502	182	14	that	that	SCONJ
ejde-502	182	15	the	the	DET
ejde-502	182	16	supersolutions	supersolution	NOUN
ejde-502	182	17	constructed	construct	VERB
ejde-502	182	18	for	for	ADP
ejde-502	182	19	(	(	PUNCT
ejde-502	182	20	1.1	1.1	NUM
ejde-502	182	21	)	)	PUNCT
ejde-502	182	22	with	with	ADP
ejde-502	182	23	such	such	DET
ejde-502	182	24	an	an	DET
ejde-502	182	25	exponent	exponent	NOUN
ejde-502	182	26	σ1	σ1	PROPN
ejde-502	182	27	>	>	X
ejde-502	182	28	2(1	2(1	NUM
ejde-502	182	29	−	−	NOUN
ejde-502	183	1	p)/(m	p)/(m	INTJ
ejde-502	183	2	−	−	NOUN
ejde-502	183	3	1	1	NUM
ejde-502	183	4	)	)	PUNCT
ejde-502	183	5	as	as	ADP
ejde-502	183	6	above	above	ADV
ejde-502	183	7	,	,	PUNCT
ejde-502	183	8	will	will	AUX
ejde-502	183	9	serve	serve	VERB
ejde-502	183	10	also	also	ADV
ejde-502	183	11	as	as	ADP
ejde-502	183	12	supersolutions	supersolution	NOUN
ejde-502	183	13	to	to	ADP
ejde-502	183	14	(	(	PUNCT
ejde-502	183	15	1.1	1.1	NUM
ejde-502	183	16	)	)	PUNCT
ejde-502	183	17	with	with	ADP
ejde-502	183	18	exponents	exponent	NOUN
ejde-502	183	19	σ	σ	X
ejde-502	183	20	smaller	small	ADJ
ejde-502	183	21	.	.	PUNCT
ejde-502	184	1	since	since	SCONJ
ejde-502	184	2	t	t	PROPN
ejde-502	184	3	is	be	AUX
ejde-502	184	4	a	a	DET
ejde-502	184	5	free	free	ADJ
ejde-502	184	6	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	184	7	reaction	reaction	NOUN
ejde-502	184	8	-	-	PUNCT
ejde-502	184	9	diffusion	diffusion	NOUN
ejde-502	184	10	with	with	ADP
ejde-502	184	11	weighted	weight	VERB
ejde-502	184	12	strong	strong	ADJ
ejde-502	184	13	reaction	reaction	NOUN
ejde-502	184	14	9	9	NUM
ejde-502	184	15	parameter	parameter	NOUN
ejde-502	184	16	and	and	CCONJ
ejde-502	184	17	u0	u0	PROPN
ejde-502	184	18	is	be	AUX
ejde-502	184	19	bounded	bound	VERB
ejde-502	184	20	and	and	CCONJ
ejde-502	184	21	compactly	compactly	ADV
ejde-502	184	22	supported	support	VERB
ejde-502	184	23	,	,	PUNCT
ejde-502	184	24	we	we	PRON
ejde-502	184	25	can	can	AUX
ejde-502	184	26	choose	choose	VERB
ejde-502	184	27	some	some	DET
ejde-502	184	28	t0	t0	PROPN
ejde-502	184	29	>	>	X
ejde-502	184	30	0	0	PUNCT
ejde-502	185	1	sufficiently	sufficiently	ADV
ejde-502	185	2	small	small	ADJ
ejde-502	185	3	such	such	ADJ
ejde-502	185	4	that	that	SCONJ
ejde-502	185	5	z(x	z(x	NUM
ejde-502	185	6	,	,	PUNCT
ejde-502	185	7	0	0	NUM
ejde-502	185	8	)	)	PUNCT
ejde-502	185	9	=	=	NOUN
ejde-502	185	10	t−α0	t−α0	X
ejde-502	185	11	f((1	f((1	PROPN
ejde-502	185	12	+	+	CCONJ
ejde-502	185	13	|x|)t	|x|)t	PROPN
ejde-502	185	14	β0	β0	PROPN
ejde-502	185	15	)	)	PUNCT
ejde-502	185	16	≥	≥	PROPN
ejde-502	185	17	‖u0‖∞	‖u0‖∞	PROPN
ejde-502	185	18	≥	≥	X
ejde-502	185	19	u0(x	u0(x	NUM
ejde-502	185	20	)	)	PUNCT
ejde-502	185	21	(	(	PUNCT
ejde-502	185	22	2.5	2.5	NUM
ejde-502	185	23	)	)	PUNCT
ejde-502	185	24	for	for	ADP
ejde-502	185	25	any	any	DET
ejde-502	185	26	x	x	SYM
ejde-502	185	27	∈	∈	PROPN
ejde-502	185	28	suppu0	suppu0	NOUN
ejde-502	185	29	.	.	PUNCT
ejde-502	186	1	this	this	PRON
ejde-502	186	2	,	,	PUNCT
ejde-502	186	3	together	together	ADV
ejde-502	186	4	with	with	ADP
ejde-502	186	5	the	the	DET
ejde-502	186	6	fact	fact	NOUN
ejde-502	186	7	that	that	SCONJ
ejde-502	186	8	any	any	DET
ejde-502	186	9	function	function	NOUN
ejde-502	186	10	z	z	NOUN
ejde-502	186	11	as	as	ADP
ejde-502	186	12	in	in	ADP
ejde-502	186	13	(	(	PUNCT
ejde-502	186	14	2.4	2.4	NUM
ejde-502	186	15	)	)	PUNCT
ejde-502	186	16	is	be	AUX
ejde-502	186	17	a	a	DET
ejde-502	186	18	supersolution	supersolution	NOUN
ejde-502	186	19	to	to	ADP
ejde-502	186	20	(	(	PUNCT
ejde-502	186	21	1.1	1.1	NUM
ejde-502	186	22	)	)	PUNCT
ejde-502	186	23	,	,	PUNCT
ejde-502	186	24	gives	give	VERB
ejde-502	186	25	that	that	PRON
ejde-502	186	26	z	z	NOUN
ejde-502	186	27	is	be	AUX
ejde-502	186	28	also	also	ADV
ejde-502	186	29	a	a	DET
ejde-502	186	30	supersolution	supersolution	NOUN
ejde-502	186	31	to	to	ADP
ejde-502	186	32	the	the	DET
ejde-502	186	33	cauchy	cauchy	ADJ
ejde-502	186	34	problem	problem	NOUN
ejde-502	186	35	(	(	PUNCT
ejde-502	186	36	2.1	2.1	NUM
ejde-502	186	37	)	)	PUNCT
ejde-502	186	38	for	for	ADP
ejde-502	186	39	any	any	DET
ejde-502	186	40	k	k	PROPN
ejde-502	186	41	≥	≥	NUM
ejde-502	186	42	1	1	NUM
ejde-502	186	43	for	for	ADP
ejde-502	186	44	which	which	PRON
ejde-502	186	45	the	the	DET
ejde-502	186	46	comparison	comparison	NOUN
ejde-502	186	47	principle	principle	NOUN
ejde-502	186	48	applies	apply	VERB
ejde-502	186	49	to	to	PART
ejde-502	186	50	give	give	VERB
ejde-502	186	51	that	that	PRON
ejde-502	186	52	wk(x	wk(x	NOUN
ejde-502	186	53	,	,	PUNCT
ejde-502	186	54	t	t	PROPN
ejde-502	186	55	)	)	PUNCT
ejde-502	186	56	≤	≤	NOUN
ejde-502	186	57	z(x	z(x	NUM
ejde-502	186	58	,	,	PUNCT
ejde-502	186	59	t	t	PROPN
ejde-502	186	60	)	)	PUNCT
ejde-502	186	61	,	,	PUNCT
ejde-502	186	62	for	for	ADP
ejde-502	186	63	all	all	DET
ejde-502	186	64	(	(	PUNCT
ejde-502	186	65	x	x	NOUN
ejde-502	186	66	,	,	PUNCT
ejde-502	186	67	t	t	PROPN
ejde-502	186	68	)	)	PUNCT
ejde-502	186	69	∈	∈	PROPN
ejde-502	186	70	r×	r×	NOUN
ejde-502	186	71	(	(	PUNCT
ejde-502	186	72	0	0	NUM
ejde-502	186	73	,	,	PUNCT
ejde-502	186	74	t0	t0	NOUN
ejde-502	186	75	)	)	PUNCT
ejde-502	186	76	.	.	PUNCT
ejde-502	187	1	in	in	ADP
ejde-502	187	2	particular	particular	ADJ
ejde-502	187	3	we	we	PRON
ejde-502	187	4	infer	infer	VERB
ejde-502	187	5	on	on	ADP
ejde-502	187	6	the	the	DET
ejde-502	187	7	one	one	NUM
ejde-502	187	8	hand	hand	NOUN
ejde-502	187	9	that	that	PRON
ejde-502	187	10	t∞	t∞	VERB
ejde-502	187	11	≥	≥	NUM
ejde-502	187	12	t0	t0	X
ejde-502	187	13	>	>	X
ejde-502	187	14	0	0	PUNCT
ejde-502	187	15	as	as	SCONJ
ejde-502	187	16	claimed	claim	VERB
ejde-502	187	17	,	,	PUNCT
ejde-502	187	18	and	and	CCONJ
ejde-502	187	19	on	on	ADP
ejde-502	187	20	the	the	DET
ejde-502	187	21	other	other	ADJ
ejde-502	187	22	hand	hand	NOUN
ejde-502	187	23	by	by	ADP
ejde-502	187	24	passing	pass	VERB
ejde-502	187	25	to	to	ADP
ejde-502	187	26	the	the	DET
ejde-502	187	27	limit	limit	NOUN
ejde-502	187	28	as	as	SCONJ
ejde-502	187	29	k	k	PROPN
ejde-502	187	30	→∞	→∞	PROPN
ejde-502	187	31	we	we	PRON
ejde-502	187	32	obtain	obtain	VERB
ejde-502	187	33	u(x	u(x	NOUN
ejde-502	187	34	,	,	PUNCT
ejde-502	187	35	t	t	NOUN
ejde-502	187	36	)	)	PUNCT
ejde-502	187	37	≤	≤	NOUN
ejde-502	187	38	z(x	z(x	NUM
ejde-502	187	39	,	,	PUNCT
ejde-502	187	40	t	t	PROPN
ejde-502	187	41	)	)	PUNCT
ejde-502	187	42	,	,	PUNCT
ejde-502	187	43	for	for	ADP
ejde-502	187	44	all	all	DET
ejde-502	187	45	(	(	PUNCT
ejde-502	187	46	x	x	NOUN
ejde-502	187	47	,	,	PUNCT
ejde-502	187	48	t	t	PROPN
ejde-502	187	49	)	)	PUNCT
ejde-502	187	50	∈	∈	PROPN
ejde-502	187	51	r×	r×	NOUN
ejde-502	187	52	(	(	PUNCT
ejde-502	187	53	0	0	NUM
ejde-502	187	54	,	,	PUNCT
ejde-502	187	55	t0	t0	PROPN
ejde-502	187	56	)	)	PUNCT
ejde-502	187	57	,	,	PUNCT
ejde-502	187	58	which	which	PRON
ejde-502	187	59	proves	prove	VERB
ejde-502	187	60	the	the	DET
ejde-502	187	61	finite	finite	ADJ
ejde-502	187	62	speed	speed	NOUN
ejde-502	187	63	of	of	ADP
ejde-502	187	64	propagation	propagation	NOUN
ejde-502	187	65	of	of	ADP
ejde-502	187	66	the	the	DET
ejde-502	187	67	support	support	NOUN
ejde-502	187	68	of	of	ADP
ejde-502	187	69	u	u	NOUN
ejde-502	187	70	at	at	ADP
ejde-502	187	71	least	least	ADJ
ejde-502	187	72	for	for	ADP
ejde-502	187	73	some	some	DET
ejde-502	187	74	short	short	ADJ
ejde-502	187	75	interval	interval	NOUN
ejde-502	187	76	of	of	ADP
ejde-502	187	77	time	time	NOUN
ejde-502	187	78	.	.	PUNCT
ejde-502	188	1	step	step	NOUN
ejde-502	188	2	3	3	NUM
ejde-502	188	3	.	.	PUNCT
ejde-502	188	4	supersolutions	supersolution	NOUN
ejde-502	188	5	in	in	ADP
ejde-502	188	6	dimension	dimension	NOUN
ejde-502	188	7	n	n	CCONJ
ejde-502	188	8	≥	≥	NOUN
ejde-502	188	9	2	2	NUM
ejde-502	188	10	.	.	PUNCT
ejde-502	189	1	in	in	ADP
ejde-502	189	2	this	this	DET
ejde-502	189	3	case	case	NOUN
ejde-502	189	4	,	,	PUNCT
ejde-502	189	5	there	there	PRON
ejde-502	189	6	are	be	VERB
ejde-502	189	7	no	no	ADV
ejde-502	189	8	longer	long	ADV
ejde-502	189	9	bounded	bound	VERB
ejde-502	189	10	and	and	CCONJ
ejde-502	189	11	decreasing	decrease	VERB
ejde-502	189	12	supersolutions	supersolution	NOUN
ejde-502	189	13	in	in	ADP
ejde-502	189	14	the	the	DET
ejde-502	189	15	self	self	NOUN
ejde-502	189	16	-	-	PUNCT
ejde-502	189	17	similar	similar	ADJ
ejde-502	189	18	form	form	NOUN
ejde-502	189	19	used	use	VERB
ejde-502	189	20	in	in	ADP
ejde-502	189	21	step	step	NOUN
ejde-502	189	22	2	2	NUM
ejde-502	189	23	.	.	PUNCT
ejde-502	190	1	we	we	PRON
ejde-502	190	2	thus	thus	ADV
ejde-502	190	3	construct	construct	VERB
ejde-502	190	4	suitable	suitable	ADJ
ejde-502	190	5	supersolutions	supersolution	NOUN
ejde-502	190	6	to	to	ADP
ejde-502	190	7	(	(	PUNCT
ejde-502	190	8	1.7	1.7	NUM
ejde-502	190	9	)	)	PUNCT
ejde-502	190	10	by	by	ADP
ejde-502	190	11	joining	join	VERB
ejde-502	190	12	two	two	NUM
ejde-502	190	13	different	different	ADJ
ejde-502	190	14	self	self	NOUN
ejde-502	190	15	-	-	PUNCT
ejde-502	190	16	similar	similar	ADJ
ejde-502	190	17	profiles	profile	NOUN
ejde-502	190	18	.	.	PUNCT
ejde-502	191	1	we	we	PRON
ejde-502	191	2	begin	begin	VERB
ejde-502	191	3	again	again	ADV
ejde-502	191	4	by	by	ADP
ejde-502	191	5	fixing	fix	VERB
ejde-502	191	6	σ	σ	PROPN
ejde-502	191	7	>	>	PUNCT
ejde-502	191	8	2(1	2(1	NUM
ejde-502	191	9	−	−	NOUN
ejde-502	192	1	p)/(m	p)/(m	INTJ
ejde-502	192	2	−	−	NOUN
ejde-502	192	3	1	1	NUM
ejde-502	192	4	)	)	PUNCT
ejde-502	192	5	as	as	ADP
ejde-502	192	6	a	a	DET
ejde-502	192	7	first	first	ADJ
ejde-502	192	8	case	case	NOUN
ejde-502	192	9	.	.	PUNCT
ejde-502	193	1	we	we	PRON
ejde-502	193	2	look	look	VERB
ejde-502	193	3	for	for	ADP
ejde-502	193	4	self	self	NOUN
ejde-502	193	5	-	-	PUNCT
ejde-502	193	6	similar	similar	ADJ
ejde-502	193	7	solutions	solution	NOUN
ejde-502	193	8	to	to	ADP
ejde-502	193	9	(	(	PUNCT
ejde-502	193	10	1.7	1.7	NUM
ejde-502	193	11	)	)	PUNCT
ejde-502	193	12	in	in	ADP
ejde-502	193	13	the	the	DET
ejde-502	193	14	same	same	ADJ
ejde-502	193	15	form	form	NOUN
ejde-502	193	16	(	(	PUNCT
ejde-502	193	17	2.2	2.2	NUM
ejde-502	193	18	)	)	PUNCT
ejde-502	193	19	as	as	ADP
ejde-502	193	20	above	above	ADV
ejde-502	193	21	,	,	PUNCT
ejde-502	193	22	with	with	ADP
ejde-502	193	23	the	the	DET
ejde-502	193	24	same	same	ADJ
ejde-502	193	25	exponents	exponent	NOUN
ejde-502	193	26	α	α	PROPN
ejde-502	193	27	and	and	CCONJ
ejde-502	193	28	β	β	NOUN
ejde-502	193	29	,	,	PUNCT
ejde-502	193	30	but	but	CCONJ
ejde-502	193	31	whose	whose	DET
ejde-502	193	32	profiles	profile	NOUN
ejde-502	193	33	solve	solve	VERB
ejde-502	193	34	the	the	DET
ejde-502	193	35	differential	differential	ADJ
ejde-502	193	36	equation	equation	NOUN
ejde-502	193	37	(	(	PUNCT
ejde-502	193	38	fm)′′(ξ	fm)′′(ξ	NOUN
ejde-502	193	39	)	)	PUNCT
ejde-502	193	40	+	+	CCONJ
ejde-502	193	41	n	n	CCONJ
ejde-502	193	42	−	−	PROPN
ejde-502	193	43	1	1	NUM
ejde-502	193	44	ξ	ξ	X
ejde-502	193	45	(	(	PUNCT
ejde-502	193	46	fm)′(ξ)−	fm)′(ξ)−	NOUN
ejde-502	193	47	αf(ξ	αf(ξ	PUNCT
ejde-502	193	48	)	)	PUNCT
ejde-502	193	49	+	+	CCONJ
ejde-502	193	50	βξf	βξf	PRON
ejde-502	193	51	′(ξ	′(ξ	NOUN
ejde-502	193	52	)	)	PUNCT
ejde-502	193	53	+	+	CCONJ
ejde-502	193	54	ξσf(ξ)p	ξσf(ξ)p	VERB
ejde-502	193	55	=	=	SYM
ejde-502	193	56	0	0	NUM
ejde-502	193	57	,	,	PUNCT
ejde-502	193	58	ξ	ξ	X
ejde-502	193	59	=	=	SYM
ejde-502	193	60	|x|σ(t	|x|σ(t	PROPN
ejde-502	193	61	−	−	PROPN
ejde-502	193	62	t)β	t)β	NOUN
ejde-502	193	63	.	.	PUNCT
ejde-502	194	1	(	(	PUNCT
ejde-502	194	2	2.6	2.6	NUM
ejde-502	194	3	)	)	PUNCT
ejde-502	194	4	on	on	ADP
ejde-502	194	5	the	the	DET
ejde-502	194	6	one	one	NUM
ejde-502	194	7	hand	hand	NOUN
ejde-502	194	8	,	,	PUNCT
ejde-502	194	9	the	the	DET
ejde-502	194	10	analysis	analysis	NOUN
ejde-502	194	11	in	in	ADP
ejde-502	194	12	[	[	X
ejde-502	194	13	16	16	NUM
ejde-502	194	14	,	,	PUNCT
ejde-502	194	15	proposition	proposition	NOUN
ejde-502	194	16	4.1	4.1	NUM
ejde-502	194	17	]	]	PUNCT
ejde-502	194	18	if	if	SCONJ
ejde-502	194	19	m	m	PROPN
ejde-502	194	20	+	+	NOUN
ejde-502	194	21	p	p	X
ejde-502	194	22	>	>	X
ejde-502	194	23	2	2	NUM
ejde-502	194	24	,	,	PUNCT
ejde-502	194	25	respectively	respectively	ADV
ejde-502	194	26	[	[	X
ejde-502	194	27	16	16	NUM
ejde-502	194	28	,	,	PUNCT
ejde-502	194	29	proposition	proposition	NOUN
ejde-502	194	30	4.2	4.2	NUM
ejde-502	194	31	and	and	CCONJ
ejde-502	194	32	lemma	lemma	PROPN
ejde-502	194	33	4.3	4.3	NUM
ejde-502	194	34	]	]	PUNCT
ejde-502	194	35	if	if	SCONJ
ejde-502	194	36	m	m	VERB
ejde-502	194	37	+	+	NOUN
ejde-502	194	38	p	p	X
ejde-502	194	39	=	=	SYM
ejde-502	194	40	2	2	NUM
ejde-502	194	41	,	,	PUNCT
ejde-502	194	42	ensure	ensure	VERB
ejde-502	194	43	that	that	SCONJ
ejde-502	194	44	,	,	PUNCT
ejde-502	194	45	given	give	VERB
ejde-502	194	46	ξ0	ξ0	PROPN
ejde-502	194	47	∈	∈	PROPN
ejde-502	194	48	(	(	PUNCT
ejde-502	194	49	0	0	NUM
ejde-502	194	50	,	,	PUNCT
ejde-502	194	51	ξ∗	ξ∗	ADJ
ejde-502	194	52	)	)	PUNCT
ejde-502	194	53	sufficiently	sufficiently	ADV
ejde-502	194	54	small	small	ADJ
ejde-502	194	55	,	,	PUNCT
ejde-502	194	56	there	there	PRON
ejde-502	194	57	exists	exist	VERB
ejde-502	194	58	a	a	DET
ejde-502	194	59	decreasing	decrease	VERB
ejde-502	194	60	self	self	NOUN
ejde-502	194	61	-	-	PUNCT
ejde-502	194	62	similar	similar	ADJ
ejde-502	194	63	profile	profile	NOUN
ejde-502	194	64	f2(ξ	f2(ξ	NOUN
ejde-502	194	65	)	)	PUNCT
ejde-502	194	66	solution	solution	NOUN
ejde-502	194	67	to	to	ADP
ejde-502	194	68	the	the	DET
ejde-502	194	69	differential	differential	ADJ
ejde-502	194	70	equation	equation	NOUN
ejde-502	194	71	(	(	PUNCT
ejde-502	194	72	2.6	2.6	NUM
ejde-502	194	73	)	)	PUNCT
ejde-502	194	74	having	have	VERB
ejde-502	194	75	an	an	DET
ejde-502	194	76	interface	interface	NOUN
ejde-502	194	77	exactly	exactly	ADV
ejde-502	194	78	at	at	ADP
ejde-502	194	79	ξ	ξ	X
ejde-502	194	80	=	=	SYM
ejde-502	194	81	ξ0	ξ0	PROPN
ejde-502	194	82	and	and	CCONJ
ejde-502	194	83	a	a	DET
ejde-502	194	84	vertical	vertical	ADJ
ejde-502	194	85	asymptote	asymptote	NOUN
ejde-502	194	86	as	as	ADP
ejde-502	194	87	ξ	ξ	PROPN
ejde-502	194	88	→	→	SYM
ejde-502	194	89	0	0	NUM
ejde-502	194	90	with	with	ADP
ejde-502	194	91	local	local	ADJ
ejde-502	194	92	behavior	behavior	NOUN
ejde-502	194	93	f2(ξ	f2(ξ	NOUN
ejde-502	194	94	)	)	PUNCT
ejde-502	194	95	∼	∼	NOUN
ejde-502	194	96	{	{	PUNCT
ejde-502	194	97	cξ−(n−2)/m	cξ−(n−2)/m	PROPN
ejde-502	194	98	,	,	PUNCT
ejde-502	194	99	if	if	SCONJ
ejde-502	194	100	n	n	PRON
ejde-502	194	101	≥	≥	NOUN
ejde-502	194	102	3	3	NUM
ejde-502	194	103	,	,	PUNCT
ejde-502	194	104	c(−	c(−	NOUN
ejde-502	194	105	ln	ln	ADJ
ejde-502	194	106	ξ)1	ξ)1	PROPN
ejde-502	194	107	/	/	SYM
ejde-502	194	108	m	m	PROPN
ejde-502	194	109	,	,	PUNCT
ejde-502	194	110	if	if	SCONJ
ejde-502	194	111	n	n	NOUN
ejde-502	194	112	=	=	SYM
ejde-502	194	113	2	2	NUM
ejde-502	194	114	,	,	PUNCT
ejde-502	194	115	as	as	SCONJ
ejde-502	194	116	it	it	PRON
ejde-502	194	117	follows	follow	VERB
ejde-502	194	118	from	from	ADP
ejde-502	194	119	[	[	X
ejde-502	194	120	16	16	NUM
ejde-502	194	121	,	,	PUNCT
ejde-502	194	122	lemma	lemma	PROPN
ejde-502	194	123	3.2	3.2	NUM
ejde-502	194	124	and	and	CCONJ
ejde-502	194	125	lemma	lemma	PROPN
ejde-502	194	126	3.5	3.5	NUM
ejde-502	194	127	]	]	PUNCT
ejde-502	194	128	,	,	PUNCT
ejde-502	194	129	where	where	SCONJ
ejde-502	194	130	c	c	AUX
ejde-502	194	131	>	>	X
ejde-502	194	132	0	0	NUM
ejde-502	194	133	designs	design	VERB
ejde-502	194	134	a	a	DET
ejde-502	194	135	positive	positive	ADJ
ejde-502	194	136	constant	constant	NOUN
ejde-502	194	137	that	that	PRON
ejde-502	194	138	might	might	AUX
ejde-502	194	139	change	change	VERB
ejde-502	194	140	from	from	ADP
ejde-502	194	141	one	one	NUM
ejde-502	194	142	case	case	NOUN
ejde-502	194	143	to	to	ADP
ejde-502	194	144	another	another	PRON
ejde-502	194	145	.	.	PUNCT
ejde-502	195	1	on	on	ADP
ejde-502	195	2	the	the	DET
ejde-502	195	3	other	other	ADJ
ejde-502	195	4	hand	hand	NOUN
ejde-502	195	5	,	,	PUNCT
ejde-502	195	6	the	the	DET
ejde-502	195	7	analysis	analysis	NOUN
ejde-502	195	8	performed	perform	VERB
ejde-502	195	9	in	in	ADP
ejde-502	195	10	[	[	X
ejde-502	195	11	16	16	NUM
ejde-502	195	12	,	,	PUNCT
ejde-502	195	13	lemma	lemma	PROPN
ejde-502	195	14	3.1	3.1	NUM
ejde-502	195	15	]	]	PUNCT
ejde-502	195	16	implies	imply	VERB
ejde-502	195	17	that	that	SCONJ
ejde-502	195	18	,	,	PUNCT
ejde-502	195	19	for	for	ADP
ejde-502	195	20	any	any	DET
ejde-502	195	21	a	a	DET
ejde-502	195	22	>	>	X
ejde-502	195	23	0	0	NUM
ejde-502	195	24	,	,	PUNCT
ejde-502	195	25	there	there	PRON
ejde-502	195	26	exists	exist	VERB
ejde-502	195	27	a	a	DET
ejde-502	195	28	profile	profile	NOUN
ejde-502	195	29	f1(ξ;a	f1(ξ;a	NOUN
ejde-502	195	30	)	)	PUNCT
ejde-502	195	31	local	local	ADJ
ejde-502	195	32	solution	solution	NOUN
ejde-502	195	33	to	to	ADP
ejde-502	195	34	(	(	PUNCT
ejde-502	195	35	2.6	2.6	NUM
ejde-502	195	36	)	)	PUNCT
ejde-502	195	37	such	such	ADJ
ejde-502	195	38	that	that	DET
ejde-502	195	39	f1(0;a	f1(0;a	NOUN
ejde-502	195	40	)	)	PUNCT
ejde-502	195	41	=	=	SYM
ejde-502	195	42	a	a	PRON
ejde-502	195	43	,	,	PUNCT
ejde-502	195	44	f1(ξ;a	f1(ξ;a	ADJ
ejde-502	195	45	)	)	PUNCT
ejde-502	195	46	∼	∼	NOUN
ejde-502	195	47	[	[	PUNCT
ejde-502	195	48	am−1	am−1	NOUN
ejde-502	195	49	+	+	CCONJ
ejde-502	195	50	α(m−	α(m−	NUM
ejde-502	195	51	1	1	NUM
ejde-502	195	52	)	)	PUNCT
ejde-502	195	53	2mn	2mn	PROPN
ejde-502	196	1	ξ2	ξ2	NOUN
ejde-502	196	2	]	]	PUNCT
ejde-502	196	3	1/(m−1	1/(m−1	NUM
ejde-502	196	4	)	)	PUNCT
ejde-502	196	5	,	,	PUNCT
ejde-502	196	6	as	as	ADP
ejde-502	196	7	ξ	ξ	X
ejde-502	196	8	→	→	SYM
ejde-502	196	9	0	0	NUM
ejde-502	196	10	,	,	PUNCT
ejde-502	196	11	which	which	PRON
ejde-502	196	12	is	be	AUX
ejde-502	196	13	increasing	increase	VERB
ejde-502	196	14	in	in	ADP
ejde-502	196	15	a	a	DET
ejde-502	196	16	right	right	ADJ
ejde-502	196	17	-	-	PUNCT
ejde-502	196	18	neighborhood	neighborhood	NOUN
ejde-502	196	19	of	of	ADP
ejde-502	196	20	the	the	DET
ejde-502	196	21	origin	origin	NOUN
ejde-502	196	22	up	up	ADP
ejde-502	196	23	to	to	ADP
ejde-502	196	24	some	some	DET
ejde-502	196	25	maximum	maximum	ADJ
ejde-502	196	26	point	point	NOUN
ejde-502	196	27	ξ1(a	ξ1(a	NUM
ejde-502	196	28	)	)	PUNCT
ejde-502	196	29	>	>	X
ejde-502	197	1	0	0	X
ejde-502	197	2	.	.	PUNCT
ejde-502	198	1	thus	thus	ADV
ejde-502	198	2	,	,	PUNCT
ejde-502	198	3	given	give	VERB
ejde-502	198	4	an	an	DET
ejde-502	198	5	initial	initial	ADJ
ejde-502	198	6	condition	condition	NOUN
ejde-502	198	7	u0	u0	ADJ
ejde-502	198	8	as	as	ADP
ejde-502	198	9	in	in	ADP
ejde-502	198	10	(	(	PUNCT
ejde-502	198	11	1.4	1.4	NUM
ejde-502	198	12	)	)	PUNCT
ejde-502	198	13	,	,	PUNCT
ejde-502	198	14	one	one	PRON
ejde-502	198	15	can	can	AUX
ejde-502	198	16	choose	choose	VERB
ejde-502	198	17	for	for	ADP
ejde-502	198	18	example	example	NOUN
ejde-502	198	19	a	a	DET
ejde-502	198	20	=	=	X
ejde-502	198	21	‖u0‖∞	‖u0‖∞	PROPN
ejde-502	198	22	and	and	CCONJ
ejde-502	198	23	fix	fix	VERB
ejde-502	198	24	some	some	DET
ejde-502	198	25	ξ0	ξ0	ADJ
ejde-502	198	26	∈	∈	PROPN
ejde-502	198	27	(	(	PUNCT
ejde-502	198	28	0	0	NUM
ejde-502	198	29	,	,	PUNCT
ejde-502	198	30	ξ1(a	ξ1(a	NUM
ejde-502	198	31	)	)	PUNCT
ejde-502	198	32	)	)	PUNCT
ejde-502	198	33	∩	∩	NOUN
ejde-502	198	34	(	(	PUNCT
ejde-502	198	35	0	0	NUM
ejde-502	198	36	,	,	PUNCT
ejde-502	198	37	ξ∗	ξ∗	NOUN
ejde-502	198	38	)	)	PUNCT
ejde-502	198	39	such	such	ADJ
ejde-502	198	40	that	that	SCONJ
ejde-502	198	41	there	there	PRON
ejde-502	198	42	exists	exist	VERB
ejde-502	198	43	a	a	DET
ejde-502	198	44	decreasing	decrease	VERB
ejde-502	198	45	profile	profile	NOUN
ejde-502	198	46	f2(ξ	f2(ξ	ADJ
ejde-502	198	47	)	)	PUNCT
ejde-502	198	48	as	as	ADP
ejde-502	198	49	above	above	ADV
ejde-502	198	50	with	with	ADP
ejde-502	198	51	vertical	vertical	ADJ
ejde-502	198	52	asymptote	asymptote	NOUN
ejde-502	198	53	as	as	ADP
ejde-502	198	54	ξ	ξ	PROPN
ejde-502	198	55	→	→	SYM
ejde-502	198	56	0	0	NUM
ejde-502	198	57	and	and	CCONJ
ejde-502	198	58	edge	edge	NOUN
ejde-502	198	59	of	of	ADP
ejde-502	198	60	the	the	DET
ejde-502	198	61	support	support	NOUN
ejde-502	198	62	at	at	ADP
ejde-502	198	63	ξ	ξ	X
ejde-502	198	64	=	=	SYM
ejde-502	198	65	ξ0	ξ0	PROPN
ejde-502	198	66	.	.	PUNCT
ejde-502	199	1	the	the	DET
ejde-502	199	2	profiles	profile	NOUN
ejde-502	199	3	f1(·;a	f1(·;a	NOUN
ejde-502	199	4	)	)	PUNCT
ejde-502	199	5	and	and	CCONJ
ejde-502	199	6	f2	f2	PROPN
ejde-502	199	7	have	have	VERB
ejde-502	199	8	to	to	PART
ejde-502	199	9	cross	cross	VERB
ejde-502	199	10	at	at	ADP
ejde-502	199	11	some	some	DET
ejde-502	199	12	point	point	NOUN
ejde-502	199	13	ξ	ξ	X
ejde-502	199	14	∈	∈	PROPN
ejde-502	199	15	(	(	PUNCT
ejde-502	199	16	0	0	NUM
ejde-502	199	17	,	,	PUNCT
ejde-502	199	18	ξ1(a	ξ1(a	NUM
ejde-502	199	19	)	)	PUNCT
ejde-502	199	20	)	)	PUNCT
ejde-502	199	21	.	.	PUNCT
ejde-502	200	1	we	we	PRON
ejde-502	200	2	finally	finally	ADV
ejde-502	200	3	define	define	VERB
ejde-502	200	4	a	a	DET
ejde-502	200	5	self	self	NOUN
ejde-502	200	6	-	-	PUNCT
ejde-502	200	7	similar	similar	ADJ
ejde-502	200	8	supersolution	supersolution	NOUN
ejde-502	200	9	to	to	ADP
ejde-502	200	10	(	(	PUNCT
ejde-502	200	11	1.1	1.1	NUM
ejde-502	200	12	)	)	PUNCT
ejde-502	200	13	as	as	SCONJ
ejde-502	200	14	follows	follow	VERB
ejde-502	200	15	:	:	PUNCT
ejde-502	200	16	z(x	z(x	NUM
ejde-502	200	17	,	,	PUNCT
ejde-502	200	18	t	t	PROPN
ejde-502	200	19	)	)	PUNCT
ejde-502	200	20	=	=	PUNCT
ejde-502	201	1	(	(	PUNCT
ejde-502	201	2	t	t	PROPN
ejde-502	201	3	−	−	PROPN
ejde-502	201	4	t)−αf((1	t)−αf((1	PROPN
ejde-502	201	5	+	+	CCONJ
ejde-502	201	6	|x|)(t	|x|)(t	ADJ
ejde-502	201	7	−	−	NOUN
ejde-502	201	8	t)β	t)β	NOUN
ejde-502	201	9	)	)	PUNCT
ejde-502	201	10	,	,	PUNCT
ejde-502	201	11	f(ξ	f(ξ	X
ejde-502	201	12	)	)	PUNCT
ejde-502	201	13	=	=	SYM
ejde-502	201	14	{	{	PUNCT
ejde-502	201	15	f1(ξ;a	f1(ξ;a	PROPN
ejde-502	201	16	)	)	PUNCT
ejde-502	201	17	,	,	PUNCT
ejde-502	201	18	ξ	ξ	PROPN
ejde-502	201	19	∈	∈	PROPN
ejde-502	202	1	[	[	X
ejde-502	202	2	0	0	NUM
ejde-502	202	3	,	,	PUNCT
ejde-502	202	4	ξ	ξ	NOUN
ejde-502	202	5	]	]	X
ejde-502	202	6	,	,	PUNCT
ejde-502	202	7	f2(ξ	f2(ξ	PROPN
ejde-502	202	8	)	)	PUNCT
ejde-502	202	9	,	,	PUNCT
ejde-502	202	10	ξ	ξ	X
ejde-502	202	11	≥	≥	X
ejde-502	202	12	ξ	ξ	NUM
ejde-502	202	13	,	,	PUNCT
ejde-502	202	14	(	(	PUNCT
ejde-502	202	15	2.7	2.7	NUM
ejde-502	202	16	)	)	PUNCT
ejde-502	202	17	10	10	NUM
ejde-502	202	18	r.	r.	PROPN
ejde-502	202	19	g.	g.	PROPN
ejde-502	202	20	iagar	iagar	PROPN
ejde-502	202	21	,	,	PUNCT
ejde-502	202	22	a.	a.	NOUN
ejde-502	202	23	i.	i.	PROPN
ejde-502	202	24	muñoz	muñoz	PROPN
ejde-502	202	25	,	,	PUNCT
ejde-502	202	26	a.	a.	NOUN
ejde-502	202	27	sánchez	sánchez	PROPN
ejde-502	202	28	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	202	29	and	and	CCONJ
ejde-502	202	30	notice	notice	VERB
ejde-502	202	31	that	that	SCONJ
ejde-502	202	32	this	this	PRON
ejde-502	202	33	is	be	AUX
ejde-502	202	34	indeed	indeed	ADV
ejde-502	202	35	a	a	DET
ejde-502	202	36	supersolution	supersolution	NOUN
ejde-502	202	37	for	for	ADP
ejde-502	202	38	any	any	DET
ejde-502	202	39	t	t	NOUN
ejde-502	202	40	>	>	X
ejde-502	202	41	0	0	NUM
ejde-502	202	42	,	,	PUNCT
ejde-502	202	43	as	as	SCONJ
ejde-502	202	44	it	it	PRON
ejde-502	202	45	can	can	AUX
ejde-502	202	46	be	be	AUX
ejde-502	202	47	described	describe	VERB
ejde-502	202	48	alternatively	alternatively	ADV
ejde-502	202	49	as	as	ADP
ejde-502	202	50	z(x	z(x	NUM
ejde-502	202	51	,	,	PUNCT
ejde-502	202	52	t	t	NOUN
ejde-502	202	53	)	)	PUNCT
ejde-502	202	54	=	=	SYM
ejde-502	202	55	min{z1(x	min{z1(x	PROPN
ejde-502	202	56	,	,	PUNCT
ejde-502	202	57	t	t	PROPN
ejde-502	202	58	)	)	PUNCT
ejde-502	202	59	,	,	PUNCT
ejde-502	202	60	z2(x	z2(x	PROPN
ejde-502	202	61	,	,	PUNCT
ejde-502	202	62	t	t	PROPN
ejde-502	202	63	)	)	PUNCT
ejde-502	202	64	}	}	PUNCT
ejde-502	202	65	,	,	PUNCT
ejde-502	202	66	zi(x	zi(x	NUM
ejde-502	202	67	,	,	PUNCT
ejde-502	202	68	t	t	PROPN
ejde-502	202	69	)	)	PUNCT
ejde-502	202	70	=	=	PUNCT
ejde-502	202	71	(	(	PUNCT
ejde-502	202	72	t	t	PROPN
ejde-502	202	73	−	−	PROPN
ejde-502	202	74	t)−αfi((1	t)−αfi((1	PROPN
ejde-502	202	75	+	+	CCONJ
ejde-502	202	76	|x|)(t	|x|)(t	ADJ
ejde-502	202	77	−	−	NOUN
ejde-502	202	78	t)β	t)β	NOUN
ejde-502	202	79	)	)	PUNCT
ejde-502	202	80	,	,	PUNCT
ejde-502	202	81	for	for	ADP
ejde-502	202	82	i	i	PROPN
ejde-502	202	83	=	=	SYM
ejde-502	202	84	1	1	NUM
ejde-502	202	85	,	,	PUNCT
ejde-502	202	86	2	2	NUM
ejde-502	202	87	,	,	PUNCT
ejde-502	202	88	and	and	CCONJ
ejde-502	202	89	z1	z1	VERB
ejde-502	202	90	,	,	PUNCT
ejde-502	202	91	z2	z2	PROPN
ejde-502	202	92	are	be	AUX
ejde-502	202	93	in	in	ADP
ejde-502	202	94	fact	fact	NOUN
ejde-502	202	95	solutions	solution	NOUN
ejde-502	202	96	.	.	PUNCT
ejde-502	203	1	the	the	DET
ejde-502	203	2	same	same	ADJ
ejde-502	203	3	considerations	consideration	NOUN
ejde-502	203	4	about	about	ADP
ejde-502	203	5	the	the	DET
ejde-502	203	6	magnitude	magnitude	NOUN
ejde-502	203	7	of	of	ADP
ejde-502	203	8	σ	σ	NOUN
ejde-502	203	9	as	as	ADP
ejde-502	203	10	in	in	ADP
ejde-502	203	11	the	the	DET
ejde-502	203	12	end	end	NOUN
ejde-502	203	13	of	of	ADP
ejde-502	203	14	step	step	NOUN
ejde-502	203	15	2	2	NUM
ejde-502	203	16	ensure	ensure	VERB
ejde-502	203	17	that	that	SCONJ
ejde-502	203	18	the	the	DET
ejde-502	203	19	supersolution	supersolution	NOUN
ejde-502	203	20	defined	define	VERB
ejde-502	203	21	in	in	ADP
ejde-502	203	22	(	(	PUNCT
ejde-502	203	23	2.7	2.7	NUM
ejde-502	203	24	)	)	PUNCT
ejde-502	203	25	works	work	NOUN
ejde-502	203	26	also	also	ADV
ejde-502	203	27	for	for	ADP
ejde-502	203	28	values	value	NOUN
ejde-502	203	29	of	of	ADP
ejde-502	203	30	σ	σ	X
ejde-502	203	31	smaller	small	ADJ
ejde-502	203	32	than	than	ADP
ejde-502	203	33	2(1−p)/(m−1	2(1−p)/(m−1	NUM
ejde-502	203	34	)	)	PUNCT
ejde-502	203	35	.	.	PUNCT
ejde-502	204	1	we	we	PRON
ejde-502	204	2	thus	thus	ADV
ejde-502	204	3	complete	complete	VERB
ejde-502	204	4	the	the	DET
ejde-502	204	5	proof	proof	NOUN
ejde-502	204	6	by	by	ADP
ejde-502	204	7	choosing	choose	VERB
ejde-502	204	8	a	a	DET
ejde-502	204	9	sufficiently	sufficiently	ADV
ejde-502	204	10	small	small	ADJ
ejde-502	204	11	t0	t0	PROPN
ejde-502	204	12	>	>	X
ejde-502	204	13	0	0	NUM
ejde-502	205	1	such	such	ADJ
ejde-502	205	2	that	that	SCONJ
ejde-502	205	3	(	(	PUNCT
ejde-502	205	4	2.5	2.5	NUM
ejde-502	205	5	)	)	PUNCT
ejde-502	205	6	holds	hold	VERB
ejde-502	205	7	true	true	ADJ
ejde-502	205	8	on	on	ADP
ejde-502	205	9	the	the	DET
ejde-502	205	10	support	support	NOUN
ejde-502	205	11	of	of	ADP
ejde-502	205	12	u0	u0	ADJ
ejde-502	205	13	,	,	PUNCT
ejde-502	205	14	and	and	CCONJ
ejde-502	205	15	notice	notice	VERB
ejde-502	205	16	that	that	SCONJ
ejde-502	205	17	z	z	NOUN
ejde-502	205	18	is	be	AUX
ejde-502	205	19	a	a	DET
ejde-502	205	20	supersolution	supersolution	NOUN
ejde-502	205	21	to	to	ADP
ejde-502	205	22	the	the	DET
ejde-502	205	23	approximating	approximate	VERB
ejde-502	205	24	problems	problem	NOUN
ejde-502	205	25	(	(	PUNCT
ejde-502	205	26	2.1	2.1	NUM
ejde-502	205	27	)	)	PUNCT
ejde-502	205	28	leading	lead	VERB
ejde-502	205	29	to	to	ADP
ejde-502	205	30	the	the	DET
ejde-502	205	31	minimal	minimal	ADJ
ejde-502	205	32	solution	solution	NOUN
ejde-502	205	33	.	.	PUNCT
ejde-502	206	1	this	this	PRON
ejde-502	206	2	again	again	ADV
ejde-502	206	3	implies	imply	VERB
ejde-502	206	4	that	that	SCONJ
ejde-502	206	5	t∞	t∞	PROPN
ejde-502	206	6	≥	≥	NUM
ejde-502	206	7	t0	t0	X
ejde-502	206	8	>	>	X
ejde-502	206	9	0	0	NUM
ejde-502	206	10	,	,	PUNCT
ejde-502	206	11	completing	complete	VERB
ejde-502	206	12	the	the	DET
ejde-502	206	13	proof	proof	NOUN
ejde-502	206	14	.	.	PUNCT
ejde-502	207	1	�	�	PROPN
ejde-502	207	2	the	the	DET
ejde-502	207	3	solution	solution	NOUN
ejde-502	207	4	u	u	NOUN
ejde-502	207	5	constructed	construct	VERB
ejde-502	207	6	in	in	ADP
ejde-502	207	7	the	the	DET
ejde-502	207	8	proof	proof	NOUN
ejde-502	207	9	of	of	ADP
ejde-502	207	10	proposition	proposition	NOUN
ejde-502	207	11	2.1	2.1	NUM
ejde-502	207	12	will	will	AUX
ejde-502	207	13	be	be	AUX
ejde-502	207	14	referred	refer	VERB
ejde-502	207	15	as	as	ADP
ejde-502	207	16	the	the	DET
ejde-502	207	17	minimal	minimal	ADJ
ejde-502	207	18	solution	solution	NOUN
ejde-502	207	19	to	to	ADP
ejde-502	207	20	the	the	DET
ejde-502	207	21	cauchy	cauchy	ADJ
ejde-502	207	22	problem	problem	NOUN
ejde-502	207	23	(	(	PUNCT
ejde-502	207	24	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	207	25	)	)	PUNCT
ejde-502	207	26	and	and	CCONJ
ejde-502	207	27	denoted	denote	VERB
ejde-502	207	28	by	by	ADP
ejde-502	207	29	m(u0	m(u0	NOUN
ejde-502	207	30	)	)	PUNCT
ejde-502	207	31	in	in	ADP
ejde-502	207	32	the	the	DET
ejde-502	207	33	sequel	sequel	NOUN
ejde-502	207	34	.	.	PUNCT
ejde-502	208	1	notice	notice	VERB
ejde-502	208	2	that	that	SCONJ
ejde-502	208	3	the	the	DET
ejde-502	208	4	above	above	ADJ
ejde-502	208	5	proof	proof	NOUN
ejde-502	208	6	does	do	AUX
ejde-502	208	7	not	not	PART
ejde-502	208	8	imply	imply	VERB
ejde-502	208	9	that	that	SCONJ
ejde-502	208	10	necessarily	necessarily	ADV
ejde-502	208	11	the	the	DET
ejde-502	208	12	minimal	minimal	ADJ
ejde-502	208	13	solution	solution	NOUN
ejde-502	208	14	blows	blow	VERB
ejde-502	208	15	up	up	ADP
ejde-502	208	16	in	in	ADP
ejde-502	208	17	finite	finite	ADJ
ejde-502	208	18	time	time	NOUN
ejde-502	208	19	.	.	PUNCT
ejde-502	209	1	in	in	ADP
ejde-502	209	2	fact	fact	NOUN
ejde-502	209	3	,	,	PUNCT
ejde-502	209	4	it	it	PRON
ejde-502	209	5	might	might	AUX
ejde-502	209	6	blow	blow	VERB
ejde-502	209	7	up	up	ADP
ejde-502	209	8	or	or	CCONJ
ejde-502	209	9	not	not	PART
ejde-502	209	10	according	accord	VERB
ejde-502	209	11	to	to	ADP
ejde-502	209	12	whether	whether	SCONJ
ejde-502	209	13	σ	σ	NOUN
ejde-502	209	14	is	be	AUX
ejde-502	209	15	larger	large	ADJ
ejde-502	209	16	or	or	CCONJ
ejde-502	209	17	smaller	small	ADJ
ejde-502	209	18	than	than	ADP
ejde-502	209	19	2(1−	2(1−	NUM
ejde-502	209	20	p)/(m−	p)/(m−	NOUN
ejde-502	209	21	1	1	NUM
ejde-502	209	22	)	)	PUNCT
ejde-502	209	23	,	,	PUNCT
ejde-502	209	24	but	but	CCONJ
ejde-502	209	25	this	this	PRON
ejde-502	209	26	is	be	AUX
ejde-502	209	27	not	not	PART
ejde-502	209	28	easy	easy	ADJ
ejde-502	209	29	to	to	PART
ejde-502	209	30	prove	prove	VERB
ejde-502	209	31	once	once	SCONJ
ejde-502	209	32	we	we	PRON
ejde-502	209	33	miss	miss	VERB
ejde-502	209	34	a	a	DET
ejde-502	209	35	comparison	comparison	NOUN
ejde-502	209	36	principle	principle	NOUN
ejde-502	209	37	,	,	PUNCT
ejde-502	209	38	and	and	CCONJ
ejde-502	209	39	we	we	PRON
ejde-502	209	40	refer	refer	VERB
ejde-502	209	41	the	the	DET
ejde-502	209	42	reader	reader	NOUN
ejde-502	209	43	to	to	ADP
ejde-502	209	44	the	the	DET
ejde-502	209	45	section	section	NOUN
ejde-502	209	46	of	of	ADP
ejde-502	209	47	open	open	ADJ
ejde-502	209	48	problems	problem	NOUN
ejde-502	209	49	at	at	ADP
ejde-502	209	50	the	the	DET
ejde-502	209	51	end	end	NOUN
ejde-502	209	52	.	.	PUNCT
ejde-502	210	1	3	3	X
ejde-502	210	2	.	.	X
ejde-502	210	3	non	non	ADJ
ejde-502	210	4	-	-	NOUN
ejde-502	210	5	uniqueness	uniqueness	NOUN
ejde-502	210	6	for	for	ADP
ejde-502	210	7	m+	m+	NUM
ejde-502	210	8	p	p	NOUN
ejde-502	210	9	≥	≥	NUM
ejde-502	210	10	2	2	NUM
ejde-502	210	11	a	a	DET
ejde-502	210	12	natural	natural	ADJ
ejde-502	210	13	question	question	NOUN
ejde-502	210	14	raised	raise	VERB
ejde-502	210	15	by	by	ADP
ejde-502	210	16	the	the	DET
ejde-502	210	17	previous	previous	ADJ
ejde-502	210	18	section	section	NOUN
ejde-502	210	19	is	be	AUX
ejde-502	211	1	whether	whether	SCONJ
ejde-502	211	2	in	in	ADP
ejde-502	211	3	the	the	DET
ejde-502	211	4	case	case	NOUN
ejde-502	211	5	m+	m+	NUM
ejde-502	211	6	p	p	NOUN
ejde-502	211	7	≥	≥	NUM
ejde-502	211	8	2	2	NUM
ejde-502	211	9	and	and	CCONJ
ejde-502	211	10	for	for	SCONJ
ejde-502	211	11	compactly	compactly	ADV
ejde-502	211	12	supported	support	VERB
ejde-502	211	13	and	and	CCONJ
ejde-502	211	14	continuous	continuous	ADJ
ejde-502	211	15	data	datum	NOUN
ejde-502	211	16	u0	u0	VERB
ejde-502	211	17	the	the	DET
ejde-502	211	18	minimal	minimal	ADJ
ejde-502	211	19	solution	solution	NOUN
ejde-502	211	20	u	u	NOUN
ejde-502	211	21	constructed	construct	VERB
ejde-502	211	22	in	in	ADP
ejde-502	211	23	proposition	proposition	NOUN
ejde-502	211	24	2.1	2.1	NUM
ejde-502	211	25	is	be	AUX
ejde-502	211	26	the	the	DET
ejde-502	211	27	only	only	ADJ
ejde-502	211	28	solution	solution	NOUN
ejde-502	211	29	to	to	ADP
ejde-502	211	30	the	the	DET
ejde-502	211	31	cauchy	cauchy	ADJ
ejde-502	211	32	problem	problem	NOUN
ejde-502	211	33	(	(	PUNCT
ejde-502	211	34	1.1)(1.2	1.1)(1.2	NUM
ejde-502	211	35	)	)	PUNCT
ejde-502	211	36	.	.	PUNCT
ejde-502	212	1	for	for	ADP
ejde-502	212	2	σ	σ	PROPN
ejde-502	212	3	=	=	SYM
ejde-502	212	4	0	0	NUM
ejde-502	212	5	non	non	ADJ
ejde-502	212	6	-	-	ADJ
ejde-502	212	7	uniqueness	uniqueness	NOUN
ejde-502	212	8	follows	follow	VERB
ejde-502	212	9	easily	easily	ADV
ejde-502	212	10	from	from	ADP
ejde-502	212	11	the	the	DET
ejde-502	212	12	construction	construction	NOUN
ejde-502	212	13	of	of	ADP
ejde-502	212	14	a	a	DET
ejde-502	212	15	different	different	ADJ
ejde-502	212	16	solution	solution	NOUN
ejde-502	212	17	called	call	VERB
ejde-502	212	18	maximal	maximal	ADJ
ejde-502	212	19	,	,	PUNCT
ejde-502	212	20	which	which	PRON
ejde-502	212	21	is	be	AUX
ejde-502	212	22	shown	show	VERB
ejde-502	212	23	to	to	PART
ejde-502	212	24	be	be	AUX
ejde-502	212	25	strictly	strictly	ADV
ejde-502	212	26	positive	positive	ADJ
ejde-502	212	27	for	for	ADP
ejde-502	212	28	any	any	DET
ejde-502	212	29	t	t	NOUN
ejde-502	212	30	>	>	X
ejde-502	212	31	0	0	PUNCT
ejde-502	212	32	and	and	CCONJ
ejde-502	212	33	thus	thus	ADV
ejde-502	212	34	different	different	ADJ
ejde-502	212	35	from	from	ADP
ejde-502	212	36	u	u	NOUN
ejde-502	212	37	,	,	PUNCT
ejde-502	212	38	see	see	VERB
ejde-502	212	39	[	[	X
ejde-502	212	40	27	27	NUM
ejde-502	212	41	]	]	PUNCT
ejde-502	212	42	.	.	PUNCT
ejde-502	213	1	we	we	PRON
ejde-502	213	2	can	can	AUX
ejde-502	213	3	not	not	PART
ejde-502	213	4	construct	construct	VERB
ejde-502	213	5	such	such	DET
ejde-502	213	6	a	a	DET
ejde-502	213	7	maximal	maximal	ADJ
ejde-502	213	8	solution	solution	NOUN
ejde-502	213	9	to	to	ADP
ejde-502	213	10	(	(	PUNCT
ejde-502	213	11	1.1	1.1	NUM
ejde-502	213	12	)	)	PUNCT
ejde-502	213	13	,	,	PUNCT
ejde-502	213	14	since	since	SCONJ
ejde-502	213	15	strictly	strictly	ADV
ejde-502	213	16	positive	positive	ADJ
ejde-502	213	17	solutions	solution	NOUN
ejde-502	213	18	might	might	AUX
ejde-502	213	19	not	not	PART
ejde-502	213	20	even	even	ADV
ejde-502	213	21	exist	exist	VERB
ejde-502	213	22	at	at	ADV
ejde-502	213	23	all	all	ADV
ejde-502	213	24	in	in	ADP
ejde-502	213	25	some	some	DET
ejde-502	213	26	ranges	range	NOUN
ejde-502	213	27	of	of	ADP
ejde-502	213	28	σ	σ	NOUN
ejde-502	213	29	(	(	PUNCT
ejde-502	213	30	see	see	VERB
ejde-502	213	31	a	a	DET
ejde-502	213	32	comment	comment	NOUN
ejde-502	213	33	with	with	ADP
ejde-502	213	34	a	a	DET
ejde-502	213	35	formal	formal	ADJ
ejde-502	213	36	intuition	intuition	NOUN
ejde-502	213	37	for	for	ADP
ejde-502	213	38	such	such	ADJ
ejde-502	213	39	non	non	ADJ
ejde-502	213	40	-	-	NOUN
ejde-502	213	41	existence	existence	NOUN
ejde-502	213	42	in	in	ADP
ejde-502	213	43	the	the	DET
ejde-502	213	44	final	final	ADJ
ejde-502	213	45	section	section	NOUN
ejde-502	213	46	of	of	ADP
ejde-502	213	47	this	this	DET
ejde-502	213	48	work	work	NOUN
ejde-502	213	49	)	)	PUNCT
ejde-502	213	50	.	.	PUNCT
ejde-502	214	1	in	in	ADP
ejde-502	214	2	order	order	NOUN
ejde-502	214	3	thus	thus	ADV
ejde-502	214	4	to	to	PART
ejde-502	214	5	prove	prove	VERB
ejde-502	214	6	the	the	DET
ejde-502	214	7	existence	existence	NOUN
ejde-502	214	8	of	of	ADP
ejde-502	214	9	multiple	multiple	ADJ
ejde-502	214	10	solutions	solution	NOUN
ejde-502	214	11	(	(	PUNCT
ejde-502	214	12	and	and	CCONJ
ejde-502	214	13	in	in	ADP
ejde-502	214	14	fact	fact	NOUN
ejde-502	214	15	an	an	DET
ejde-502	214	16	infinite	infinite	ADJ
ejde-502	214	17	number	number	NOUN
ejde-502	214	18	of	of	ADP
ejde-502	214	19	them	they	PRON
ejde-502	214	20	)	)	PUNCT
ejde-502	214	21	we	we	PRON
ejde-502	214	22	adapt	adapt	VERB
ejde-502	214	23	to	to	ADP
ejde-502	214	24	our	our	PRON
ejde-502	214	25	case	case	NOUN
ejde-502	214	26	the	the	DET
ejde-502	214	27	deeper	deep	ADJ
ejde-502	214	28	results	result	NOUN
ejde-502	214	29	in	in	ADP
ejde-502	214	30	[	[	X
ejde-502	214	31	28	28	NUM
ejde-502	214	32	]	]	PUNCT
ejde-502	214	33	,	,	PUNCT
ejde-502	214	34	where	where	SCONJ
ejde-502	214	35	infinitely	infinitely	ADV
ejde-502	214	36	many	many	ADJ
ejde-502	214	37	solutions	solution	NOUN
ejde-502	214	38	to	to	ADP
ejde-502	214	39	the	the	DET
ejde-502	214	40	cauchy	cauchy	ADJ
ejde-502	214	41	problem	problem	NOUN
ejde-502	214	42	(	(	PUNCT
ejde-502	214	43	1.5)-(1.2	1.5)-(1.2	NOUN
ejde-502	214	44	)	)	PUNCT
ejde-502	214	45	are	be	AUX
ejde-502	214	46	constructed	construct	VERB
ejde-502	214	47	,	,	PUNCT
ejde-502	214	48	based	base	VERB
ejde-502	214	49	on	on	ADP
ejde-502	214	50	a	a	DET
ejde-502	214	51	prescribed	prescribed	ADJ
ejde-502	214	52	evolution	evolution	NOUN
ejde-502	214	53	of	of	ADP
ejde-502	214	54	the	the	DET
ejde-502	214	55	interface	interface	NOUN
ejde-502	214	56	of	of	ADP
ejde-502	214	57	them	they	PRON
ejde-502	214	58	.	.	PUNCT
ejde-502	215	1	as	as	ADP
ejde-502	215	2	a	a	DET
ejde-502	215	3	preliminary	preliminary	ADJ
ejde-502	215	4	fact	fact	NOUN
ejde-502	215	5	,	,	PUNCT
ejde-502	215	6	let	let	VERB
ejde-502	215	7	us	we	PRON
ejde-502	215	8	recall	recall	VERB
ejde-502	215	9	that	that	SCONJ
ejde-502	215	10	(	(	PUNCT
ejde-502	215	11	1.5	1.5	NUM
ejde-502	215	12	)	)	PUNCT
ejde-502	215	13	admits	admit	VERB
ejde-502	215	14	in	in	ADP
ejde-502	215	15	dimension	dimension	NOUN
ejde-502	215	16	n	n	NOUN
ejde-502	215	17	=	=	SYM
ejde-502	215	18	1	1	NUM
ejde-502	215	19	an	an	DET
ejde-502	215	20	absolute	absolute	ADJ
ejde-502	215	21	minimal	minimal	ADJ
ejde-502	215	22	solution	solution	NOUN
ejde-502	215	23	in	in	ADP
ejde-502	215	24	self	self	NOUN
ejde-502	215	25	-	-	PUNCT
ejde-502	215	26	similar	similar	ADJ
ejde-502	215	27	form	form	NOUN
ejde-502	215	28	e(x	e(x	NUM
ejde-502	215	29	,	,	PUNCT
ejde-502	215	30	t	t	NOUN
ejde-502	215	31	)	)	PUNCT
ejde-502	216	1	=	=	SYM
ejde-502	216	2	t1/(1−p)ϕ(|x|t−γ	t1/(1−p)ϕ(|x|t−γ	NUM
ejde-502	216	3	)	)	PUNCT
ejde-502	216	4	,	,	PUNCT
ejde-502	216	5	γ	γ	X
ejde-502	216	6	=	=	SYM
ejde-502	216	7	m−	m−	PROPN
ejde-502	217	1	p	p	NOUN
ejde-502	217	2	2(1−	2(1−	PROPN
ejde-502	217	3	p	p	X
ejde-502	217	4	)	)	PUNCT
ejde-502	217	5	(	(	PUNCT
ejde-502	217	6	3.1	3.1	NUM
ejde-502	217	7	)	)	PUNCT
ejde-502	217	8	with	with	ADP
ejde-502	217	9	zero	zero	NUM
ejde-502	217	10	initial	initial	ADJ
ejde-502	217	11	condition	condition	NOUN
ejde-502	217	12	e(x	e(x	NUM
ejde-502	217	13	,	,	PUNCT
ejde-502	217	14	0	0	NUM
ejde-502	217	15	)	)	PUNCT
ejde-502	217	16	=	=	SYM
ejde-502	217	17	0	0	NUM
ejde-502	218	1	for	for	ADP
ejde-502	218	2	any	any	DET
ejde-502	218	3	x	x	SYM
ejde-502	218	4	∈	∈	PROPN
ejde-502	218	5	r	r	NOUN
ejde-502	218	6	,	,	PUNCT
ejde-502	218	7	according	accord	VERB
ejde-502	218	8	to	to	ADP
ejde-502	218	9	[	[	X
ejde-502	218	10	26	26	NUM
ejde-502	218	11	]	]	PUNCT
ejde-502	218	12	.	.	PUNCT
ejde-502	219	1	it	it	PRON
ejde-502	219	2	is	be	AUX
ejde-502	219	3	also	also	ADV
ejde-502	219	4	shown	show	VERB
ejde-502	219	5	in	in	ADP
ejde-502	219	6	[	[	X
ejde-502	219	7	26	26	NUM
ejde-502	219	8	]	]	PUNCT
ejde-502	219	9	that	that	SCONJ
ejde-502	219	10	such	such	ADJ
ejde-502	219	11	solution	solution	NOUN
ejde-502	219	12	lies	lie	VERB
ejde-502	219	13	below	below	ADP
ejde-502	219	14	any	any	DET
ejde-502	219	15	solution	solution	NOUN
ejde-502	219	16	(	(	PUNCT
ejde-502	219	17	and	and	CCONJ
ejde-502	219	18	supersolution	supersolution	NOUN
ejde-502	219	19	)	)	PUNCT
ejde-502	219	20	to	to	ADP
ejde-502	219	21	(	(	PUNCT
ejde-502	219	22	1.5	1.5	NUM
ejde-502	219	23	)	)	PUNCT
ejde-502	219	24	and	and	CCONJ
ejde-502	219	25	consequently	consequently	ADV
ejde-502	219	26	also	also	ADV
ejde-502	219	27	below	below	ADP
ejde-502	219	28	any	any	DET
ejde-502	219	29	solution	solution	NOUN
ejde-502	219	30	to	to	ADP
ejde-502	219	31	(	(	PUNCT
ejde-502	219	32	1.1	1.1	NUM
ejde-502	219	33	)	)	PUNCT
ejde-502	219	34	(	(	PUNCT
ejde-502	219	35	which	which	PRON
ejde-502	219	36	is	be	AUX
ejde-502	219	37	a	a	DET
ejde-502	219	38	strict	strict	ADJ
ejde-502	219	39	supersolution	supersolution	NOUN
ejde-502	219	40	to	to	ADP
ejde-502	219	41	(	(	PUNCT
ejde-502	219	42	1.5	1.5	NUM
ejde-502	219	43	)	)	PUNCT
ejde-502	219	44	)	)	PUNCT
ejde-502	219	45	.	.	PUNCT
ejde-502	220	1	moreover	moreover	ADV
ejde-502	220	2	,	,	PUNCT
ejde-502	220	3	the	the	DET
ejde-502	220	4	profile	profile	ADJ
ejde-502	220	5	ϕ	ϕ	PROPN
ejde-502	220	6	of	of	ADP
ejde-502	220	7	e	e	PROPN
ejde-502	220	8	is	be	AUX
ejde-502	220	9	non	non	ADJ
ejde-502	220	10	-	-	ADJ
ejde-502	220	11	increasing	increase	VERB
ejde-502	220	12	and	and	CCONJ
ejde-502	220	13	compactly	compactly	ADV
ejde-502	220	14	supported	support	VERB
ejde-502	220	15	,	,	PUNCT
ejde-502	220	16	thus	thus	ADV
ejde-502	220	17	suppe	suppe	NOUN
ejde-502	220	18	⊆	⊆	NUM
ejde-502	220	19	[	[	X
ejde-502	220	20	−%0	−%0	PROPN
ejde-502	220	21	t	t	PROPN
ejde-502	220	22	γ	γ	X
ejde-502	220	23	,	,	PUNCT
ejde-502	221	1	%	%	NOUN
ejde-502	221	2	0	0	NUM
ejde-502	221	3	t	t	NOUN
ejde-502	221	4	γ	γ	X
ejde-502	221	5	]	]	PUNCT
ejde-502	222	1	where	where	SCONJ
ejde-502	222	2	%	%	NOUN
ejde-502	222	3	0	0	SYM
ejde-502	222	4	>	>	X
ejde-502	222	5	0	0	NUM
ejde-502	222	6	is	be	AUX
ejde-502	222	7	the	the	DET
ejde-502	222	8	right	right	ADJ
ejde-502	222	9	-	-	PUNCT
ejde-502	222	10	interface	interface	NOUN
ejde-502	222	11	point	point	NOUN
ejde-502	222	12	of	of	ADP
ejde-502	222	13	the	the	DET
ejde-502	222	14	profile	profile	NOUN
ejde-502	222	15	ϕ.	ϕ.	PROPN
ejde-502	222	16	with	with	ADP
ejde-502	222	17	these	these	DET
ejde-502	222	18	elements	element	NOUN
ejde-502	222	19	and	and	CCONJ
ejde-502	222	20	notation	notation	NOUN
ejde-502	222	21	in	in	ADP
ejde-502	222	22	mind	mind	NOUN
ejde-502	222	23	,	,	PUNCT
ejde-502	222	24	we	we	PRON
ejde-502	222	25	can	can	AUX
ejde-502	222	26	prove	prove	VERB
ejde-502	222	27	theorem	theorem	ADJ
ejde-502	222	28	1.3	1.3	NUM
ejde-502	222	29	as	as	ADP
ejde-502	222	30	an	an	DET
ejde-502	222	31	immediate	immediate	ADJ
ejde-502	222	32	consequence	consequence	NOUN
ejde-502	222	33	of	of	ADP
ejde-502	222	34	a	a	DET
ejde-502	222	35	stronger	strong	ADJ
ejde-502	222	36	result	result	NOUN
ejde-502	222	37	adapting	adapt	VERB
ejde-502	222	38	a	a	DET
ejde-502	222	39	construction	construction	NOUN
ejde-502	222	40	from	from	ADP
ejde-502	222	41	[	[	X
ejde-502	222	42	28	28	NUM
ejde-502	222	43	]	]	PUNCT
ejde-502	222	44	.	.	PUNCT
ejde-502	223	1	more	more	ADV
ejde-502	223	2	precisely	precisely	ADV
ejde-502	223	3	,	,	PUNCT
ejde-502	223	4	by	by	ADP
ejde-502	223	5	prescribing	prescribe	VERB
ejde-502	223	6	the	the	DET
ejde-502	223	7	behavior	behavior	NOUN
ejde-502	223	8	of	of	ADP
ejde-502	223	9	the	the	DET
ejde-502	223	10	interface	interface	NOUN
ejde-502	223	11	(	(	PUNCT
ejde-502	223	12	under	under	ADP
ejde-502	223	13	some	some	DET
ejde-502	223	14	limitations	limitation	NOUN
ejde-502	223	15	)	)	PUNCT
ejde-502	223	16	we	we	PRON
ejde-502	223	17	can	can	AUX
ejde-502	223	18	obtain	obtain	VERB
ejde-502	223	19	a	a	DET
ejde-502	223	20	solution	solution	NOUN
ejde-502	223	21	to	to	ADP
ejde-502	223	22	the	the	DET
ejde-502	223	23	cauchy	cauchy	ADJ
ejde-502	223	24	problem	problem	NOUN
ejde-502	223	25	(	(	PUNCT
ejde-502	223	26	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	223	27	)	)	PUNCT
ejde-502	223	28	with	with	ADP
ejde-502	223	29	exactly	exactly	ADV
ejde-502	223	30	that	that	PRON
ejde-502	223	31	given	give	VERB
ejde-502	223	32	interface	interface	NOUN
ejde-502	223	33	at	at	ADP
ejde-502	223	34	every	every	DET
ejde-502	223	35	(	(	PUNCT
ejde-502	223	36	small	small	ADJ
ejde-502	223	37	)	)	PUNCT
ejde-502	223	38	time	time	NOUN
ejde-502	223	39	t	t	PROPN
ejde-502	223	40	∈	∈	PROPN
ejde-502	223	41	(	(	PUNCT
ejde-502	223	42	0	0	NUM
ejde-502	223	43	,	,	PUNCT
ejde-502	223	44	t	t	NOUN
ejde-502	223	45	)	)	PUNCT
ejde-502	223	46	.	.	PUNCT
ejde-502	224	1	we	we	PRON
ejde-502	224	2	formalize	formalize	VERB
ejde-502	224	3	this	this	PRON
ejde-502	224	4	below	below	ADV
ejde-502	224	5	in	in	ADP
ejde-502	224	6	dimension	dimension	NOUN
ejde-502	224	7	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	224	8	reaction	reaction	NOUN
ejde-502	224	9	-	-	PUNCT
ejde-502	224	10	diffusion	diffusion	NOUN
ejde-502	224	11	with	with	ADP
ejde-502	224	12	weighted	weight	VERB
ejde-502	224	13	strong	strong	ADJ
ejde-502	224	14	reaction	reaction	NOUN
ejde-502	224	15	11	11	NUM
ejde-502	224	16	n	n	NOUN
ejde-502	224	17	=	=	SYM
ejde-502	224	18	1	1	NUM
ejde-502	224	19	,	,	PUNCT
ejde-502	224	20	but	but	CCONJ
ejde-502	224	21	the	the	DET
ejde-502	224	22	result	result	NOUN
ejde-502	224	23	for	for	ADP
ejde-502	224	24	radially	radially	ADV
ejde-502	224	25	symmetric	symmetric	ADJ
ejde-502	224	26	solutions	solution	NOUN
ejde-502	224	27	in	in	ADP
ejde-502	224	28	dimension	dimension	NOUN
ejde-502	224	29	n	n	CCONJ
ejde-502	224	30	≥	≥	NUM
ejde-502	224	31	2	2	NUM
ejde-502	224	32	will	will	AUX
ejde-502	224	33	be	be	AUX
ejde-502	224	34	then	then	ADV
ejde-502	224	35	completely	completely	ADV
ejde-502	224	36	analogous	analogous	ADJ
ejde-502	224	37	.	.	PUNCT
ejde-502	225	1	proposition	proposition	NOUN
ejde-502	225	2	3.1	3.1	NUM
ejde-502	225	3	.	.	PUNCT
ejde-502	226	1	let	let	VERB
ejde-502	226	2	u0	u0	PROPN
ejde-502	226	3	∈	∈	PROPN
ejde-502	226	4	c0(r	c0(r	PROPN
ejde-502	226	5	)	)	PUNCT
ejde-502	226	6	be	be	VERB
ejde-502	226	7	a	a	DET
ejde-502	226	8	compactly	compactly	ADV
ejde-502	226	9	supported	support	VERB
ejde-502	226	10	initial	initial	ADJ
ejde-502	226	11	condition	condition	NOUN
ejde-502	226	12	such	such	ADJ
ejde-502	226	13	that	that	DET
ejde-502	226	14	suppu0	suppu0	NOUN
ejde-502	227	1	=	=	PUNCT
ejde-502	228	1	[	[	X
ejde-502	228	2	r0	r0	NOUN
ejde-502	228	3	,	,	PUNCT
ejde-502	228	4	r0	r0	NOUN
ejde-502	228	5	]	]	PUNCT
ejde-502	228	6	for	for	ADP
ejde-502	228	7	some	some	DET
ejde-502	228	8	r0	r0	NOUN
ejde-502	228	9	,	,	PUNCT
ejde-502	228	10	r0	r0	PROPN
ejde-502	228	11	∈	∈	PROPN
ejde-502	228	12	r.	r.	PROPN
ejde-502	228	13	let	let	VERB
ejde-502	228	14	m(u0	m(u0	NOUN
ejde-502	228	15	)	)	PUNCT
ejde-502	228	16	be	be	VERB
ejde-502	228	17	the	the	DET
ejde-502	228	18	minimal	minimal	ADJ
ejde-502	228	19	solution	solution	NOUN
ejde-502	228	20	to	to	ADP
ejde-502	228	21	the	the	DET
ejde-502	228	22	cauchy	cauchy	ADJ
ejde-502	228	23	problem	problem	NOUN
ejde-502	228	24	(	(	PUNCT
ejde-502	228	25	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	228	26	)	)	PUNCT
ejde-502	228	27	with	with	ADP
ejde-502	228	28	initial	initial	ADJ
ejde-502	228	29	condition	condition	NOUN
ejde-502	228	30	u0	u0	NOUN
ejde-502	228	31	defined	define	VERB
ejde-502	228	32	for	for	ADP
ejde-502	228	33	t	t	PROPN
ejde-502	228	34	∈	∈	PROPN
ejde-502	228	35	(	(	PUNCT
ejde-502	228	36	0	0	NUM
ejde-502	228	37	,	,	PUNCT
ejde-502	228	38	t0	t0	PROPN
ejde-502	228	39	)	)	PUNCT
ejde-502	228	40	and	and	CCONJ
ejde-502	228	41	denote	denote	VERB
ejde-502	228	42	by	by	ADP
ejde-502	228	43	sl(t	sl(t	NOUN
ejde-502	228	44	)	)	PUNCT
ejde-502	228	45	,	,	PUNCT
ejde-502	228	46	sr(t	sr(t	ADP
ejde-502	228	47	)	)	PUNCT
ejde-502	228	48	the	the	DET
ejde-502	228	49	left	left	ADJ
ejde-502	228	50	and	and	CCONJ
ejde-502	228	51	right	right	ADJ
ejde-502	228	52	interfaces	interface	NOUN
ejde-502	228	53	of	of	ADP
ejde-502	228	54	m(u0)(t	m(u0)(t	NOUN
ejde-502	228	55	)	)	PUNCT
ejde-502	228	56	for	for	ADP
ejde-502	228	57	t	t	PROPN
ejde-502	228	58	∈	∈	PROPN
ejde-502	228	59	(	(	PUNCT
ejde-502	228	60	0	0	NUM
ejde-502	228	61	,	,	PUNCT
ejde-502	228	62	t0	t0	PROPN
ejde-502	228	63	)	)	PUNCT
ejde-502	228	64	,	,	PUNCT
ejde-502	228	65	that	that	ADV
ejde-502	228	66	is	is	ADV
ejde-502	228	67	,	,	PUNCT
ejde-502	228	68	suppm(u0)(t	suppm(u0)(t	PROPN
ejde-502	228	69	)	)	PUNCT
ejde-502	228	70	=	=	PUNCT
ejde-502	229	1	[	[	X
ejde-502	229	2	sl(t	sl(t	X
ejde-502	229	3	)	)	PUNCT
ejde-502	229	4	,	,	PUNCT
ejde-502	229	5	sr(t	sr(t	NOUN
ejde-502	229	6	)	)	PUNCT
ejde-502	229	7	]	]	PUNCT
ejde-502	229	8	.	.	PUNCT
ejde-502	230	1	let	let	VERB
ejde-502	230	2	ξl(t	ξl(t	NOUN
ejde-502	230	3	)	)	PUNCT
ejde-502	230	4	,	,	PUNCT
ejde-502	230	5	ξr(t	ξr(t	NUM
ejde-502	230	6	)	)	PUNCT
ejde-502	230	7	be	be	AUX
ejde-502	230	8	two	two	NUM
ejde-502	230	9	continuous	continuous	ADJ
ejde-502	230	10	functions	function	NOUN
ejde-502	230	11	of	of	ADP
ejde-502	230	12	time	time	NOUN
ejde-502	230	13	such	such	ADJ
ejde-502	230	14	that	that	SCONJ
ejde-502	230	15	ξl(0	ξl(0	NOUN
ejde-502	230	16	)	)	PUNCT
ejde-502	230	17	≤	≤	NOUN
ejde-502	230	18	r0	r0	NOUN
ejde-502	230	19	,	,	PUNCT
ejde-502	230	20	ξr(0	ξr(0	PROPN
ejde-502	230	21	)	)	PUNCT
ejde-502	230	22	≥	≥	NOUN
ejde-502	230	23	r0	r0	NOUN
ejde-502	230	24	and	and	CCONJ
ejde-502	230	25	ξl(t1)−	ξl(t1)−	ADJ
ejde-502	230	26	ξl(t2	ξl(t2	NOUN
ejde-502	230	27	)	)	PUNCT
ejde-502	230	28	≥	≥	NOUN
ejde-502	230	29	sl(t1)−	sl(t1)−	ADJ
ejde-502	230	30	sl(t2	sl(t2	NOUN
ejde-502	230	31	)	)	PUNCT
ejde-502	230	32	,	,	PUNCT
ejde-502	230	33	ξr(t2)−	ξr(t2)−	NUM
ejde-502	230	34	ξr(t1	ξr(t1	NOUN
ejde-502	230	35	)	)	PUNCT
ejde-502	230	36	≥	≥	NOUN
ejde-502	230	37	sr(t2)−	sr(t2)−	VERB
ejde-502	230	38	sr(t1	sr(t1	NOUN
ejde-502	230	39	)	)	PUNCT
ejde-502	230	40	,	,	PUNCT
ejde-502	230	41	for	for	ADP
ejde-502	230	42	any	any	DET
ejde-502	230	43	t1	t1	NOUN
ejde-502	230	44	,	,	PUNCT
ejde-502	230	45	t2	t2	PROPN
ejde-502	230	46	∈	∈	PROPN
ejde-502	230	47	(	(	PUNCT
ejde-502	230	48	0	0	NUM
ejde-502	230	49	,	,	PUNCT
ejde-502	230	50	t0	t0	PROPN
ejde-502	230	51	)	)	PUNCT
ejde-502	230	52	such	such	ADJ
ejde-502	230	53	that	that	SCONJ
ejde-502	230	54	t1	t1	NOUN
ejde-502	230	55	<	<	X
ejde-502	230	56	t2	t2	PROPN
ejde-502	230	57	.	.	PUNCT
ejde-502	231	1	then	then	ADV
ejde-502	231	2	there	there	PRON
ejde-502	231	3	exists	exist	VERB
ejde-502	231	4	at	at	ADP
ejde-502	231	5	least	least	ADJ
ejde-502	231	6	a	a	DET
ejde-502	231	7	shorter	short	ADJ
ejde-502	231	8	time	time	NOUN
ejde-502	231	9	interval	interval	NOUN
ejde-502	231	10	(	(	PUNCT
ejde-502	231	11	0	0	NUM
ejde-502	231	12	,	,	PUNCT
ejde-502	231	13	t	t	NOUN
ejde-502	231	14	)	)	PUNCT
ejde-502	232	1	⊂	⊂	PROPN
ejde-502	232	2	(	(	PUNCT
ejde-502	232	3	0	0	NUM
ejde-502	232	4	,	,	PUNCT
ejde-502	232	5	t0	t0	PROPN
ejde-502	232	6	)	)	PUNCT
ejde-502	232	7	and	and	CCONJ
ejde-502	232	8	a	a	DET
ejde-502	232	9	solution	solution	NOUN
ejde-502	232	10	u	u	NOUN
ejde-502	232	11	to	to	ADP
ejde-502	232	12	the	the	DET
ejde-502	232	13	cauchy	cauchy	ADJ
ejde-502	232	14	problem	problem	NOUN
ejde-502	232	15	(	(	PUNCT
ejde-502	232	16	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	232	17	)	)	PUNCT
ejde-502	232	18	with	with	ADP
ejde-502	232	19	initial	initial	ADJ
ejde-502	232	20	condition	condition	NOUN
ejde-502	232	21	u0	u0	NOUN
ejde-502	232	22	defined	define	VERB
ejde-502	232	23	for	for	ADP
ejde-502	232	24	t	t	PROPN
ejde-502	232	25	∈	∈	PROPN
ejde-502	232	26	(	(	PUNCT
ejde-502	232	27	0	0	NUM
ejde-502	232	28	,	,	PUNCT
ejde-502	232	29	t	t	NOUN
ejde-502	232	30	)	)	PUNCT
ejde-502	232	31	such	such	ADJ
ejde-502	232	32	that	that	SCONJ
ejde-502	232	33	its	its	PRON
ejde-502	232	34	left	left	ADJ
ejde-502	232	35	and	and	CCONJ
ejde-502	232	36	right	right	ADJ
ejde-502	232	37	interfaces	interface	NOUN
ejde-502	232	38	are	be	AUX
ejde-502	232	39	given	give	VERB
ejde-502	232	40	exactly	exactly	ADV
ejde-502	232	41	by	by	ADP
ejde-502	232	42	ξl(t	ξl(t	NOUN
ejde-502	232	43	)	)	PUNCT
ejde-502	232	44	and	and	CCONJ
ejde-502	232	45	ξr(t	ξr(t	NUM
ejde-502	232	46	)	)	PUNCT
ejde-502	232	47	for	for	ADP
ejde-502	232	48	any	any	DET
ejde-502	232	49	t	t	NOUN
ejde-502	232	50	∈	∈	PROPN
ejde-502	232	51	(	(	PUNCT
ejde-502	232	52	0	0	NUM
ejde-502	232	53	,	,	PUNCT
ejde-502	232	54	t	t	NOUN
ejde-502	232	55	)	)	PUNCT
ejde-502	232	56	.	.	PUNCT
ejde-502	233	1	proof	proof	NOUN
ejde-502	233	2	.	.	PUNCT
ejde-502	234	1	we	we	PRON
ejde-502	234	2	divide	divide	VERB
ejde-502	234	3	the	the	DET
ejde-502	234	4	proof	proof	NOUN
ejde-502	234	5	into	into	ADP
ejde-502	234	6	two	two	NUM
ejde-502	234	7	steps	step	NOUN
ejde-502	234	8	.	.	PUNCT
ejde-502	235	1	step	step	NOUN
ejde-502	235	2	1	1	NUM
ejde-502	235	3	.	.	PUNCT
ejde-502	235	4	fundamental	fundamental	ADJ
ejde-502	235	5	extension	extension	NOUN
ejde-502	235	6	.	.	PUNCT
ejde-502	236	1	this	this	DET
ejde-502	236	2	step	step	NOUN
ejde-502	236	3	adapts	adapt	VERB
ejde-502	236	4	to	to	ADP
ejde-502	236	5	our	our	PRON
ejde-502	236	6	problem	problem	NOUN
ejde-502	236	7	step	step	NOUN
ejde-502	236	8	1	1	NUM
ejde-502	236	9	in	in	ADP
ejde-502	236	10	[	[	X
ejde-502	236	11	28	28	NUM
ejde-502	236	12	,	,	PUNCT
ejde-502	236	13	proof	proof	NOUN
ejde-502	236	14	of	of	ADP
ejde-502	236	15	corollary	corollary	ADJ
ejde-502	236	16	1.2	1.2	NUM
ejde-502	236	17	]	]	PUNCT
ejde-502	236	18	and	and	CCONJ
ejde-502	236	19	at	at	ADP
ejde-502	236	20	its	its	PRON
ejde-502	236	21	end	end	NOUN
ejde-502	236	22	in	in	ADP
ejde-502	236	23	fact	fact	NOUN
ejde-502	236	24	we	we	PRON
ejde-502	236	25	already	already	ADV
ejde-502	236	26	have	have	VERB
ejde-502	236	27	the	the	DET
ejde-502	236	28	proof	proof	NOUN
ejde-502	236	29	of	of	ADP
ejde-502	236	30	theorem	theorem	NOUN
ejde-502	236	31	1.3	1.3	NUM
ejde-502	236	32	.	.	PUNCT
ejde-502	237	1	the	the	DET
ejde-502	237	2	goal	goal	NOUN
ejde-502	237	3	here	here	ADV
ejde-502	237	4	is	be	AUX
ejde-502	237	5	to	to	PART
ejde-502	237	6	construct	construct	VERB
ejde-502	237	7	a	a	DET
ejde-502	237	8	solution	solution	NOUN
ejde-502	237	9	to	to	ADP
ejde-502	237	10	our	our	PRON
ejde-502	237	11	cauchy	cauchy	ADJ
ejde-502	237	12	problem	problem	NOUN
ejde-502	237	13	whose	whose	DET
ejde-502	237	14	support	support	NOUN
ejde-502	237	15	has	have	VERB
ejde-502	237	16	an	an	DET
ejde-502	237	17	instantaneous	instantaneous	ADJ
ejde-502	237	18	jump	jump	NOUN
ejde-502	237	19	to	to	ADP
ejde-502	237	20	the	the	DET
ejde-502	237	21	right	right	NOUN
ejde-502	237	22	from	from	ADP
ejde-502	237	23	r0	r0	NOUN
ejde-502	237	24	=	=	SYM
ejde-502	237	25	sr(0	sr(0	PROPN
ejde-502	237	26	)	)	PUNCT
ejde-502	237	27	to	to	PART
ejde-502	237	28	r0	r0	VERB
ejde-502	237	29	+	+	CCONJ
ejde-502	237	30	r	r	NOUN
ejde-502	237	31	at	at	ADP
ejde-502	237	32	time	time	NOUN
ejde-502	237	33	t	t	PROPN
ejde-502	237	34	=	=	SYM
ejde-502	237	35	0	0	NUM
ejde-502	237	36	for	for	ADP
ejde-502	237	37	a	a	DET
ejde-502	237	38	given	give	VERB
ejde-502	237	39	(	(	PUNCT
ejde-502	237	40	fixed	fix	VERB
ejde-502	237	41	)	)	PUNCT
ejde-502	237	42	r	r	NOUN
ejde-502	237	43	>	>	X
ejde-502	237	44	0	0	PUNCT
ejde-502	238	1	(	(	PUNCT
ejde-502	238	2	and	and	CCONJ
ejde-502	238	3	of	of	ADP
ejde-502	238	4	course	course	NOUN
ejde-502	238	5	,	,	PUNCT
ejde-502	238	6	a	a	DET
ejde-502	238	7	perfectly	perfectly	ADV
ejde-502	238	8	similar	similar	ADJ
ejde-502	238	9	construction	construction	NOUN
ejde-502	238	10	can	can	AUX
ejde-502	238	11	be	be	AUX
ejde-502	238	12	done	do	VERB
ejde-502	238	13	for	for	ADP
ejde-502	238	14	the	the	DET
ejde-502	238	15	left	left	ADJ
ejde-502	238	16	interface	interface	NOUN
ejde-502	238	17	)	)	PUNCT
ejde-502	238	18	.	.	PUNCT
ejde-502	239	1	let	let	VERB
ejde-502	239	2	then	then	ADV
ejde-502	239	3	r	r	VERB
ejde-502	239	4	>	>	X
ejde-502	239	5	0	0	NUM
ejde-502	239	6	be	be	AUX
ejde-502	239	7	given	give	VERB
ejde-502	239	8	.	.	PUNCT
ejde-502	240	1	for	for	ADP
ejde-502	240	2	any	any	DET
ejde-502	240	3	ε	ε	PROPN
ejde-502	240	4	>	>	X
ejde-502	240	5	0	0	PUNCT
ejde-502	240	6	sufficiently	sufficiently	ADV
ejde-502	240	7	small	small	ADJ
ejde-502	240	8	(	(	PUNCT
ejde-502	240	9	it	it	PRON
ejde-502	240	10	is	be	AUX
ejde-502	240	11	enough	enough	ADJ
ejde-502	240	12	to	to	PART
ejde-502	240	13	start	start	VERB
ejde-502	240	14	for	for	ADP
ejde-502	240	15	example	example	NOUN
ejde-502	240	16	from	from	ADP
ejde-502	240	17	ε	ε	PROPN
ejde-502	240	18	<	<	X
ejde-502	240	19	‖u0‖∞/2	‖u0‖∞/2	PROPN
ejde-502	240	20	)	)	PUNCT
ejde-502	240	21	there	there	PRON
ejde-502	240	22	exists	exist	VERB
ejde-502	240	23	a	a	DET
ejde-502	240	24	last	last	ADJ
ejde-502	240	25	,	,	PUNCT
ejde-502	240	26	closest	close	ADJ
ejde-502	240	27	point	point	NOUN
ejde-502	240	28	(	(	PUNCT
ejde-502	240	29	that	that	SCONJ
ejde-502	240	30	we	we	PRON
ejde-502	240	31	denote	denote	VERB
ejde-502	240	32	by	by	ADP
ejde-502	240	33	rε	rε	NOUN
ejde-502	240	34	)	)	PUNCT
ejde-502	240	35	to	to	ADP
ejde-502	240	36	the	the	DET
ejde-502	240	37	right	right	ADJ
ejde-502	240	38	interface	interface	NOUN
ejde-502	240	39	r0	r0	NOUN
ejde-502	240	40	of	of	ADP
ejde-502	240	41	u0	u0	ADJ
ejde-502	240	42	such	such	ADJ
ejde-502	240	43	that	that	SCONJ
ejde-502	240	44	u0(rε	u0(rε	NOUN
ejde-502	240	45	)	)	PUNCT
ejde-502	240	46	=	=	SYM
ejde-502	240	47	ε	ε	PROPN
ejde-502	240	48	.	.	PUNCT
ejde-502	240	49	let	let	VERB
ejde-502	240	50	us	we	PRON
ejde-502	240	51	introduce	introduce	VERB
ejde-502	240	52	the	the	DET
ejde-502	240	53	following	follow	VERB
ejde-502	240	54	compactly	compactly	ADV
ejde-502	240	55	supported	support	VERB
ejde-502	240	56	and	and	CCONJ
ejde-502	240	57	continuous	continuous	ADJ
ejde-502	240	58	initial	initial	ADJ
ejde-502	240	59	condition	condition	NOUN
ejde-502	240	60	uε,0(x	uε,0(x	NOUN
ejde-502	240	61	)	)	PUNCT
ejde-502	241	1	=	=	PUNCT
ejde-502	242	1			PRON
ejde-502	242	2	u0(x	u0(x	NUM
ejde-502	242	3	)	)	PUNCT
ejde-502	242	4	,	,	PUNCT
ejde-502	242	5	for	for	ADP
ejde-502	242	6	x	x	PROPN
ejde-502	242	7	∈	∈	PROPN
ejde-502	242	8	(	(	PUNCT
ejde-502	242	9	−∞	−∞	NOUN
ejde-502	242	10	,	,	PUNCT
ejde-502	242	11	rε	rε	NOUN
ejde-502	242	12	)	)	PUNCT
ejde-502	242	13	,	,	PUNCT
ejde-502	242	14	ε(r0+r−x	ε(r0+r−x	NOUN
ejde-502	242	15	)	)	PUNCT
ejde-502	242	16	r0+r−rε	r0+r−rε	NOUN
ejde-502	242	17	,	,	PUNCT
ejde-502	242	18	for	for	ADP
ejde-502	242	19	x	x	PROPN
ejde-502	242	20	∈	∈	PROPN
ejde-502	242	21	[	[	X
ejde-502	242	22	rε	rε	NOUN
ejde-502	242	23	,	,	PUNCT
ejde-502	242	24	r0	r0	NOUN
ejde-502	242	25	+	+	CCONJ
ejde-502	242	26	r	r	X
ejde-502	242	27	]	]	X
ejde-502	242	28	,	,	PUNCT
ejde-502	242	29	0	0	NUM
ejde-502	242	30	,	,	PUNCT
ejde-502	242	31	for	for	SCONJ
ejde-502	242	32	x	x	PROPN
ejde-502	242	33	∈	∈	PROPN
ejde-502	242	34	(	(	PUNCT
ejde-502	242	35	r0	r0	NOUN
ejde-502	242	36	+	+	CCONJ
ejde-502	242	37	r,∞	r,∞	NOUN
ejde-502	242	38	)	)	PUNCT
ejde-502	242	39	,	,	PUNCT
ejde-502	242	40	(	(	PUNCT
ejde-502	242	41	3.2	3.2	NUM
ejde-502	242	42	)	)	PUNCT
ejde-502	242	43	that	that	PRON
ejde-502	242	44	is	be	AUX
ejde-502	242	45	,	,	PUNCT
ejde-502	242	46	adding	add	VERB
ejde-502	242	47	a	a	DET
ejde-502	242	48	linear	linear	ADJ
ejde-502	242	49	extension	extension	NOUN
ejde-502	242	50	to	to	PART
ejde-502	242	51	u0	u0	VERB
ejde-502	242	52	from	from	ADP
ejde-502	242	53	rε	rε	PRON
ejde-502	242	54	to	to	ADP
ejde-502	242	55	the	the	DET
ejde-502	242	56	new	new	ADJ
ejde-502	242	57	edge	edge	NOUN
ejde-502	242	58	of	of	ADP
ejde-502	242	59	the	the	DET
ejde-502	242	60	support	support	NOUN
ejde-502	242	61	r0	r0	NOUN
ejde-502	242	62	+	+	CCONJ
ejde-502	242	63	r.	r.	PROPN
ejde-502	242	64	let	let	VERB
ejde-502	242	65	uε	uε	PROPN
ejde-502	242	66	=	=	PUNCT
ejde-502	242	67	m(uε,0	m(uε,0	PROPN
ejde-502	242	68	)	)	PUNCT
ejde-502	242	69	be	be	VERB
ejde-502	242	70	the	the	DET
ejde-502	242	71	minimal	minimal	ADJ
ejde-502	242	72	solution	solution	NOUN
ejde-502	242	73	associated	associate	VERB
ejde-502	242	74	to	to	ADP
ejde-502	242	75	this	this	DET
ejde-502	242	76	new	new	ADJ
ejde-502	242	77	condition	condition	NOUN
ejde-502	242	78	.	.	PUNCT
ejde-502	243	1	by	by	ADP
ejde-502	243	2	comparison	comparison	NOUN
ejde-502	243	3	between	between	ADP
ejde-502	243	4	minimal	minimal	ADJ
ejde-502	243	5	solutions	solution	NOUN
ejde-502	243	6	(	(	PUNCT
ejde-502	243	7	which	which	PRON
ejde-502	243	8	holds	hold	VERB
ejde-502	243	9	since	since	SCONJ
ejde-502	243	10	it	it	PRON
ejde-502	243	11	is	be	AUX
ejde-502	243	12	obvious	obvious	ADJ
ejde-502	243	13	that	that	SCONJ
ejde-502	243	14	any	any	DET
ejde-502	243	15	solution	solution	NOUN
ejde-502	243	16	to	to	ADP
ejde-502	243	17	(	(	PUNCT
ejde-502	243	18	1.1	1.1	NUM
ejde-502	243	19	)	)	PUNCT
ejde-502	243	20	with	with	ADP
ejde-502	243	21	a	a	DET
ejde-502	243	22	larger	large	ADJ
ejde-502	243	23	initial	initial	ADJ
ejde-502	243	24	condition	condition	NOUN
ejde-502	243	25	than	than	SCONJ
ejde-502	243	26	the	the	DET
ejde-502	243	27	given	give	VERB
ejde-502	243	28	u0	u0	NOUN
ejde-502	243	29	becomes	become	VERB
ejde-502	243	30	a	a	DET
ejde-502	243	31	supersolution	supersolution	NOUN
ejde-502	243	32	to	to	ADP
ejde-502	243	33	any	any	PRON
ejde-502	243	34	of	of	ADP
ejde-502	243	35	the	the	DET
ejde-502	243	36	problems	problem	NOUN
ejde-502	243	37	(	(	PUNCT
ejde-502	243	38	2.1	2.1	NUM
ejde-502	243	39	)	)	PUNCT
ejde-502	243	40	approximating	approximate	VERB
ejde-502	243	41	the	the	DET
ejde-502	243	42	minimal	minimal	ADJ
ejde-502	243	43	solution	solution	NOUN
ejde-502	243	44	m(u0	m(u0	NOUN
ejde-502	243	45	)	)	PUNCT
ejde-502	243	46	)	)	PUNCT
ejde-502	244	1	it	it	PRON
ejde-502	244	2	readily	readily	ADV
ejde-502	244	3	follows	follow	VERB
ejde-502	244	4	that	that	SCONJ
ejde-502	244	5	uε2(x	uε2(x	PROPN
ejde-502	244	6	,	,	PUNCT
ejde-502	244	7	t	t	PROPN
ejde-502	244	8	)	)	PUNCT
ejde-502	244	9	≥	≥	NOUN
ejde-502	244	10	uε1(x	uε1(x	NOUN
ejde-502	244	11	,	,	PUNCT
ejde-502	244	12	t	t	PROPN
ejde-502	244	13	)	)	PUNCT
ejde-502	244	14	for	for	ADP
ejde-502	244	15	any	any	DET
ejde-502	244	16	ε2	ε2	NOUN
ejde-502	244	17	>	>	X
ejde-502	244	18	ε1	ε1	VERB
ejde-502	244	19	>	>	X
ejde-502	244	20	0	0	PUNCT
ejde-502	245	1	and	and	CCONJ
ejde-502	245	2	at	at	ADP
ejde-502	245	3	any	any	DET
ejde-502	245	4	(	(	PUNCT
ejde-502	245	5	x	x	NOUN
ejde-502	245	6	,	,	PUNCT
ejde-502	245	7	t	t	PROPN
ejde-502	245	8	)	)	PUNCT
ejde-502	245	9	∈	∈	PROPN
ejde-502	245	10	r	r	NOUN
ejde-502	245	11	×	×	NOUN
ejde-502	245	12	(	(	PUNCT
ejde-502	245	13	0	0	NUM
ejde-502	245	14	,	,	PUNCT
ejde-502	245	15	t	t	PROPN
ejde-502	245	16	)	)	PUNCT
ejde-502	245	17	where	where	SCONJ
ejde-502	245	18	t	t	PROPN
ejde-502	245	19	>	>	X
ejde-502	245	20	0	0	NUM
ejde-502	245	21	is	be	AUX
ejde-502	245	22	for	for	ADP
ejde-502	245	23	example	example	NOUN
ejde-502	245	24	the	the	DET
ejde-502	245	25	lifetime	lifetime	NOUN
ejde-502	245	26	of	of	ADP
ejde-502	245	27	the	the	DET
ejde-502	245	28	solution	solution	NOUN
ejde-502	245	29	with	with	ADP
ejde-502	245	30	the	the	DET
ejde-502	245	31	biggest	big	ADJ
ejde-502	245	32	ε	ε	PROPN
ejde-502	245	33	chosen	choose	VERB
ejde-502	245	34	.	.	PUNCT
ejde-502	246	1	recalling	recall	VERB
ejde-502	246	2	the	the	DET
ejde-502	246	3	construction	construction	NOUN
ejde-502	246	4	of	of	ADP
ejde-502	246	5	the	the	DET
ejde-502	246	6	absolute	absolute	ADJ
ejde-502	246	7	minimal	minimal	ADJ
ejde-502	246	8	solution	solution	NOUN
ejde-502	246	9	e(x	e(x	NUM
ejde-502	246	10	,	,	PUNCT
ejde-502	246	11	t	t	PROPN
ejde-502	246	12	)	)	PUNCT
ejde-502	246	13	to	to	ADP
ejde-502	246	14	(	(	PUNCT
ejde-502	246	15	1.5	1.5	NUM
ejde-502	246	16	)	)	PUNCT
ejde-502	246	17	(	(	PUNCT
ejde-502	246	18	see	see	VERB
ejde-502	246	19	the	the	DET
ejde-502	246	20	details	detail	NOUN
ejde-502	246	21	in	in	ADP
ejde-502	246	22	[	[	X
ejde-502	246	23	26	26	NUM
ejde-502	246	24	,	,	PUNCT
ejde-502	246	25	theorem	theorem	VERB
ejde-502	246	26	4	4	NUM
ejde-502	246	27	]	]	PUNCT
ejde-502	246	28	)	)	PUNCT
ejde-502	246	29	and	and	CCONJ
ejde-502	246	30	the	the	DET
ejde-502	246	31	fact	fact	NOUN
ejde-502	246	32	that	that	SCONJ
ejde-502	246	33	any	any	DET
ejde-502	246	34	non	non	ADJ
ejde-502	246	35	-	-	ADJ
ejde-502	246	36	trivial	trivial	ADJ
ejde-502	246	37	solution	solution	NOUN
ejde-502	246	38	to	to	ADP
ejde-502	246	39	our	our	PRON
ejde-502	246	40	eq	eq	NOUN
ejde-502	246	41	.	.	PUNCT
ejde-502	247	1	(	(	PUNCT
ejde-502	247	2	1.1	1.1	NUM
ejde-502	247	3	)	)	PUNCT
ejde-502	247	4	is	be	AUX
ejde-502	247	5	a	a	DET
ejde-502	247	6	strict	strict	ADJ
ejde-502	247	7	supersolution	supersolution	NOUN
ejde-502	247	8	to	to	ADP
ejde-502	247	9	(	(	PUNCT
ejde-502	247	10	1.5	1.5	NUM
ejde-502	247	11	)	)	PUNCT
ejde-502	247	12	we	we	PRON
ejde-502	247	13	easily	easily	ADV
ejde-502	247	14	conclude	conclude	VERB
ejde-502	247	15	that	that	SCONJ
ejde-502	247	16	for	for	ADP
ejde-502	247	17	any	any	DET
ejde-502	247	18	(	(	PUNCT
ejde-502	247	19	x0	x0	PROPN
ejde-502	247	20	,	,	PUNCT
ejde-502	247	21	t0	t0	PROPN
ejde-502	247	22	)	)	PUNCT
ejde-502	247	23	∈	∈	PROPN
ejde-502	247	24	r×	r×	NOUN
ejde-502	247	25	(	(	PUNCT
ejde-502	247	26	0	0	NUM
ejde-502	247	27	,	,	PUNCT
ejde-502	247	28	t	t	NOUN
ejde-502	247	29	)	)	PUNCT
ejde-502	247	30	with	with	ADP
ejde-502	247	31	x0	x0	PROPN
ejde-502	247	32	∈	∈	PROPN
ejde-502	247	33	[	[	X
ejde-502	247	34	rε	rε	NOUN
ejde-502	247	35	,	,	PUNCT
ejde-502	247	36	r0	r0	NOUN
ejde-502	247	37	+	+	CCONJ
ejde-502	247	38	r	r	X
ejde-502	247	39	]	]	X
ejde-502	247	40	we	we	PRON
ejde-502	247	41	have	have	VERB
ejde-502	247	42	uε(x	uε(x	NOUN
ejde-502	247	43	,	,	PUNCT
ejde-502	247	44	t	t	PROPN
ejde-502	247	45	)	)	PUNCT
ejde-502	247	46	≥	≥	NOUN
ejde-502	247	47	e(x−	e(x−	PROPN
ejde-502	247	48	x0	x0	PROPN
ejde-502	247	49	,	,	PUNCT
ejde-502	247	50	t−	t−	PROPN
ejde-502	247	51	t0	t0	PROPN
ejde-502	247	52	)	)	PUNCT
ejde-502	247	53	,	,	PUNCT
ejde-502	247	54	t	t	PROPN
ejde-502	247	55	>	>	X
ejde-502	247	56	t0	t0	PROPN
ejde-502	247	57	.	.	PUNCT
ejde-502	248	1	(	(	PUNCT
ejde-502	248	2	3.3	3.3	NUM
ejde-502	248	3	)	)	PUNCT
ejde-502	248	4	we	we	PRON
ejde-502	248	5	can	can	AUX
ejde-502	248	6	then	then	ADV
ejde-502	248	7	define	define	VERB
ejde-502	248	8	u(x	u(x	NOUN
ejde-502	248	9	,	,	PUNCT
ejde-502	248	10	t	t	NOUN
ejde-502	248	11	)	)	PUNCT
ejde-502	248	12	=	=	SYM
ejde-502	249	1	lim	lim	PROPN
ejde-502	249	2	ε→0	ε→0	NOUN
ejde-502	249	3	uε(x	uε(x	SYM
ejde-502	249	4	,	,	PUNCT
ejde-502	249	5	t	t	PROPN
ejde-502	249	6	)	)	PUNCT
ejde-502	249	7	,	,	PUNCT
ejde-502	249	8	(	(	PUNCT
ejde-502	249	9	x	x	X
ejde-502	249	10	,	,	PUNCT
ejde-502	249	11	t	t	PROPN
ejde-502	249	12	)	)	PUNCT
ejde-502	249	13	∈	∈	PROPN
ejde-502	249	14	r×	r×	NOUN
ejde-502	249	15	(	(	PUNCT
ejde-502	249	16	0	0	NUM
ejde-502	249	17	,	,	PUNCT
ejde-502	249	18	t	t	PROPN
ejde-502	249	19	)	)	PUNCT
ejde-502	249	20	,	,	PUNCT
ejde-502	249	21	which	which	PRON
ejde-502	249	22	is	be	AUX
ejde-502	249	23	a	a	DET
ejde-502	249	24	monotone	monotone	ADJ
ejde-502	249	25	limit	limit	NOUN
ejde-502	250	1	and	and	CCONJ
ejde-502	250	2	it	it	PRON
ejde-502	250	3	is	be	AUX
ejde-502	250	4	easy	easy	ADJ
ejde-502	250	5	to	to	PART
ejde-502	250	6	see	see	VERB
ejde-502	250	7	that	that	SCONJ
ejde-502	250	8	u	u	NOUN
ejde-502	250	9	is	be	AUX
ejde-502	250	10	a	a	DET
ejde-502	250	11	weak	weak	ADJ
ejde-502	250	12	solution	solution	NOUN
ejde-502	250	13	to	to	ADP
ejde-502	250	14	(	(	PUNCT
ejde-502	250	15	1.1	1.1	NUM
ejde-502	250	16	)	)	PUNCT
ejde-502	250	17	using	use	VERB
ejde-502	250	18	the	the	DET
ejde-502	250	19	monotone	monotone	ADJ
ejde-502	250	20	convergence	convergence	NOUN
ejde-502	250	21	theorem	theorem	NOUN
ejde-502	250	22	(	(	PUNCT
ejde-502	250	23	in	in	ADP
ejde-502	250	24	fact	fact	NOUN
ejde-502	250	25	we	we	PRON
ejde-502	250	26	even	even	ADV
ejde-502	250	27	have	have	AUX
ejde-502	250	28	uniform	uniform	ADJ
ejde-502	250	29	convergence	convergence	NOUN
ejde-502	250	30	by	by	ADP
ejde-502	250	31	dini	dini	NOUN
ejde-502	250	32	’s	’s	PART
ejde-502	250	33	theorem	theorem	NOUN
ejde-502	250	34	since	since	SCONJ
ejde-502	250	35	the	the	DET
ejde-502	250	36	limit	limit	NOUN
ejde-502	250	37	is	be	AUX
ejde-502	250	38	continuous	continuous	ADJ
ejde-502	250	39	)	)	PUNCT
ejde-502	250	40	.	.	PUNCT
ejde-502	251	1	let	let	VERB
ejde-502	251	2	us	we	PRON
ejde-502	251	3	stress	stress	VERB
ejde-502	251	4	here	here	ADV
ejde-502	251	5	that	that	SCONJ
ejde-502	251	6	we	we	PRON
ejde-502	251	7	do	do	VERB
ejde-502	251	8	not	not	PART
ejde-502	251	9	12	12	NUM
ejde-502	251	10	r.	r.	PROPN
ejde-502	251	11	g.	g.	PROPN
ejde-502	251	12	iagar	iagar	PROPN
ejde-502	251	13	,	,	PUNCT
ejde-502	251	14	a.	a.	NOUN
ejde-502	251	15	i.	i.	PROPN
ejde-502	251	16	muñoz	muñoz	PROPN
ejde-502	251	17	,	,	PUNCT
ejde-502	251	18	a.	a.	NOUN
ejde-502	251	19	sánchez	sánchez	PROPN
ejde-502	251	20	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	251	21	need	need	VERB
ejde-502	251	22	in	in	ADP
ejde-502	251	23	this	this	DET
ejde-502	251	24	construction	construction	NOUN
ejde-502	251	25	a	a	DET
ejde-502	251	26	bound	bind	VERB
ejde-502	251	27	from	from	ADP
ejde-502	251	28	above	above	ADV
ejde-502	251	29	to	to	PART
ejde-502	251	30	guarantee	guarantee	VERB
ejde-502	251	31	that	that	SCONJ
ejde-502	251	32	the	the	DET
ejde-502	251	33	limit	limit	NOUN
ejde-502	251	34	is	be	AUX
ejde-502	251	35	finite	finite	ADJ
ejde-502	251	36	,	,	PUNCT
ejde-502	251	37	since	since	SCONJ
ejde-502	251	38	we	we	PRON
ejde-502	251	39	are	be	AUX
ejde-502	251	40	dealing	deal	VERB
ejde-502	251	41	with	with	ADP
ejde-502	251	42	a	a	DET
ejde-502	251	43	decreasing	decrease	VERB
ejde-502	251	44	limit	limit	NOUN
ejde-502	251	45	.	.	PUNCT
ejde-502	252	1	it	it	PRON
ejde-502	252	2	is	be	AUX
ejde-502	252	3	then	then	ADV
ejde-502	252	4	obvious	obvious	ADJ
ejde-502	252	5	that	that	SCONJ
ejde-502	252	6	u(x	u(x	NOUN
ejde-502	252	7	,	,	PUNCT
ejde-502	252	8	0	0	NUM
ejde-502	252	9	)	)	PUNCT
ejde-502	253	1	=	=	VERB
ejde-502	253	2	lim	lim	NOUN
ejde-502	253	3	ε→0	ε→0	ADJ
ejde-502	253	4	uε,0(x	uε,0(x	NOUN
ejde-502	253	5	)	)	PUNCT
ejde-502	253	6	=	=	SYM
ejde-502	253	7	u0(x	u0(x	NOUN
ejde-502	253	8	)	)	PUNCT
ejde-502	253	9	and	and	CCONJ
ejde-502	253	10	the	the	DET
ejde-502	253	11	comparison	comparison	NOUN
ejde-502	253	12	from	from	ADP
ejde-502	253	13	above	above	ADV
ejde-502	253	14	with	with	ADP
ejde-502	253	15	e	e	NOUN
ejde-502	253	16	in	in	ADP
ejde-502	253	17	(	(	PUNCT
ejde-502	253	18	3.3	3.3	NUM
ejde-502	253	19	)	)	PUNCT
ejde-502	253	20	transfers	transfer	NOUN
ejde-502	253	21	to	to	ADP
ejde-502	253	22	the	the	DET
ejde-502	253	23	limit	limit	NOUN
ejde-502	253	24	u	u	NOUN
ejde-502	253	25	,	,	PUNCT
ejde-502	253	26	proving	prove	VERB
ejde-502	253	27	that	that	SCONJ
ejde-502	253	28	for	for	ADP
ejde-502	253	29	any	any	DET
ejde-502	253	30	t	t	NOUN
ejde-502	253	31	>	>	X
ejde-502	253	32	0	0	NUM
ejde-502	253	33	we	we	PRON
ejde-502	253	34	have	have	VERB
ejde-502	253	35	u(x	u(x	NOUN
ejde-502	253	36	,	,	PUNCT
ejde-502	253	37	t	t	PROPN
ejde-502	253	38	)	)	PUNCT
ejde-502	253	39	>	>	X
ejde-502	253	40	0	0	PUNCT
ejde-502	254	1	for	for	ADP
ejde-502	254	2	x	x	PROPN
ejde-502	254	3	∈	∈	PROPN
ejde-502	254	4	(	(	PUNCT
ejde-502	254	5	r0	r0	NOUN
ejde-502	254	6	,	,	PUNCT
ejde-502	254	7	r0	r0	NOUN
ejde-502	254	8	+	+	CCONJ
ejde-502	254	9	r	r	NOUN
ejde-502	254	10	)	)	PUNCT
ejde-502	254	11	.	.	PUNCT
ejde-502	255	1	step	step	NOUN
ejde-502	255	2	2	2	NUM
ejde-502	255	3	.	.	PUNCT
ejde-502	255	4	iterative	iterative	NOUN
ejde-502	255	5	construction	construction	NOUN
ejde-502	255	6	.	.	PUNCT
ejde-502	256	1	we	we	PRON
ejde-502	256	2	perform	perform	VERB
ejde-502	256	3	now	now	ADV
ejde-502	256	4	an	an	DET
ejde-502	256	5	iterative	iterative	NOUN
ejde-502	256	6	construction	construction	NOUN
ejde-502	256	7	based	base	VERB
ejde-502	256	8	on	on	ADP
ejde-502	256	9	a	a	DET
ejde-502	256	10	discretization	discretization	NOUN
ejde-502	256	11	in	in	ADP
ejde-502	256	12	time	time	NOUN
ejde-502	256	13	and	and	CCONJ
ejde-502	256	14	the	the	DET
ejde-502	256	15	application	application	NOUN
ejde-502	256	16	of	of	ADP
ejde-502	256	17	step	step	NOUN
ejde-502	256	18	1	1	NUM
ejde-502	256	19	in	in	ADP
ejde-502	256	20	each	each	DET
ejde-502	256	21	interval	interval	NOUN
ejde-502	256	22	of	of	ADP
ejde-502	256	23	this	this	DET
ejde-502	256	24	discretization	discretization	NOUN
ejde-502	256	25	to	to	PART
ejde-502	256	26	perform	perform	VERB
ejde-502	256	27	an	an	DET
ejde-502	256	28	instantaneous	instantaneous	ADJ
ejde-502	256	29	jump	jump	NOUN
ejde-502	256	30	in	in	ADP
ejde-502	256	31	the	the	DET
ejde-502	256	32	supports	support	NOUN
ejde-502	256	33	.	.	PUNCT
ejde-502	257	1	let	let	AUX
ejde-502	257	2	then	then	ADV
ejde-502	257	3	ξl(t	ξl(t	NOUN
ejde-502	257	4	)	)	PUNCT
ejde-502	257	5	and	and	CCONJ
ejde-502	257	6	ξr(t	ξr(t	NUM
ejde-502	257	7	)	)	PUNCT
ejde-502	257	8	be	be	AUX
ejde-502	257	9	two	two	NUM
ejde-502	257	10	continuous	continuous	ADJ
ejde-502	257	11	,	,	PUNCT
ejde-502	257	12	increasing	increase	VERB
ejde-502	257	13	functions	function	NOUN
ejde-502	257	14	of	of	ADP
ejde-502	257	15	time	time	NOUN
ejde-502	257	16	as	as	ADP
ejde-502	257	17	in	in	ADP
ejde-502	257	18	the	the	DET
ejde-502	257	19	statement	statement	NOUN
ejde-502	257	20	of	of	ADP
ejde-502	257	21	proposition	proposition	NOUN
ejde-502	257	22	3.1	3.1	NUM
ejde-502	257	23	.	.	PUNCT
ejde-502	258	1	fix	fix	VERB
ejde-502	258	2	a	a	DET
ejde-502	258	3	time	time	NOUN
ejde-502	258	4	t	t	X
ejde-502	258	5	∈	∈	PROPN
ejde-502	258	6	(	(	PUNCT
ejde-502	258	7	0	0	NUM
ejde-502	258	8	,	,	PUNCT
ejde-502	258	9	t	t	PROPN
ejde-502	258	10	)	)	PUNCT
ejde-502	258	11	for	for	ADP
ejde-502	258	12	a	a	DET
ejde-502	258	13	t	t	NOUN
ejde-502	258	14	>	>	X
ejde-502	258	15	0	0	PUNCT
ejde-502	258	16	sufficiently	sufficiently	ADV
ejde-502	258	17	small	small	ADJ
ejde-502	258	18	(	(	PUNCT
ejde-502	258	19	that	that	PRON
ejde-502	258	20	will	will	AUX
ejde-502	258	21	be	be	AUX
ejde-502	258	22	chosen	choose	VERB
ejde-502	258	23	later	later	ADV
ejde-502	258	24	)	)	PUNCT
ejde-502	258	25	.	.	PUNCT
ejde-502	259	1	for	for	ADP
ejde-502	259	2	any	any	DET
ejde-502	259	3	positive	positive	ADJ
ejde-502	259	4	integer	integer	NOUN
ejde-502	259	5	n	n	CCONJ
ejde-502	259	6	we	we	PRON
ejde-502	259	7	construct	construct	VERB
ejde-502	259	8	an	an	DET
ejde-502	259	9	approximate	approximate	ADJ
ejde-502	259	10	solution	solution	NOUN
ejde-502	259	11	un	un	PROPN
ejde-502	259	12	as	as	ADP
ejde-502	259	13	in	in	ADP
ejde-502	259	14	[	[	X
ejde-502	259	15	28	28	NUM
ejde-502	259	16	,	,	PUNCT
ejde-502	259	17	proof	proof	NOUN
ejde-502	259	18	of	of	ADP
ejde-502	259	19	corollary	corollary	ADJ
ejde-502	259	20	1.2	1.2	NUM
ejde-502	259	21	]	]	PUNCT
ejde-502	259	22	.	.	PUNCT
ejde-502	260	1	we	we	PRON
ejde-502	260	2	briefly	briefly	VERB
ejde-502	260	3	and	and	CCONJ
ejde-502	260	4	sketchy	sketchy	ADJ
ejde-502	260	5	describe	describe	VERB
ejde-502	260	6	the	the	DET
ejde-502	260	7	construction	construction	NOUN
ejde-502	260	8	here	here	ADV
ejde-502	260	9	for	for	ADP
ejde-502	260	10	the	the	DET
ejde-502	260	11	sake	sake	NOUN
ejde-502	260	12	of	of	ADP
ejde-502	260	13	completeness	completeness	NOUN
ejde-502	260	14	.	.	PUNCT
ejde-502	261	1	consider	consider	VERB
ejde-502	261	2	a	a	DET
ejde-502	261	3	partition	partition	NOUN
ejde-502	261	4	0	0	NUM
ejde-502	262	1	=	=	SYM
ejde-502	262	2	t0	t0	PROPN
ejde-502	262	3	<	<	X
ejde-502	262	4	t1	t1	NOUN
ejde-502	262	5	<	<	X
ejde-502	262	6	t2	t2	PROPN
ejde-502	262	7	<	<	X
ejde-502	262	8	·	·	PUNCT
ejde-502	262	9	·	·	PUNCT
ejde-502	262	10	·	·	PUNCT
ejde-502	263	1	<	<	X
ejde-502	263	2	tn−1	tn−1	PROPN
ejde-502	263	3	<	<	X
ejde-502	263	4	tn	tn	PROPN
ejde-502	263	5	=	=	SYM
ejde-502	263	6	t	t	PROPN
ejde-502	263	7	,	,	PUNCT
ejde-502	263	8	tj	tj	X
ejde-502	263	9	=	=	PROPN
ejde-502	263	10	jt	jt	PROPN
ejde-502	263	11	n	n	PROPN
ejde-502	263	12	and	and	CCONJ
ejde-502	263	13	construct	construct	VERB
ejde-502	263	14	the	the	DET
ejde-502	263	15	function	function	NOUN
ejde-502	263	16	un	un	PROPN
ejde-502	263	17	by	by	ADP
ejde-502	263	18	induction	induction	NOUN
ejde-502	263	19	.	.	PUNCT
ejde-502	264	1	more	more	ADV
ejde-502	264	2	precisely	precisely	ADV
ejde-502	264	3	,	,	PUNCT
ejde-502	264	4	assume	assume	VERB
ejde-502	264	5	first	first	ADV
ejde-502	264	6	that	that	SCONJ
ejde-502	264	7	un	un	PROPN
ejde-502	264	8	is	be	AUX
ejde-502	264	9	already	already	ADV
ejde-502	264	10	constructed	construct	VERB
ejde-502	264	11	for	for	ADP
ejde-502	264	12	t	t	PROPN
ejde-502	264	13	∈	∈	PROPN
ejde-502	265	1	[	[	X
ejde-502	265	2	0	0	NUM
ejde-502	265	3	,	,	PUNCT
ejde-502	265	4	tj	tj	NOUN
ejde-502	265	5	]	]	PUNCT
ejde-502	265	6	and	and	CCONJ
ejde-502	265	7	we	we	PRON
ejde-502	265	8	want	want	VERB
ejde-502	265	9	to	to	PART
ejde-502	265	10	pass	pass	VERB
ejde-502	265	11	to	to	ADP
ejde-502	265	12	tj+1	tj+1	NUM
ejde-502	265	13	.	.	PUNCT
ejde-502	266	1	to	to	ADP
ejde-502	266	2	this	this	DET
ejde-502	266	3	end	end	NOUN
ejde-502	266	4	,	,	PUNCT
ejde-502	266	5	we	we	PRON
ejde-502	266	6	begin	begin	VERB
ejde-502	266	7	from	from	ADP
ejde-502	266	8	the	the	DET
ejde-502	266	9	edges	edge	NOUN
ejde-502	266	10	of	of	ADP
ejde-502	266	11	the	the	DET
ejde-502	266	12	support	support	NOUN
ejde-502	266	13	of	of	ADP
ejde-502	266	14	un(tj	un(tj	PROPN
ejde-502	266	15	)	)	PUNCT
ejde-502	266	16	and	and	CCONJ
ejde-502	266	17	we	we	PRON
ejde-502	266	18	perform	perform	VERB
ejde-502	266	19	a	a	DET
ejde-502	266	20	jump	jump	NOUN
ejde-502	266	21	of	of	ADP
ejde-502	266	22	them	they	PRON
ejde-502	266	23	by	by	ADP
ejde-502	266	24	applying	apply	VERB
ejde-502	266	25	step	step	NOUN
ejde-502	266	26	1	1	NUM
ejde-502	266	27	both	both	CCONJ
ejde-502	266	28	to	to	ADP
ejde-502	266	29	the	the	DET
ejde-502	266	30	right	right	NOUN
ejde-502	266	31	(	(	PUNCT
ejde-502	266	32	with	with	ADP
ejde-502	266	33	r	r	NOUN
ejde-502	266	34	=	=	SYM
ejde-502	266	35	ξr(tj+1	ξr(tj+1	PROPN
ejde-502	266	36	)	)	PUNCT
ejde-502	266	37	−	−	PROPN
ejde-502	266	38	ξr(tj	ξr(tj	PROPN
ejde-502	266	39	)	)	PUNCT
ejde-502	266	40	)	)	PUNCT
ejde-502	266	41	and	and	CCONJ
ejde-502	266	42	to	to	ADP
ejde-502	266	43	the	the	DET
ejde-502	266	44	left	left	NOUN
ejde-502	266	45	(	(	PUNCT
ejde-502	266	46	with	with	ADP
ejde-502	266	47	r	r	NOUN
ejde-502	266	48	=	=	SYM
ejde-502	266	49	ξl(tj	ξl(tj	NOUN
ejde-502	266	50	)	)	PUNCT
ejde-502	266	51	−	−	PROPN
ejde-502	267	1	ξl(tj+1	ξl(tj+1	ADP
ejde-502	267	2	)	)	PUNCT
ejde-502	267	3	)	)	PUNCT
ejde-502	267	4	,	,	PUNCT
ejde-502	267	5	using	use	VERB
ejde-502	267	6	here	here	ADV
ejde-502	267	7	the	the	DET
ejde-502	267	8	fact	fact	NOUN
ejde-502	267	9	that	that	SCONJ
ejde-502	267	10	the	the	DET
ejde-502	267	11	speed	speed	NOUN
ejde-502	267	12	of	of	ADP
ejde-502	267	13	advance	advance	NOUN
ejde-502	267	14	of	of	ADP
ejde-502	267	15	the	the	DET
ejde-502	267	16	prescribed	prescribed	ADJ
ejde-502	267	17	interfaces	interface	NOUN
ejde-502	267	18	to	to	ADP
ejde-502	267	19	the	the	DET
ejde-502	267	20	left	left	NOUN
ejde-502	267	21	and	and	CCONJ
ejde-502	267	22	to	to	ADP
ejde-502	267	23	the	the	DET
ejde-502	267	24	right	right	NOUN
ejde-502	267	25	is	be	AUX
ejde-502	267	26	higher	high	ADJ
ejde-502	267	27	than	than	ADP
ejde-502	267	28	the	the	DET
ejde-502	267	29	ones	one	NOUN
ejde-502	267	30	of	of	ADP
ejde-502	267	31	the	the	DET
ejde-502	267	32	minimal	minimal	ADJ
ejde-502	267	33	solution	solution	NOUN
ejde-502	267	34	with	with	ADP
ejde-502	267	35	initial	initial	ADJ
ejde-502	267	36	condition	condition	NOUN
ejde-502	267	37	u0	u0	ADJ
ejde-502	267	38	.	.	PUNCT
ejde-502	268	1	the	the	DET
ejde-502	268	2	precise	precise	ADJ
ejde-502	268	3	details	detail	NOUN
ejde-502	268	4	are	be	AUX
ejde-502	268	5	easy	easy	ADJ
ejde-502	268	6	and	and	CCONJ
ejde-502	268	7	given	give	VERB
ejde-502	268	8	in	in	ADP
ejde-502	268	9	the	the	DET
ejde-502	268	10	above	above	ADJ
ejde-502	268	11	mentioned	mention	VERB
ejde-502	268	12	proof	proof	NOUN
ejde-502	268	13	of	of	ADP
ejde-502	268	14	[	[	X
ejde-502	268	15	28	28	NUM
ejde-502	268	16	,	,	PUNCT
ejde-502	268	17	corollary	corollary	ADJ
ejde-502	268	18	1.2	1.2	NUM
ejde-502	268	19	]	]	PUNCT
ejde-502	268	20	.	.	PUNCT
ejde-502	269	1	to	to	PART
ejde-502	269	2	pass	pass	VERB
ejde-502	269	3	to	to	ADP
ejde-502	269	4	the	the	DET
ejde-502	269	5	limit	limit	NOUN
ejde-502	269	6	as	as	ADP
ejde-502	269	7	n→∞	n→∞	NUM
ejde-502	269	8	in	in	ADP
ejde-502	269	9	the	the	DET
ejde-502	269	10	iteration	iteration	NOUN
ejde-502	269	11	we	we	PRON
ejde-502	269	12	need	need	VERB
ejde-502	269	13	an	an	DET
ejde-502	269	14	uniform	uniform	NOUN
ejde-502	269	15	bound	bind	VERB
ejde-502	269	16	from	from	ADP
ejde-502	269	17	above	above	ADV
ejde-502	269	18	to	to	PART
ejde-502	269	19	show	show	VERB
ejde-502	269	20	that	that	SCONJ
ejde-502	269	21	the	the	DET
ejde-502	269	22	limit	limit	NOUN
ejde-502	269	23	solution	solution	NOUN
ejde-502	269	24	does	do	AUX
ejde-502	269	25	not	not	PART
ejde-502	269	26	escape	escape	VERB
ejde-502	269	27	to	to	ADP
ejde-502	269	28	infinity	infinity	NOUN
ejde-502	269	29	.	.	PUNCT
ejde-502	270	1	we	we	PRON
ejde-502	270	2	can	can	AUX
ejde-502	270	3	not	not	PART
ejde-502	270	4	use	use	VERB
ejde-502	270	5	a	a	DET
ejde-502	270	6	translation	translation	NOUN
ejde-502	270	7	of	of	ADP
ejde-502	270	8	a	a	DET
ejde-502	270	9	minimal	minimal	ADJ
ejde-502	270	10	solution	solution	NOUN
ejde-502	270	11	or	or	CCONJ
ejde-502	270	12	a	a	DET
ejde-502	270	13	construction	construction	NOUN
ejde-502	270	14	based	base	VERB
ejde-502	270	15	on	on	ADP
ejde-502	270	16	it	it	PRON
ejde-502	270	17	(	(	PUNCT
ejde-502	270	18	as	as	SCONJ
ejde-502	270	19	it	it	PRON
ejde-502	270	20	was	be	AUX
ejde-502	270	21	done	do	VERB
ejde-502	270	22	in	in	ADP
ejde-502	270	23	[	[	X
ejde-502	270	24	28	28	NUM
ejde-502	270	25	,	,	PUNCT
ejde-502	270	26	lemma	lemma	PROPN
ejde-502	270	27	2.4	2.4	NUM
ejde-502	270	28	]	]	PUNCT
ejde-502	270	29	)	)	PUNCT
ejde-502	270	30	since	since	SCONJ
ejde-502	270	31	our	our	PRON
ejde-502	270	32	equation	equation	NOUN
ejde-502	270	33	is	be	AUX
ejde-502	270	34	not	not	PART
ejde-502	270	35	invariant	invariant	ADJ
ejde-502	270	36	to	to	ADP
ejde-502	270	37	translations	translation	NOUN
ejde-502	270	38	,	,	PUNCT
ejde-502	270	39	but	but	CCONJ
ejde-502	270	40	instead	instead	ADV
ejde-502	270	41	we	we	PRON
ejde-502	270	42	can	can	AUX
ejde-502	270	43	bound	bind	VERB
ejde-502	270	44	uniformly	uniformly	ADV
ejde-502	270	45	from	from	ADP
ejde-502	270	46	above	above	ADP
ejde-502	270	47	the	the	DET
ejde-502	270	48	iterated	iterated	ADJ
ejde-502	270	49	solutions	solution	NOUN
ejde-502	270	50	by	by	ADP
ejde-502	270	51	a	a	DET
ejde-502	270	52	sufficiently	sufficiently	ADV
ejde-502	270	53	big	big	ADJ
ejde-502	270	54	non	non	ADJ
ejde-502	270	55	-	-	ADJ
ejde-502	270	56	increasing	increasing	ADJ
ejde-502	270	57	supersolution	supersolution	NOUN
ejde-502	270	58	in	in	ADP
ejde-502	270	59	self	self	NOUN
ejde-502	270	60	-	-	PUNCT
ejde-502	270	61	similar	similar	ADJ
ejde-502	270	62	form	form	NOUN
ejde-502	270	63	u(x	u(x	NOUN
ejde-502	270	64	,	,	PUNCT
ejde-502	270	65	t	t	NOUN
ejde-502	270	66	)	)	PUNCT
ejde-502	270	67	=	=	PUNCT
ejde-502	271	1	(	(	PUNCT
ejde-502	271	2	t	t	PROPN
ejde-502	271	3	−	−	PROPN
ejde-502	271	4	t)−αf((1	t)−αf((1	PROPN
ejde-502	271	5	+	+	CCONJ
ejde-502	271	6	|x|)(t	|x|)(t	ADJ
ejde-502	271	7	−	−	NOUN
ejde-502	271	8	t)β	t)β	NOUN
ejde-502	271	9	)	)	PUNCT
ejde-502	271	10	,	,	PUNCT
ejde-502	271	11	α	α	X
ejde-502	271	12	=	=	PUNCT
ejde-502	271	13	σ	σ	PROPN
ejde-502	272	1	+	+	NUM
ejde-502	272	2	2	2	NUM
ejde-502	272	3	l	l	NOUN
ejde-502	272	4	,	,	PUNCT
ejde-502	272	5	β	β	X
ejde-502	272	6	=	=	PUNCT
ejde-502	272	7	m−	m−	PROPN
ejde-502	272	8	p	p	X
ejde-502	272	9	l	l	NOUN
ejde-502	272	10	similar	similar	ADJ
ejde-502	272	11	to	to	ADP
ejde-502	272	12	the	the	DET
ejde-502	272	13	ones	one	NOUN
ejde-502	272	14	introduced	introduce	VERB
ejde-502	272	15	in	in	ADP
ejde-502	272	16	(	(	PUNCT
ejde-502	272	17	2.2	2.2	NUM
ejde-502	272	18	)	)	PUNCT
ejde-502	272	19	with	with	ADP
ejde-502	272	20	a	a	DET
ejde-502	272	21	profile	profile	NOUN
ejde-502	272	22	f(ξ	f(ξ	NOUN
ejde-502	272	23	)	)	PUNCT
ejde-502	272	24	solving	solving	NOUN
ejde-502	272	25	(	(	PUNCT
ejde-502	272	26	2.3	2.3	NUM
ejde-502	272	27	)	)	PUNCT
ejde-502	272	28	and	and	CCONJ
ejde-502	272	29	having	have	VERB
ejde-502	272	30	a	a	DET
ejde-502	272	31	right	right	ADJ
ejde-502	272	32	interface	interface	NOUN
ejde-502	272	33	at	at	ADP
ejde-502	272	34	ξ0	ξ0	PROPN
ejde-502	272	35	∈	∈	PROPN
ejde-502	272	36	(	(	PUNCT
ejde-502	272	37	0,∞	0,∞	NOUN
ejde-502	272	38	)	)	PUNCT
ejde-502	272	39	.	.	PUNCT
ejde-502	273	1	indeed	indeed	ADV
ejde-502	273	2	,	,	PUNCT
ejde-502	273	3	comparison	comparison	NOUN
ejde-502	273	4	from	from	ADP
ejde-502	273	5	above	above	ADV
ejde-502	273	6	with	with	ADP
ejde-502	273	7	such	such	DET
ejde-502	273	8	a	a	DET
ejde-502	273	9	supersolution	supersolution	NOUN
ejde-502	273	10	u	u	NOUN
ejde-502	273	11	can	can	AUX
ejde-502	273	12	be	be	AUX
ejde-502	273	13	performed	perform	VERB
ejde-502	273	14	as	as	ADP
ejde-502	273	15	the	the	DET
ejde-502	273	16	iterative	iterative	NOUN
ejde-502	273	17	construction	construction	NOUN
ejde-502	273	18	of	of	ADP
ejde-502	273	19	un	un	PROPN
ejde-502	273	20	is	be	AUX
ejde-502	273	21	based	base	VERB
ejde-502	273	22	on	on	ADP
ejde-502	273	23	adding	add	VERB
ejde-502	273	24	up	up	ADP
ejde-502	273	25	at	at	ADP
ejde-502	273	26	each	each	DET
ejde-502	273	27	iteration	iteration	NOUN
ejde-502	273	28	step	step	NOUN
ejde-502	273	29	only	only	ADV
ejde-502	273	30	minimal	minimal	ADJ
ejde-502	273	31	solutions	solution	NOUN
ejde-502	273	32	,	,	PUNCT
ejde-502	273	33	provided	provide	VERB
ejde-502	273	34	that	that	SCONJ
ejde-502	273	35	at	at	ADP
ejde-502	273	36	our	our	PRON
ejde-502	273	37	fixed	fix	VERB
ejde-502	273	38	time	time	NOUN
ejde-502	273	39	t	t	PROPN
ejde-502	273	40	>	>	X
ejde-502	273	41	0	0	PUNCT
ejde-502	273	42	for	for	ADP
ejde-502	273	43	which	which	PRON
ejde-502	273	44	we	we	PRON
ejde-502	273	45	have	have	AUX
ejde-502	273	46	built	build	VERB
ejde-502	273	47	un	un	PROPN
ejde-502	273	48	we	we	PRON
ejde-502	273	49	have	have	AUX
ejde-502	273	50	ordered	order	VERB
ejde-502	273	51	supports	support	NOUN
ejde-502	273	52	between	between	ADP
ejde-502	273	53	un	un	PROPN
ejde-502	273	54	and	and	CCONJ
ejde-502	273	55	u	u	PROPN
ejde-502	273	56	,	,	PUNCT
ejde-502	273	57	that	that	ADV
ejde-502	273	58	is	be	AUX
ejde-502	273	59	ξr(t	ξr(t	NOUN
ejde-502	273	60	)	)	PUNCT
ejde-502	273	61	<	<	X
ejde-502	274	1	ξ0(t	ξ0(t	X
ejde-502	274	2	−	−	X
ejde-502	274	3	t)−β	t)−β	PROPN
ejde-502	274	4	,	,	PUNCT
ejde-502	274	5	ξl(t	ξl(t	NOUN
ejde-502	274	6	)	)	PUNCT
ejde-502	274	7	>	>	X
ejde-502	275	1	−ξ0(t	−ξ0(t	PROPN
ejde-502	275	2	−	−	PROPN
ejde-502	275	3	t)−β	t)−β	ADP
ejde-502	275	4	,	,	PUNCT
ejde-502	275	5	(	(	PUNCT
ejde-502	275	6	3.4	3.4	NUM
ejde-502	275	7	)	)	PUNCT
ejde-502	275	8	which	which	PRON
ejde-502	275	9	also	also	ADV
ejde-502	275	10	gives	give	VERB
ejde-502	275	11	a	a	DET
ejde-502	275	12	limitation	limitation	NOUN
ejde-502	275	13	for	for	ADP
ejde-502	275	14	the	the	DET
ejde-502	275	15	lifetime	lifetime	NOUN
ejde-502	275	16	t	t	PROPN
ejde-502	275	17	>	>	X
ejde-502	275	18	0	0	PUNCT
ejde-502	276	1	depending	depend	VERB
ejde-502	276	2	on	on	ADP
ejde-502	276	3	the	the	DET
ejde-502	276	4	two	two	NUM
ejde-502	276	5	prescribed	prescribe	VERB
ejde-502	276	6	functions	function	NOUN
ejde-502	276	7	ξr	ξr	NOUN
ejde-502	276	8	,	,	PUNCT
ejde-502	276	9	ξl	ξl	NOUN
ejde-502	276	10	.	.	PUNCT
ejde-502	277	1	we	we	PRON
ejde-502	277	2	notice	notice	VERB
ejde-502	277	3	that	that	SCONJ
ejde-502	277	4	for	for	ADP
ejde-502	277	5	“	"	PUNCT
ejde-502	277	6	faster	fast	ADJ
ejde-502	277	7	”	"	PUNCT
ejde-502	277	8	advancing	advance	VERB
ejde-502	277	9	interface	interface	NOUN
ejde-502	277	10	functions	function	NOUN
ejde-502	277	11	ξl(t	ξl(t	NOUN
ejde-502	277	12	)	)	PUNCT
ejde-502	277	13	,	,	PUNCT
ejde-502	277	14	ξr(t	ξr(t	NUM
ejde-502	277	15	)	)	PUNCT
ejde-502	277	16	,	,	PUNCT
ejde-502	277	17	smaller	small	ADJ
ejde-502	277	18	lifetime	lifetime	NOUN
ejde-502	277	19	t	t	PROPN
ejde-502	277	20	is	be	AUX
ejde-502	277	21	expected	expect	VERB
ejde-502	277	22	.	.	PUNCT
ejde-502	278	1	once	once	ADV
ejde-502	278	2	satisfied	satisfied	ADJ
ejde-502	278	3	this	this	DET
ejde-502	278	4	condition	condition	NOUN
ejde-502	278	5	,	,	PUNCT
ejde-502	278	6	we	we	PRON
ejde-502	278	7	can	can	AUX
ejde-502	278	8	pass	pass	VERB
ejde-502	278	9	to	to	ADP
ejde-502	278	10	the	the	DET
ejde-502	278	11	limit	limit	NOUN
ejde-502	278	12	in	in	ADP
ejde-502	278	13	the	the	DET
ejde-502	278	14	discretization	discretization	NOUN
ejde-502	278	15	as	as	ADP
ejde-502	278	16	n→∞	n→∞	NUM
ejde-502	278	17	and	and	CCONJ
ejde-502	278	18	obtain	obtain	VERB
ejde-502	278	19	the	the	DET
ejde-502	278	20	desired	desire	VERB
ejde-502	278	21	weak	weak	ADJ
ejde-502	278	22	solution	solution	NOUN
ejde-502	278	23	v(x	v(x	PROPN
ejde-502	278	24	,	,	PUNCT
ejde-502	278	25	t	t	PROPN
ejde-502	278	26	)	)	PUNCT
ejde-502	278	27	=	=	VERB
ejde-502	278	28	lim	lim	PROPN
ejde-502	278	29	n→∞	n→∞	NUM
ejde-502	278	30	un(x	un(x	PROPN
ejde-502	278	31	,	,	PUNCT
ejde-502	278	32	t	t	PROPN
ejde-502	278	33	)	)	PUNCT
ejde-502	278	34	,	,	PUNCT
ejde-502	278	35	t	t	PROPN
ejde-502	278	36	∈	∈	PROPN
ejde-502	278	37	(	(	PUNCT
ejde-502	278	38	0	0	NUM
ejde-502	278	39	,	,	PUNCT
ejde-502	278	40	t	t	PROPN
ejde-502	278	41	)	)	PUNCT
ejde-502	278	42	,	,	PUNCT
ejde-502	278	43	with	with	ADP
ejde-502	278	44	t	t	PROPN
ejde-502	278	45	>	>	X
ejde-502	278	46	0	0	PUNCT
ejde-502	278	47	chosen	choose	VERB
ejde-502	278	48	sufficiently	sufficiently	ADV
ejde-502	278	49	small	small	ADJ
ejde-502	278	50	according	accord	VERB
ejde-502	278	51	to	to	ADP
ejde-502	278	52	(	(	PUNCT
ejde-502	278	53	3.4	3.4	NUM
ejde-502	278	54	)	)	PUNCT
ejde-502	278	55	.	.	PUNCT
ejde-502	279	1	the	the	DET
ejde-502	279	2	bound	bind	VERB
ejde-502	279	3	from	from	ADP
ejde-502	279	4	below	below	ADV
ejde-502	279	5	by	by	ADP
ejde-502	279	6	e	e	NOUN
ejde-502	279	7	at	at	ADP
ejde-502	279	8	every	every	DET
ejde-502	279	9	point	point	NOUN
ejde-502	279	10	of	of	ADP
ejde-502	279	11	positivity	positivity	NOUN
ejde-502	279	12	(	(	PUNCT
ejde-502	279	13	as	as	SCONJ
ejde-502	279	14	done	do	VERB
ejde-502	279	15	in	in	ADP
ejde-502	279	16	step	step	NOUN
ejde-502	279	17	1	1	NUM
ejde-502	279	18	)	)	PUNCT
ejde-502	279	19	together	together	ADV
ejde-502	279	20	with	with	ADP
ejde-502	279	21	the	the	DET
ejde-502	279	22	construction	construction	NOUN
ejde-502	279	23	and	and	CCONJ
ejde-502	279	24	the	the	DET
ejde-502	279	25	continuity	continuity	NOUN
ejde-502	279	26	of	of	ADP
ejde-502	279	27	the	the	DET
ejde-502	279	28	prescribed	prescribe	VERB
ejde-502	279	29	functions	function	NOUN
ejde-502	279	30	ξl	ξl	NOUN
ejde-502	279	31	,	,	PUNCT
ejde-502	279	32	ξr	ξr	PART
ejde-502	279	33	ensure	ensure	VERB
ejde-502	279	34	that	that	SCONJ
ejde-502	279	35	the	the	DET
ejde-502	279	36	left	left	ADJ
ejde-502	279	37	and	and	CCONJ
ejde-502	279	38	right	right	ADJ
ejde-502	279	39	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	279	40	reaction	reaction	NOUN
ejde-502	279	41	-	-	PUNCT
ejde-502	279	42	diffusion	diffusion	NOUN
ejde-502	279	43	with	with	ADP
ejde-502	279	44	weighted	weight	VERB
ejde-502	279	45	strong	strong	ADJ
ejde-502	279	46	reaction	reaction	NOUN
ejde-502	279	47	13	13	NUM
ejde-502	279	48	interfaces	interface	NOUN
ejde-502	279	49	of	of	ADP
ejde-502	279	50	v	v	NOUN
ejde-502	279	51	at	at	ADP
ejde-502	279	52	every	every	DET
ejde-502	279	53	time	time	NOUN
ejde-502	279	54	t	t	PROPN
ejde-502	279	55	∈	∈	PROPN
ejde-502	279	56	(	(	PUNCT
ejde-502	279	57	0	0	NUM
ejde-502	279	58	,	,	PUNCT
ejde-502	279	59	t	t	PROPN
ejde-502	279	60	)	)	PUNCT
ejde-502	279	61	are	be	AUX
ejde-502	279	62	given	give	VERB
ejde-502	279	63	exactly	exactly	ADV
ejde-502	279	64	by	by	ADP
ejde-502	279	65	the	the	DET
ejde-502	279	66	continuous	continuous	ADJ
ejde-502	279	67	functions	function	NOUN
ejde-502	279	68	ξl(t	ξl(t	NOUN
ejde-502	279	69	)	)	PUNCT
ejde-502	279	70	and	and	CCONJ
ejde-502	279	71	ξr(t	ξr(t	NUM
ejde-502	279	72	)	)	PUNCT
ejde-502	279	73	.	.	PUNCT
ejde-502	280	1	in	in	ADP
ejde-502	280	2	the	the	DET
ejde-502	280	3	meantime	meantime	NOUN
ejde-502	280	4	,	,	PUNCT
ejde-502	280	5	the	the	DET
ejde-502	280	6	universal	universal	NOUN
ejde-502	280	7	bound	bind	VERB
ejde-502	280	8	from	from	ADP
ejde-502	280	9	above	above	ADV
ejde-502	280	10	by	by	ADP
ejde-502	280	11	the	the	DET
ejde-502	280	12	supersolution	supersolution	NOUN
ejde-502	280	13	u	u	NOUN
ejde-502	280	14	in	in	ADP
ejde-502	280	15	self	self	NOUN
ejde-502	280	16	-	-	PUNCT
ejde-502	280	17	similar	similar	ADJ
ejde-502	280	18	form	form	NOUN
ejde-502	280	19	ensures	ensure	VERB
ejde-502	280	20	that	that	SCONJ
ejde-502	280	21	at	at	ADV
ejde-502	280	22	least	least	ADJ
ejde-502	280	23	in	in	ADP
ejde-502	280	24	the	the	DET
ejde-502	280	25	time	time	NOUN
ejde-502	280	26	interval	interval	NOUN
ejde-502	280	27	(	(	PUNCT
ejde-502	280	28	0	0	NUM
ejde-502	280	29	,	,	PUNCT
ejde-502	280	30	t	t	NOUN
ejde-502	280	31	)	)	PUNCT
ejde-502	280	32	we	we	PRON
ejde-502	280	33	have	have	VERB
ejde-502	280	34	v(x	v(x	PROPN
ejde-502	280	35	,	,	PUNCT
ejde-502	280	36	t	t	PROPN
ejde-502	280	37	)	)	PUNCT
ejde-502	280	38	<	<	X
ejde-502	280	39	∞	∞	NUM
ejde-502	280	40	for	for	ADP
ejde-502	280	41	any	any	DET
ejde-502	280	42	x	x	SYM
ejde-502	280	43	∈	∈	PROPN
ejde-502	280	44	r.	r.	PROPN
ejde-502	280	45	�	�	PROPN
ejde-502	280	46	remarks	remark	VERB
ejde-502	280	47	.	.	PUNCT
ejde-502	281	1	(	(	PUNCT
ejde-502	281	2	1	1	X
ejde-502	281	3	)	)	PUNCT
ejde-502	281	4	step	step	NOUN
ejde-502	281	5	1	1	NUM
ejde-502	281	6	above	above	ADP
ejde-502	281	7	already	already	ADV
ejde-502	281	8	completes	complete	VERB
ejde-502	281	9	the	the	DET
ejde-502	281	10	proof	proof	NOUN
ejde-502	281	11	of	of	ADP
ejde-502	281	12	theorem	theorem	ADJ
ejde-502	281	13	1.3	1.3	NUM
ejde-502	281	14	in	in	ADP
ejde-502	281	15	dimension	dimension	NOUN
ejde-502	281	16	n	n	NOUN
ejde-502	281	17	=	=	SYM
ejde-502	281	18	1	1	X
ejde-502	281	19	.	.	PUNCT
ejde-502	281	20	indeed	indeed	ADV
ejde-502	281	21	,	,	PUNCT
ejde-502	281	22	for	for	ADP
ejde-502	281	23	every	every	DET
ejde-502	281	24	r	r	NOUN
ejde-502	281	25	>	>	X
ejde-502	281	26	0	0	NUM
ejde-502	281	27	we	we	PRON
ejde-502	281	28	can	can	AUX
ejde-502	281	29	construct	construct	VERB
ejde-502	281	30	a	a	DET
ejde-502	281	31	different	different	ADJ
ejde-502	281	32	solution	solution	NOUN
ejde-502	281	33	to	to	ADP
ejde-502	281	34	the	the	DET
ejde-502	281	35	same	same	ADJ
ejde-502	281	36	cauchy	cauchy	NOUN
ejde-502	281	37	problem	problem	NOUN
ejde-502	281	38	(	(	PUNCT
ejde-502	281	39	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	281	40	)	)	PUNCT
ejde-502	281	41	.	.	PUNCT
ejde-502	282	1	a	a	DET
ejde-502	282	2	totally	totally	ADV
ejde-502	282	3	analogous	analogous	ADJ
ejde-502	282	4	extension	extension	NOUN
ejde-502	282	5	can	can	AUX
ejde-502	282	6	be	be	AUX
ejde-502	282	7	constructed	construct	VERB
ejde-502	282	8	for	for	ADP
ejde-502	282	9	radially	radially	ADV
ejde-502	282	10	symmetric	symmetric	ADJ
ejde-502	282	11	initial	initial	ADJ
ejde-502	282	12	conditions	condition	NOUN
ejde-502	282	13	u0	u0	VERB
ejde-502	282	14	∈	∈	NOUN
ejde-502	282	15	c0(rn	c0(rn	PROPN
ejde-502	282	16	)	)	PUNCT
ejde-502	282	17	by	by	ADP
ejde-502	282	18	replacing	replace	VERB
ejde-502	282	19	x	x	PUNCT
ejde-502	282	20	by	by	ADP
ejde-502	282	21	r	r	NOUN
ejde-502	282	22	=	=	SYM
ejde-502	282	23	|x|	|x|	PROPN
ejde-502	282	24	,	,	PUNCT
ejde-502	282	25	which	which	PRON
ejde-502	282	26	allows	allow	VERB
ejde-502	282	27	us	we	PRON
ejde-502	282	28	to	to	PART
ejde-502	282	29	work	work	VERB
ejde-502	282	30	with	with	ADP
ejde-502	282	31	the	the	DET
ejde-502	282	32	radial	radial	ADJ
ejde-502	282	33	variable	variable	NOUN
ejde-502	282	34	exactly	exactly	ADV
ejde-502	282	35	as	as	ADP
ejde-502	282	36	in	in	ADP
ejde-502	282	37	dimension	dimension	NOUN
ejde-502	282	38	n	n	NOUN
ejde-502	282	39	=	=	SYM
ejde-502	282	40	1	1	NUM
ejde-502	282	41	but	but	CCONJ
ejde-502	282	42	using	use	VERB
ejde-502	282	43	for	for	ADP
ejde-502	282	44	comparison	comparison	NOUN
ejde-502	282	45	from	from	ADP
ejde-502	282	46	above	above	ADP
ejde-502	282	47	the	the	DET
ejde-502	282	48	supersolutions	supersolution	NOUN
ejde-502	282	49	introduced	introduce	VERB
ejde-502	282	50	in	in	ADP
ejde-502	282	51	step	step	NOUN
ejde-502	282	52	3	3	NUM
ejde-502	282	53	of	of	ADP
ejde-502	282	54	the	the	DET
ejde-502	282	55	proof	proof	NOUN
ejde-502	282	56	of	of	ADP
ejde-502	282	57	proposition	proposition	NOUN
ejde-502	282	58	2.1	2.1	NUM
ejde-502	282	59	,	,	PUNCT
ejde-502	282	60	thus	thus	ADV
ejde-502	282	61	completing	complete	VERB
ejde-502	282	62	the	the	DET
ejde-502	282	63	proof	proof	NOUN
ejde-502	282	64	of	of	ADP
ejde-502	282	65	theorem	theorem	ADJ
ejde-502	282	66	1.3	1.3	NUM
ejde-502	282	67	also	also	ADV
ejde-502	282	68	in	in	ADP
ejde-502	282	69	dimension	dimension	NOUN
ejde-502	282	70	n	n	CCONJ
ejde-502	282	71	≥	≥	NOUN
ejde-502	282	72	2	2	NUM
ejde-502	282	73	.	.	PUNCT
ejde-502	283	1	(	(	PUNCT
ejde-502	283	2	2	2	X
ejde-502	283	3	)	)	PUNCT
ejde-502	283	4	our	our	PRON
ejde-502	283	5	proposition	proposition	NOUN
ejde-502	283	6	3.1	3.1	NUM
ejde-502	283	7	also	also	ADV
ejde-502	283	8	holds	hold	VERB
ejde-502	283	9	for	for	ADP
ejde-502	283	10	σ	σ	NOUN
ejde-502	283	11	=	=	SYM
ejde-502	283	12	0	0	NUM
ejde-502	283	13	and	and	CCONJ
ejde-502	283	14	improves	improve	VERB
ejde-502	283	15	slightly	slightly	ADV
ejde-502	283	16	the	the	DET
ejde-502	283	17	result	result	NOUN
ejde-502	283	18	for	for	ADP
ejde-502	283	19	(	(	PUNCT
ejde-502	283	20	1.5	1.5	NUM
ejde-502	283	21	)	)	PUNCT
ejde-502	283	22	in	in	ADP
ejde-502	283	23	[	[	X
ejde-502	283	24	28	28	NUM
ejde-502	283	25	,	,	PUNCT
ejde-502	283	26	theorem	theorem	VERB
ejde-502	283	27	1.1	1.1	NUM
ejde-502	283	28	and	and	CCONJ
ejde-502	283	29	theorem	theorem	VERB
ejde-502	283	30	5.1	5.1	NUM
ejde-502	283	31	]	]	PUNCT
ejde-502	283	32	since	since	SCONJ
ejde-502	283	33	we	we	PRON
ejde-502	283	34	do	do	AUX
ejde-502	283	35	not	not	PART
ejde-502	283	36	need	need	VERB
ejde-502	283	37	to	to	PART
ejde-502	283	38	consider	consider	VERB
ejde-502	283	39	the	the	DET
ejde-502	283	40	non	non	ADJ
ejde-502	283	41	-	-	ADJ
ejde-502	283	42	increasing	increasing	ADJ
ejde-502	283	43	majorant	majorant	NOUN
ejde-502	283	44	ũ0	ũ0	PROPN
ejde-502	283	45	of	of	ADP
ejde-502	283	46	the	the	DET
ejde-502	283	47	initial	initial	ADJ
ejde-502	283	48	condition	condition	NOUN
ejde-502	283	49	as	as	SCONJ
ejde-502	283	50	considered	consider	VERB
ejde-502	283	51	in	in	ADP
ejde-502	283	52	the	the	DET
ejde-502	283	53	above	above	ADJ
ejde-502	283	54	mentioned	mention	VERB
ejde-502	283	55	work	work	NOUN
ejde-502	283	56	.	.	PUNCT
ejde-502	284	1	4	4	X
ejde-502	284	2	.	.	X
ejde-502	284	3	aronson	aronson	PROPN
ejde-502	284	4	-	-	PUNCT
ejde-502	284	5	bénilan	bénilan	PROPN
ejde-502	284	6	estimates	estimate	VERB
ejde-502	284	7	when	when	SCONJ
ejde-502	284	8	m+	m+	NOUN
ejde-502	284	9	p	p	X
ejde-502	284	10	<	<	X
ejde-502	284	11	2	2	NUM
ejde-502	284	12	this	this	DET
ejde-502	284	13	section	section	NOUN
ejde-502	284	14	is	be	AUX
ejde-502	284	15	devoted	devote	VERB
ejde-502	284	16	to	to	ADP
ejde-502	284	17	the	the	DET
ejde-502	284	18	deduction	deduction	NOUN
ejde-502	284	19	of	of	ADP
ejde-502	284	20	the	the	DET
ejde-502	284	21	aronson	aronson	PROPN
ejde-502	284	22	-	-	PUNCT
ejde-502	284	23	bénilan	bénilan	PROPN
ejde-502	284	24	estimates	estimate	NOUN
ejde-502	284	25	(	(	PUNCT
ejde-502	284	26	1.12	1.12	NUM
ejde-502	284	27	)	)	PUNCT
ejde-502	284	28	in	in	ADP
ejde-502	284	29	the	the	DET
ejde-502	284	30	homogeneous	homogeneous	ADJ
ejde-502	284	31	case	case	NOUN
ejde-502	284	32	σ	σ	X
ejde-502	284	33	=	=	SYM
ejde-502	284	34	0	0	X
ejde-502	284	35	.	.	PUNCT
ejde-502	285	1	let	let	VERB
ejde-502	285	2	us	we	PRON
ejde-502	285	3	recall	recall	VERB
ejde-502	285	4	here	here	ADV
ejde-502	285	5	the	the	DET
ejde-502	285	6	pressure	pressure	NOUN
ejde-502	285	7	function	function	NOUN
ejde-502	285	8	v	v	ADP
ejde-502	285	9	=	=	NOUN
ejde-502	285	10	m	m	VERB
ejde-502	285	11	m−	m−	PROPN
ejde-502	285	12	1	1	NUM
ejde-502	285	13	um−1	um−1	PROPN
ejde-502	285	14	,	,	PUNCT
ejde-502	285	15	since	since	SCONJ
ejde-502	285	16	most	most	ADJ
ejde-502	285	17	of	of	ADP
ejde-502	285	18	the	the	DET
ejde-502	285	19	forthcoming	forthcoming	ADJ
ejde-502	285	20	work	work	NOUN
ejde-502	285	21	will	will	AUX
ejde-502	285	22	be	be	AUX
ejde-502	285	23	performed	perform	VERB
ejde-502	285	24	on	on	ADP
ejde-502	285	25	this	this	DET
ejde-502	285	26	function	function	NOUN
ejde-502	285	27	.	.	PUNCT
ejde-502	286	1	proof	proof	NOUN
ejde-502	286	2	of	of	ADP
ejde-502	286	3	theorem	theorem	ADJ
ejde-502	286	4	1.4	1.4	NUM
ejde-502	286	5	.	.	PUNCT
ejde-502	287	1	the	the	DET
ejde-502	287	2	proof	proof	NOUN
ejde-502	287	3	is	be	AUX
ejde-502	287	4	inspired	inspire	VERB
ejde-502	287	5	from	from	ADP
ejde-502	287	6	the	the	DET
ejde-502	287	7	one	one	NOUN
ejde-502	287	8	for	for	ADP
ejde-502	287	9	[	[	X
ejde-502	287	10	36	36	NUM
ejde-502	287	11	,	,	PUNCT
ejde-502	287	12	proposition	proposition	NOUN
ejde-502	287	13	9.4	9.4	NUM
ejde-502	287	14	]	]	PUNCT
ejde-502	287	15	but	but	CCONJ
ejde-502	287	16	technically	technically	ADV
ejde-502	287	17	more	more	ADV
ejde-502	287	18	involved	involved	ADJ
ejde-502	287	19	.	.	PUNCT
ejde-502	288	1	we	we	PRON
ejde-502	288	2	first	first	ADV
ejde-502	288	3	derive	derive	VERB
ejde-502	288	4	after	after	ADP
ejde-502	288	5	rather	rather	ADV
ejde-502	288	6	straightforward	straightforward	ADJ
ejde-502	288	7	calculations	calculation	NOUN
ejde-502	288	8	the	the	DET
ejde-502	288	9	pressure	pressure	NOUN
ejde-502	288	10	equation	equation	NOUN
ejde-502	288	11	which	which	PRON
ejde-502	288	12	will	will	AUX
ejde-502	288	13	be	be	AUX
ejde-502	288	14	used	use	VERB
ejde-502	288	15	further	far	ADV
ejde-502	288	16	in	in	ADP
ejde-502	288	17	the	the	DET
ejde-502	288	18	present	present	ADJ
ejde-502	288	19	work	work	NOUN
ejde-502	288	20	,	,	PUNCT
ejde-502	288	21	that	that	ADV
ejde-502	288	22	is	is	ADV
ejde-502	288	23	,	,	PUNCT
ejde-502	288	24	the	the	DET
ejde-502	288	25	parabolic	parabolic	ADJ
ejde-502	288	26	pde	pde	NOUN
ejde-502	288	27	satisfied	satisfy	VERB
ejde-502	288	28	by	by	ADP
ejde-502	288	29	the	the	DET
ejde-502	288	30	function	function	NOUN
ejde-502	288	31	v	v	NOUN
ejde-502	288	32	introduced	introduce	VERB
ejde-502	288	33	above	above	ADV
ejde-502	288	34	:	:	PUNCT
ejde-502	288	35	vt	vt	PROPN
ejde-502	288	36	=	=	SYM
ejde-502	288	37	(	(	PUNCT
ejde-502	288	38	m−	m−	PROPN
ejde-502	288	39	1)v∆v	1)v∆v	NUM
ejde-502	288	40	+	+	CCONJ
ejde-502	288	41	|∇v|2	|∇v|2	NUM
ejde-502	288	42	+	+	PROPN
ejde-502	288	43	k(m	k(m	PROPN
ejde-502	288	44	,	,	PUNCT
ejde-502	288	45	p)v(m+p−2)/(m−1	p)v(m+p−2)/(m−1	PROPN
ejde-502	288	46	)	)	PUNCT
ejde-502	288	47	,	,	PUNCT
ejde-502	288	48	k(m	k(m	PROPN
ejde-502	288	49	,	,	PUNCT
ejde-502	288	50	p	p	X
ejde-502	288	51	)	)	PUNCT
ejde-502	288	52	=	=	VERB
ejde-502	288	53	m	m	PROPN
ejde-502	288	54	(	(	PUNCT
ejde-502	288	55	m−	m−	PROPN
ejde-502	288	56	1	1	NUM
ejde-502	288	57	m	m	NOUN
ejde-502	288	58	)	)	PUNCT
ejde-502	288	59	(	(	PUNCT
ejde-502	288	60	m+p−2)/(m−1	m+p−2)/(m−1	PROPN
ejde-502	288	61	)	)	PUNCT
ejde-502	288	62	.	.	PUNCT
ejde-502	289	1	(	(	PUNCT
ejde-502	289	2	4.1	4.1	NUM
ejde-502	289	3	)	)	PUNCT
ejde-502	289	4	let	let	VERB
ejde-502	289	5	us	we	PRON
ejde-502	289	6	notice	notice	VERB
ejde-502	289	7	here	here	ADV
ejde-502	289	8	the	the	DET
ejde-502	289	9	strong	strong	ADJ
ejde-502	289	10	influence	influence	NOUN
ejde-502	289	11	of	of	ADP
ejde-502	289	12	the	the	DET
ejde-502	289	13	sign	sign	NOUN
ejde-502	289	14	of	of	ADP
ejde-502	289	15	m	m	PROPN
ejde-502	289	16	+	+	X
ejde-502	289	17	p	p	X
ejde-502	289	18	−	−	PROPN
ejde-502	289	19	2	2	NUM
ejde-502	289	20	on	on	ADP
ejde-502	289	21	this	this	DET
ejde-502	289	22	equation	equation	NOUN
ejde-502	289	23	,	,	PUNCT
ejde-502	289	24	because	because	SCONJ
ejde-502	289	25	of	of	ADP
ejde-502	289	26	its	its	PRON
ejde-502	289	27	last	last	ADJ
ejde-502	289	28	term	term	NOUN
ejde-502	289	29	.	.	PUNCT
ejde-502	290	1	to	to	PART
ejde-502	290	2	go	go	VERB
ejde-502	290	3	further	far	ADV
ejde-502	290	4	,	,	PUNCT
ejde-502	290	5	we	we	PRON
ejde-502	290	6	set	set	VERB
ejde-502	290	7	w	w	PROPN
ejde-502	290	8	=	=	PUNCT
ejde-502	290	9	∆v	∆v	PROPN
ejde-502	290	10	and	and	CCONJ
ejde-502	290	11	we	we	PRON
ejde-502	290	12	next	next	ADV
ejde-502	290	13	derive	derive	VERB
ejde-502	290	14	the	the	DET
ejde-502	290	15	partial	partial	ADJ
ejde-502	290	16	differential	differential	NOUN
ejde-502	290	17	equation	equation	NOUN
ejde-502	290	18	solved	solve	VERB
ejde-502	290	19	by	by	ADP
ejde-502	290	20	w.	w.	PROPN
ejde-502	290	21	to	to	ADP
ejde-502	290	22	this	this	DET
ejde-502	290	23	end	end	NOUN
ejde-502	290	24	,	,	PUNCT
ejde-502	290	25	we	we	PRON
ejde-502	290	26	calculate	calculate	VERB
ejde-502	290	27	the	the	DET
ejde-502	290	28	terms	term	NOUN
ejde-502	290	29	separately	separately	ADV
ejde-502	290	30	.	.	PUNCT
ejde-502	291	1	on	on	ADP
ejde-502	291	2	the	the	DET
ejde-502	291	3	one	one	NUM
ejde-502	291	4	hand	hand	NOUN
ejde-502	291	5	,	,	PUNCT
ejde-502	291	6	for	for	ADP
ejde-502	291	7	the	the	DET
ejde-502	291	8	reaction	reaction	NOUN
ejde-502	291	9	term	term	NOUN
ejde-502	291	10	we	we	PRON
ejde-502	291	11	obtain	obtain	VERB
ejde-502	291	12	∂	∂	ADJ
ejde-502	291	13	∂xi	∂xi	PROPN
ejde-502	291	14	k(m	k(m	PROPN
ejde-502	291	15	,	,	PUNCT
ejde-502	291	16	p)v(m+p−2)/(m−1	p)v(m+p−2)/(m−1	PROPN
ejde-502	291	17	)	)	PUNCT
ejde-502	292	1	=	=	SYM
ejde-502	292	2	k(m	k(m	PROPN
ejde-502	292	3	,	,	PUNCT
ejde-502	292	4	p	p	NOUN
ejde-502	292	5	)	)	PUNCT
ejde-502	293	1	[	[	X
ejde-502	293	2	m+	m+	NOUN
ejde-502	293	3	p−	p−	NOUN
ejde-502	293	4	2	2	NUM
ejde-502	293	5	m−	m−	PROPN
ejde-502	293	6	1	1	NUM
ejde-502	293	7	v(p−1)/(m−1	v(p−1)/(m−1	NOUN
ejde-502	293	8	)	)	PUNCT
ejde-502	294	1	∂v	∂v	PROPN
ejde-502	294	2	∂xi	∂xi	PROPN
ejde-502	294	3	]	]	PUNCT
ejde-502	294	4	hence	hence	ADV
ejde-502	294	5	∂2	∂2	PROPN
ejde-502	294	6	∂x2	∂x2	NOUN
ejde-502	294	7	i	i	PRON
ejde-502	294	8	k(m	k(m	PROPN
ejde-502	294	9	,	,	PUNCT
ejde-502	294	10	p)v	p)v	NOUN
ejde-502	294	11	m+p−2	m+p−2	PROPN
ejde-502	294	12	m−1	m−1	PROPN
ejde-502	294	13	=	=	SYM
ejde-502	294	14	k(m	k(m	PROPN
ejde-502	294	15	,	,	PUNCT
ejde-502	294	16	p	p	NOUN
ejde-502	294	17	)	)	PUNCT
ejde-502	295	1	[	[	X
ejde-502	295	2	m+	m+	NOUN
ejde-502	295	3	p−	p−	NOUN
ejde-502	295	4	2	2	NUM
ejde-502	295	5	m−	m−	PROPN
ejde-502	295	6	1	1	NUM
ejde-502	295	7	v	v	ADP
ejde-502	295	8	p−1	p−1	PROPN
ejde-502	295	9	m−1	m−1	PROPN
ejde-502	295	10	∂2v	∂2v	PROPN
ejde-502	295	11	∂x2	∂x2	PROPN
ejde-502	295	12	i	i	PRON
ejde-502	295	13	+	+	X
ejde-502	295	14	(	(	PUNCT
ejde-502	295	15	2−m−	2−m−	NUM
ejde-502	295	16	p)(1−	p)(1−	ADJ
ejde-502	295	17	p	p	NOUN
ejde-502	295	18	)	)	PUNCT
ejde-502	295	19	(	(	PUNCT
ejde-502	295	20	m−	m−	PROPN
ejde-502	295	21	1)2	1)2	NUM
ejde-502	295	22	v	v	NOUN
ejde-502	295	23	p−m	p−m	PRON
ejde-502	295	24	m−1	m−1	PROPN
ejde-502	295	25	(	(	PUNCT
ejde-502	295	26	∂v	∂v	PROPN
ejde-502	295	27	∂xi	∂xi	PROPN
ejde-502	295	28	)	)	PUNCT
ejde-502	295	29	2	2	NUM
ejde-502	295	30	]	]	PUNCT
ejde-502	295	31	.	.	PUNCT
ejde-502	296	1	the	the	DET
ejde-502	296	2	contribution	contribution	NOUN
ejde-502	296	3	of	of	ADP
ejde-502	296	4	the	the	DET
ejde-502	296	5	reaction	reaction	NOUN
ejde-502	296	6	term	term	NOUN
ejde-502	296	7	thus	thus	ADV
ejde-502	296	8	gives	give	VERB
ejde-502	296	9	n∑	n∑	PROPN
ejde-502	297	1	i=1	i=1	PROPN
ejde-502	298	1	∂2	∂2	PROPN
ejde-502	298	2	∂x2	∂x2	NOUN
ejde-502	298	3	i	i	PRON
ejde-502	298	4	k(m	k(m	PROPN
ejde-502	298	5	,	,	PUNCT
ejde-502	298	6	p)v	p)v	NOUN
ejde-502	299	1	m+p−2	m+p−2	PROPN
ejde-502	299	2	m−1	m−1	PROPN
ejde-502	299	3	14	14	NUM
ejde-502	299	4	r.	r.	PROPN
ejde-502	299	5	g.	g.	PROPN
ejde-502	299	6	iagar	iagar	PROPN
ejde-502	299	7	,	,	PUNCT
ejde-502	299	8	a.	a.	NOUN
ejde-502	299	9	i.	i.	PROPN
ejde-502	299	10	muñoz	muñoz	PROPN
ejde-502	299	11	,	,	PUNCT
ejde-502	299	12	a.	a.	NOUN
ejde-502	299	13	sánchez	sánchez	PROPN
ejde-502	299	14	ejde-2023/72	ejde-2023/72	PROPN
ejde-502	299	15	=	=	SYM
ejde-502	299	16	k(m	k(m	PROPN
ejde-502	299	17	,	,	PUNCT
ejde-502	299	18	p	p	NOUN
ejde-502	299	19	)	)	PUNCT
ejde-502	299	20	m+	m+	NOUN
ejde-502	299	21	p−	p−	NOUN
ejde-502	299	22	2	2	NUM
ejde-502	299	23	m−	m−	PROPN
ejde-502	299	24	1	1	NUM
ejde-502	299	25	v	v	ADP
ejde-502	299	26	p−1	p−1	PROPN
ejde-502	299	27	m−1w	m−1w	NOUN
ejde-502	299	28	+	+	PROPN
ejde-502	299	29	k(m	k(m	PROPN
ejde-502	299	30	,	,	PUNCT
ejde-502	299	31	p	p	NOUN
ejde-502	299	32	)	)	PUNCT
ejde-502	299	33	(	(	PUNCT
ejde-502	299	34	2−m−	2−m−	NUM
ejde-502	299	35	p)(1−	p)(1−	ADJ
ejde-502	299	36	p	p	NOUN
ejde-502	299	37	)	)	PUNCT
ejde-502	299	38	(	(	PUNCT
ejde-502	299	39	m−	m−	PROPN
ejde-502	299	40	1)2	1)2	NUM
ejde-502	299	41	v	v	NOUN
ejde-502	299	42	p−m	p−m	PRON
ejde-502	299	43	m−1	m−1	PROPN
ejde-502	299	44	|∇v|2	|∇v|2	X
ejde-502	300	1	=	=	SYM
ejde-502	300	2	r1(x	r1(x	NOUN
ejde-502	300	3	,	,	PUNCT
ejde-502	300	4	v)w	v)w	ADJ
ejde-502	301	1	+	+	PROPN
ejde-502	301	2	r2(x	r2(x	PROPN
ejde-502	301	3	,	,	PUNCT
ejde-502	301	4	v	v	NOUN
ejde-502	301	5	)	)	PUNCT
ejde-502	301	6	,	,	PUNCT
ejde-502	302	1	where	where	SCONJ
ejde-502	302	2	r1(x	r1(x	NOUN
ejde-502	302	3	,	,	PUNCT
ejde-502	302	4	v	v	NOUN
ejde-502	302	5	)	)	PUNCT
ejde-502	302	6	=	=	SYM
ejde-502	302	7	k(m	k(m	PROPN
ejde-502	302	8	,	,	PUNCT
ejde-502	302	9	p	p	NOUN
ejde-502	302	10	)	)	PUNCT
ejde-502	302	11	m+	m+	NOUN
ejde-502	302	12	p−	p−	NOUN
ejde-502	302	13	2	2	NUM
ejde-502	302	14	m−	m−	PROPN
ejde-502	302	15	1	1	NUM
ejde-502	302	16	v	v	ADP
ejde-502	302	17	p−1	p−1	PROPN
ejde-502	302	18	m−1	m−1	PROPN
ejde-502	302	19	<	<	X
ejde-502	302	20	0	0	NUM
ejde-502	302	21	,	,	PUNCT
ejde-502	302	22	r2(x	r2(x	NOUN
ejde-502	302	23	,	,	PUNCT
ejde-502	302	24	v	v	NOUN
ejde-502	302	25	)	)	PUNCT
ejde-502	302	26	=	=	SYM
ejde-502	302	27	k(m	k(m	PROPN
ejde-502	302	28	,	,	PUNCT
ejde-502	302	29	p	p	NOUN
ejde-502	302	30	)	)	PUNCT
ejde-502	302	31	(	(	PUNCT
ejde-502	302	32	2−m−	2−m−	NUM
ejde-502	302	33	p)(1−	p)(1−	ADJ
ejde-502	302	34	p	p	NOUN
ejde-502	302	35	)	)	PUNCT
ejde-502	302	36	(	(	PUNCT
ejde-502	302	37	m−	m−	PROPN
ejde-502	302	38	1)2	1)2	NUM
ejde-502	302	39	v	v	NOUN
ejde-502	302	40	p−m	p−m	PRON
ejde-502	302	41	m−1	m−1	PROPN
ejde-502	302	42	|∇v|2	|∇v|2	X
ejde-502	302	43	≥	≥	NOUN
ejde-502	302	44	0	0	NUM
ejde-502	302	45	.	.	PUNCT
ejde-502	303	1	on	on	ADP
ejde-502	303	2	the	the	DET
ejde-502	303	3	other	other	ADJ
ejde-502	303	4	hand	hand	NOUN
ejde-502	303	5	,	,	PUNCT
ejde-502	303	6	the	the	DET
ejde-502	303	7	diffusion	diffusion	NOUN
ejde-502	303	8	term	term	NOUN
ejde-502	303	9	can	can	AUX
ejde-502	303	10	be	be	AUX
ejde-502	303	11	worked	work	VERB
ejde-502	303	12	out	out	ADP
ejde-502	303	13	as	as	ADP
ejde-502	303	14	in	in	ADP
ejde-502	303	15	[	[	X
ejde-502	303	16	36	36	NUM
ejde-502	303	17	,	,	PUNCT
ejde-502	303	18	proposition	proposition	NOUN
ejde-502	303	19	9.4	9.4	NUM
ejde-502	303	20	]	]	PUNCT
ejde-502	303	21	to	to	PART
ejde-502	303	22	obtain	obtain	VERB
ejde-502	303	23	wt	wt	NOUN
ejde-502	303	24	=	=	PUNCT
ejde-502	303	25	(	(	PUNCT
ejde-502	303	26	m−	m−	PROPN
ejde-502	303	27	1)v∆w	1)v∆w	NUM
ejde-502	303	28	+	+	CCONJ
ejde-502	303	29	2m∇v	2m∇v	NUM
ejde-502	303	30	·	·	PUNCT
ejde-502	303	31	∇w	∇w	NOUN
ejde-502	303	32	+	+	CCONJ
ejde-502	303	33	(	(	PUNCT
ejde-502	303	34	m−	m−	PROPN
ejde-502	303	35	1)w2	1)w2	PROPN
ejde-502	303	36	+	+	CCONJ
ejde-502	303	37	2	2	NUM
ejde-502	303	38	n∑	n∑	NOUN
ejde-502	303	39	i	i	NOUN
ejde-502	303	40	,	,	PUNCT
ejde-502	303	41	j=1	j=1	PROPN
ejde-502	303	42	(	(	PUNCT
ejde-502	303	43	∂2v	∂2v	X
ejde-502	303	44	∂xi∂xj	∂xi∂xj	PROPN
ejde-502	303	45	)	)	PUNCT
ejde-502	303	46	2	2	NUM
ejde-502	304	1	+	+	NOUN
ejde-502	304	2	r1(x	r1(x	NOUN
ejde-502	304	3	,	,	PUNCT
ejde-502	304	4	v)w	v)w	ADJ
ejde-502	305	1	+	+	PROPN
ejde-502	305	2	r2(x	r2(x	PROPN
ejde-502	305	3	,	,	PUNCT
ejde-502	305	4	v	v	NOUN
ejde-502	305	5	)	)	PUNCT
ejde-502	305	6	≥	≥	NOUN
ejde-502	305	7	(	(	PUNCT
ejde-502	305	8	m−	m−	PROPN
ejde-502	305	9	1)v∆w	1)v∆w	NUM
ejde-502	305	10	+	+	CCONJ
ejde-502	306	1	2m∇v	2m∇v	NUM
ejde-502	306	2	·	·	PUNCT
ejde-502	306	3	∇w	∇w	NOUN
ejde-502	306	4	+	+	CCONJ
ejde-502	306	5	(	(	PUNCT
ejde-502	306	6	m−	m−	PROPN
ejde-502	306	7	1	1	NUM
ejde-502	306	8	+	+	CCONJ
ejde-502	306	9	2	2	NUM
ejde-502	306	10	n	n	NOUN
ejde-502	306	11	)	)	PUNCT
ejde-502	306	12	w2	w2	NOUN
ejde-502	306	13	+	+	PROPN
ejde-502	306	14	r1(x	r1(x	PROPN
ejde-502	306	15	,	,	PUNCT
ejde-502	306	16	v)w	v)w	ADJ
ejde-502	306	17	+	+	PROPN
ejde-502	306	18	r2(x	r2(x	PROPN
ejde-502	306	19	,	,	PUNCT
ejde-502	306	20	v	v	NOUN
ejde-502	306	21	)	)	PUNCT
ejde-502	306	22	,	,	PUNCT
ejde-502	306	23	after	after	ADP
ejde-502	306	24	a	a	DET
ejde-502	306	25	straightforward	straightforward	ADJ
ejde-502	306	26	application	application	NOUN
ejde-502	306	27	of	of	ADP
ejde-502	306	28	the	the	DET
ejde-502	306	29	cauchy	cauchy	PROPN
ejde-502	306	30	-	-	PUNCT
ejde-502	306	31	schwartz	schwartz	PROPN
ejde-502	306	32	inequality	inequality	NOUN
ejde-502	306	33	.	.	PUNCT
ejde-502	307	1	this	this	PRON
ejde-502	307	2	can	can	AUX
ejde-502	307	3	be	be	AUX
ejde-502	307	4	written	write	VERB
ejde-502	307	5	equivalently	equivalently	ADV
ejde-502	307	6	as	as	ADP
ejde-502	307	7	lw	lw	PROPN
ejde-502	307	8	≥	≥	PRON
ejde-502	307	9	r2(x	r2(x	PROPN
ejde-502	307	10	,	,	PUNCT
ejde-502	307	11	v	v	NOUN
ejde-502	307	12	)	)	PUNCT
ejde-502	307	13	,	,	PUNCT
ejde-502	307	14	where	where	SCONJ
ejde-502	307	15	lw	lw	NOUN
ejde-502	307	16	:	:	PUNCT
ejde-502	307	17	=	=	PUNCT
ejde-502	307	18	wt	wt	ADP
ejde-502	307	19	−	−	PROPN
ejde-502	307	20	(	(	PUNCT
ejde-502	307	21	m−	m−	PROPN
ejde-502	307	22	1)v∆w	1)v∆w	NUM
ejde-502	307	23	−	−	PROPN
ejde-502	307	24	2m∇v	2m∇v	NUM
ejde-502	307	25	·	·	SYM
ejde-502	307	26	∇w	∇w	NOUN
ejde-502	307	27	−	−	PROPN
ejde-502	308	1	(	(	PUNCT
ejde-502	308	2	m−	m−	PROPN
ejde-502	308	3	1	1	NUM
ejde-502	308	4	+	+	CCONJ
ejde-502	308	5	2	2	NUM
ejde-502	308	6	n	n	NOUN
ejde-502	308	7	)	)	PUNCT
ejde-502	308	8	w2	w2	NOUN
ejde-502	308	9	−r1(x	−r1(x	NOUN
ejde-502	308	10	,	,	PUNCT
ejde-502	308	11	v)w	v)w	ADJ
ejde-502	308	12	is	be	AUX
ejde-502	308	13	a	a	DET
ejde-502	308	14	uniformly	uniformly	ADV
ejde-502	308	15	parabolic	parabolic	ADJ
ejde-502	308	16	operator	operator	NOUN
ejde-502	308	17	.	.	PUNCT
ejde-502	309	1	since	since	SCONJ
ejde-502	309	2	r2(x	r2(x	PROPN
ejde-502	309	3	,	,	PUNCT
ejde-502	309	4	v	v	NOUN
ejde-502	309	5	)	)	PUNCT
ejde-502	309	6	≥	≥	NOUN
ejde-502	309	7	0	0	NUM
ejde-502	309	8	,	,	PUNCT
ejde-502	309	9	we	we	PRON
ejde-502	309	10	deduce	deduce	VERB
ejde-502	309	11	that	that	SCONJ
ejde-502	309	12	lw	lw	PROPN
ejde-502	309	13	≥	≥	NOUN
ejde-502	309	14	0	0	PUNCT
ejde-502	310	1	and	and	CCONJ
ejde-502	310	2	we	we	PRON
ejde-502	310	3	aim	aim	VERB
ejde-502	310	4	to	to	PART
ejde-502	310	5	find	find	VERB
ejde-502	310	6	a	a	DET
ejde-502	310	7	subsolution	subsolution	NOUN
ejde-502	310	8	for	for	ADP
ejde-502	310	9	l	l	NOUN
ejde-502	310	10	depending	depend	VERB
ejde-502	310	11	only	only	ADV
ejde-502	310	12	on	on	ADP
ejde-502	310	13	time	time	NOUN
ejde-502	310	14	.	.	PUNCT
ejde-502	311	1	we	we	PRON
ejde-502	311	2	thus	thus	ADV
ejde-502	311	3	take	take	VERB
ejde-502	311	4	for	for	ADP
ejde-502	311	5	t	t	PROPN
ejde-502	311	6	>	>	X
ejde-502	311	7	0	0	PUNCT
ejde-502	312	1	w	w	NOUN
ejde-502	312	2	(	(	PUNCT
ejde-502	312	3	x	x	PROPN
ejde-502	312	4	,	,	PUNCT
ejde-502	312	5	t	t	PROPN
ejde-502	312	6	)	)	PUNCT
ejde-502	312	7	=	=	VERB
ejde-502	312	8	−c	−c	NOUN
ejde-502	312	9	t	t	PROPN
ejde-502	312	10	,	,	PUNCT
ejde-502	312	11	c	c	PROPN
ejde-502	312	12	=	=	SYM
ejde-502	312	13	n	n	CCONJ
ejde-502	312	14	n(m−	n(m−	PROPN
ejde-502	312	15	1	1	NUM
ejde-502	312	16	)	)	PUNCT
ejde-502	312	17	+	+	CCONJ
ejde-502	312	18	2	2	NUM
ejde-502	312	19	and	and	CCONJ
ejde-502	312	20	calculate	calculate	VERB
ejde-502	312	21	lw	lw	NOUN
ejde-502	312	22	=	=	PUNCT
ejde-502	312	23	c	c	PROPN
ejde-502	312	24	t2	t2	NOUN
ejde-502	312	25	−	−	PROPN
ejde-502	312	26	c	c	PROPN
ejde-502	312	27	t2	t2	PROPN
ejde-502	312	28	+	+	CCONJ
ejde-502	312	29	cr1(x	cr1(x	NOUN
ejde-502	312	30	,	,	PUNCT
ejde-502	312	31	v	v	NOUN
ejde-502	312	32	)	)	PUNCT
ejde-502	312	33	t	t	X
ejde-502	312	34	<	<	X
ejde-502	312	35	0	0	NUM
ejde-502	312	36	,	,	PUNCT
ejde-502	312	37	since	since	SCONJ
ejde-502	312	38	m+	m+	NOUN
ejde-502	312	39	p−	p−	NOUN
ejde-502	312	40	2	2	NUM
ejde-502	312	41	<	<	X
ejde-502	312	42	0	0	NUM
ejde-502	312	43	in	in	ADP
ejde-502	312	44	our	our	PRON
ejde-502	312	45	range	range	NOUN
ejde-502	312	46	of	of	ADP
ejde-502	312	47	exponents	exponent	NOUN
ejde-502	312	48	.	.	PUNCT
ejde-502	313	1	applying	apply	VERB
ejde-502	313	2	the	the	DET
ejde-502	313	3	comparison	comparison	NOUN
ejde-502	313	4	principle	principle	NOUN
ejde-502	313	5	to	to	ADP
ejde-502	313	6	the	the	DET
ejde-502	313	7	parabolic	parabolic	ADJ
ejde-502	313	8	operator	operator	NOUN
ejde-502	313	9	l	l	NOUN
ejde-502	313	10	we	we	PRON
ejde-502	313	11	infer	infer	VERB
ejde-502	313	12	that	that	SCONJ
ejde-502	313	13	∆v(x	∆v(x	PROPN
ejde-502	313	14	,	,	PUNCT
ejde-502	313	15	t	t	PROPN
ejde-502	313	16	)	)	PUNCT
ejde-502	313	17	=	=	SYM
ejde-502	314	1	w(x	w(x	PROPN
ejde-502	314	2	,	,	PUNCT
ejde-502	314	3	t	t	PROPN
ejde-502	314	4	)	)	PUNCT
ejde-502	314	5	≥w	≥w	PROPN
ejde-502	314	6	(	(	PUNCT
ejde-502	314	7	t	t	NOUN
ejde-502	314	8	)	)	PUNCT
ejde-502	314	9	=	=	SYM
ejde-502	315	1	−	−	PROPN
ejde-502	315	2	n	n	CCONJ
ejde-502	315	3	(	(	PUNCT
ejde-502	315	4	n(m−	n(m−	PROPN
ejde-502	315	5	1	1	NUM
ejde-502	315	6	)	)	PUNCT
ejde-502	315	7	+	+	NUM
ejde-502	315	8	2)t	2)t	NUM
ejde-502	315	9	,	,	PUNCT
ejde-502	315	10	for	for	ADP
ejde-502	315	11	any	any	DET
ejde-502	315	12	(	(	PUNCT
ejde-502	315	13	x	x	NOUN
ejde-502	315	14	,	,	PUNCT
ejde-502	315	15	t	t	PROPN
ejde-502	315	16	)	)	PUNCT
ejde-502	315	17	∈	∈	PROPN
ejde-502	315	18	rn	rn	PROPN
ejde-502	315	19	×	×	PROPN
ejde-502	315	20	(	(	PUNCT
ejde-502	315	21	0,∞	0,∞	NOUN
ejde-502	315	22	)	)	PUNCT
ejde-502	315	23	,	,	PUNCT
ejde-502	315	24	as	as	SCONJ
ejde-502	315	25	stated	state	VERB
ejde-502	315	26	.	.	PUNCT
ejde-502	316	1	all	all	DET
ejde-502	316	2	the	the	DET
ejde-502	316	3	previous	previous	ADJ
ejde-502	316	4	calculations	calculation	NOUN
ejde-502	316	5	and	and	CCONJ
ejde-502	316	6	the	the	DET
ejde-502	316	7	application	application	NOUN
ejde-502	316	8	of	of	ADP
ejde-502	316	9	the	the	DET
ejde-502	316	10	maximum	maximum	ADJ
ejde-502	316	11	principle	principle	NOUN
ejde-502	316	12	to	to	ADP
ejde-502	316	13	the	the	DET
ejde-502	316	14	operator	operator	NOUN
ejde-502	316	15	l	l	NOUN
ejde-502	316	16	are	be	AUX
ejde-502	316	17	fully	fully	ADV
ejde-502	316	18	justified	justify	VERB
ejde-502	316	19	for	for	SCONJ
ejde-502	316	20	solutions	solution	NOUN
ejde-502	316	21	u	u	NOUN
ejde-502	316	22	such	such	ADJ
ejde-502	316	23	that	that	SCONJ
ejde-502	316	24	(	(	PUNCT
ejde-502	316	25	in	in	ADP
ejde-502	316	26	the	the	DET
ejde-502	316	27	pressure	pressure	NOUN
ejde-502	316	28	variable	variable	NOUN
ejde-502	316	29	)	)	PUNCT
ejde-502	316	30	l	l	NOUN
ejde-502	316	31	is	be	AUX
ejde-502	316	32	uniformly	uniformly	ADV
ejde-502	316	33	parabolic	parabolic	ADJ
ejde-502	316	34	,	,	PUNCT
ejde-502	316	35	that	that	ADV
ejde-502	316	36	is	is	ADV
ejde-502	316	37	,	,	PUNCT
ejde-502	316	38	when	when	SCONJ
ejde-502	316	39	v	v	NOUN
ejde-502	316	40	,	,	PUNCT
ejde-502	316	41	∇v	∇v	PROPN
ejde-502	316	42	are	be	AUX
ejde-502	316	43	bounded	bound	VERB
ejde-502	316	44	and	and	CCONJ
ejde-502	316	45	v	v	ADP
ejde-502	316	46	>	>	X
ejde-502	316	47	0	0	PUNCT
ejde-502	316	48	uniformly	uniformly	ADV
ejde-502	316	49	.	.	PUNCT
ejde-502	317	1	in	in	ADP
ejde-502	317	2	order	order	NOUN
ejde-502	317	3	to	to	PART
ejde-502	317	4	extend	extend	VERB
ejde-502	317	5	the	the	DET
ejde-502	317	6	aronson	aronson	PROPN
ejde-502	317	7	-	-	PUNCT
ejde-502	317	8	bénilan	bénilan	PROPN
ejde-502	317	9	estimates	estimate	VERB
ejde-502	317	10	to	to	ADP
ejde-502	317	11	general	general	ADJ
ejde-502	317	12	weak	weak	ADJ
ejde-502	317	13	solutions	solution	NOUN
ejde-502	317	14	,	,	PUNCT
ejde-502	317	15	we	we	PRON
ejde-502	317	16	proceed	proceed	VERB
ejde-502	317	17	by	by	ADP
ejde-502	317	18	approximation	approximation	NOUN
ejde-502	317	19	.	.	PUNCT
ejde-502	318	1	let	let	VERB
ejde-502	318	2	us	we	PRON
ejde-502	318	3	first	first	ADV
ejde-502	318	4	consider	consider	VERB
ejde-502	318	5	u0	u0	ADJ
ejde-502	318	6	to	to	PART
ejde-502	318	7	be	be	AUX
ejde-502	318	8	an	an	DET
ejde-502	318	9	initial	initial	ADJ
ejde-502	318	10	condition	condition	NOUN
ejde-502	318	11	as	as	ADP
ejde-502	318	12	in	in	ADP
ejde-502	318	13	(	(	PUNCT
ejde-502	318	14	1.4	1.4	NUM
ejde-502	318	15	)	)	PUNCT
ejde-502	318	16	and	and	CCONJ
ejde-502	318	17	continuous	continuous	ADJ
ejde-502	318	18	.	.	PUNCT
ejde-502	319	1	then	then	ADV
ejde-502	319	2	,	,	PUNCT
ejde-502	319	3	there	there	PRON
ejde-502	319	4	exists	exist	VERB
ejde-502	319	5	a	a	DET
ejde-502	319	6	unique	unique	ADJ
ejde-502	319	7	solution	solution	NOUN
ejde-502	319	8	u	u	NOUN
ejde-502	319	9	to	to	ADP
ejde-502	319	10	the	the	DET
ejde-502	319	11	cauchy	cauchy	ADJ
ejde-502	319	12	problem	problem	NOUN
ejde-502	319	13	(	(	PUNCT
ejde-502	319	14	1.5)-(1.2	1.5)-(1.2	NOUN
ejde-502	319	15	)	)	PUNCT
ejde-502	319	16	in	in	ADP
ejde-502	319	17	a	a	DET
ejde-502	319	18	time	time	NOUN
ejde-502	319	19	interval	interval	NOUN
ejde-502	319	20	(	(	PUNCT
ejde-502	319	21	0	0	NUM
ejde-502	319	22	,	,	PUNCT
ejde-502	319	23	t	t	PROPN
ejde-502	319	24	)	)	PUNCT
ejde-502	319	25	,	,	PUNCT
ejde-502	319	26	according	accord	VERB
ejde-502	319	27	to	to	ADP
ejde-502	319	28	[	[	X
ejde-502	319	29	27	27	NUM
ejde-502	319	30	]	]	PUNCT
ejde-502	319	31	.	.	PUNCT
ejde-502	320	1	let	let	VERB
ejde-502	320	2	uk	uk	PROPN
ejde-502	320	3	be	be	AUX
ejde-502	320	4	the	the	DET
ejde-502	320	5	solution	solution	NOUN
ejde-502	320	6	to	to	ADP
ejde-502	320	7	the	the	DET
ejde-502	320	8	cauchy	cauchy	ADJ
ejde-502	320	9	problem	problem	NOUN
ejde-502	320	10	with	with	ADP
ejde-502	320	11	initial	initial	ADJ
ejde-502	320	12	condition	condition	NOUN
ejde-502	320	13	u0,k(x	u0,k(x	VERB
ejde-502	320	14	)	)	PUNCT
ejde-502	320	15	=	=	SYM
ejde-502	320	16	u0(x	u0(x	NOUN
ejde-502	320	17	)	)	PUNCT
ejde-502	320	18	+	+	CCONJ
ejde-502	321	1	1	1	NUM
ejde-502	321	2	k	k	NOUN
ejde-502	321	3	,	,	PUNCT
ejde-502	321	4	for	for	ADP
ejde-502	321	5	each	each	DET
ejde-502	321	6	positive	positive	ADJ
ejde-502	321	7	integer	integer	NOUN
ejde-502	321	8	k.	k.	PROPN
ejde-502	322	1	we	we	PRON
ejde-502	322	2	infer	infer	VERB
ejde-502	322	3	from	from	ADP
ejde-502	322	4	[	[	X
ejde-502	322	5	27	27	NUM
ejde-502	322	6	,	,	PUNCT
ejde-502	322	7	theorem	theorem	VERB
ejde-502	322	8	2.1	2.1	NUM
ejde-502	322	9	]	]	PUNCT
ejde-502	322	10	and	and	CCONJ
ejde-502	322	11	its	its	PRON
ejde-502	322	12	proof	proof	NOUN
ejde-502	322	13	that	that	SCONJ
ejde-502	322	14	there	there	PRON
ejde-502	322	15	exists	exist	VERB
ejde-502	322	16	a	a	DET
ejde-502	322	17	unique	unique	ADJ
ejde-502	322	18	solution	solution	NOUN
ejde-502	322	19	uk	uk	PROPN
ejde-502	322	20	to	to	ADP
ejde-502	322	21	(	(	PUNCT
ejde-502	322	22	1.5	1.5	NUM
ejde-502	322	23	)	)	PUNCT
ejde-502	322	24	with	with	ADP
ejde-502	322	25	initial	initial	ADJ
ejde-502	322	26	condition	condition	NOUN
ejde-502	322	27	u0,k	u0,k	PROPN
ejde-502	322	28	,	,	PUNCT
ejde-502	322	29	and	and	CCONJ
ejde-502	322	30	it	it	PRON
ejde-502	322	31	satisfies	satisfy	VERB
ejde-502	322	32	uk(x	uk(x	ADP
ejde-502	322	33	,	,	PUNCT
ejde-502	322	34	t	t	PROPN
ejde-502	322	35	)	)	PUNCT
ejde-502	322	36	≥	≥	NOUN
ejde-502	322	37	1	1	NUM
ejde-502	322	38	/	/	SYM
ejde-502	322	39	k	k	PROPN
ejde-502	322	40	for	for	ADP
ejde-502	322	41	all	all	DET
ejde-502	322	42	x	x	SYM
ejde-502	322	43	∈	∈	PROPN
ejde-502	322	44	rn	rn	PROPN
ejde-502	322	45	,	,	PUNCT
ejde-502	322	46	t	t	PROPN
ejde-502	322	47	>	>	X
ejde-502	322	48	0	0	X
ejde-502	322	49	.	.	PUNCT
ejde-502	323	1	we	we	PRON
ejde-502	323	2	further	far	ADV
ejde-502	323	3	find	find	VERB
ejde-502	323	4	from	from	ADP
ejde-502	323	5	[	[	X
ejde-502	323	6	23	23	NUM
ejde-502	323	7	,	,	PUNCT
ejde-502	323	8	theorem	theorem	VERB
ejde-502	323	9	8.1	8.1	NUM
ejde-502	323	10	,	,	PUNCT
ejde-502	323	11	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	323	12	reaction	reaction	NOUN
ejde-502	323	13	-	-	PUNCT
ejde-502	323	14	diffusion	diffusion	NOUN
ejde-502	323	15	with	with	ADP
ejde-502	323	16	weighted	weight	VERB
ejde-502	323	17	strong	strong	ADJ
ejde-502	323	18	reaction	reaction	NOUN
ejde-502	323	19	15	15	NUM
ejde-502	323	20	chapter	chapter	NOUN
ejde-502	323	21	v	v	NOUN
ejde-502	323	22	]	]	X
ejde-502	323	23	(	(	PUNCT
ejde-502	323	24	which	which	PRON
ejde-502	323	25	applies	apply	VERB
ejde-502	323	26	for	for	ADP
ejde-502	323	27	our	our	PRON
ejde-502	323	28	approximating	approximate	VERB
ejde-502	323	29	solutions	solution	NOUN
ejde-502	323	30	uk	uk	PROPN
ejde-502	323	31	since	since	SCONJ
ejde-502	323	32	they	they	PRON
ejde-502	323	33	are	be	AUX
ejde-502	323	34	now	now	ADV
ejde-502	323	35	bounded	bound	VERB
ejde-502	323	36	from	from	ADP
ejde-502	323	37	below	below	ADV
ejde-502	323	38	by	by	ADP
ejde-502	323	39	a	a	DET
ejde-502	323	40	positive	positive	ADJ
ejde-502	323	41	constant	constant	NOUN
ejde-502	323	42	)	)	PUNCT
ejde-502	323	43	that	that	SCONJ
ejde-502	323	44	uk	uk	PROPN
ejde-502	323	45	has	have	VERB
ejde-502	323	46	the	the	DET
ejde-502	323	47	regularity	regularity	NOUN
ejde-502	323	48	required	require	VERB
ejde-502	323	49	for	for	ADP
ejde-502	323	50	(	(	PUNCT
ejde-502	323	51	1.5	1.5	NUM
ejde-502	323	52	)	)	PUNCT
ejde-502	323	53	to	to	PART
ejde-502	323	54	hold	hold	VERB
ejde-502	323	55	in	in	ADP
ejde-502	323	56	a	a	DET
ejde-502	323	57	classical	classical	ADJ
ejde-502	323	58	sense	sense	NOUN
ejde-502	323	59	and	and	CCONJ
ejde-502	323	60	all	all	DET
ejde-502	323	61	the	the	DET
ejde-502	323	62	space	space	NOUN
ejde-502	323	63	derivatives	derivative	NOUN
ejde-502	323	64	of	of	ADP
ejde-502	323	65	uk	uk	PROPN
ejde-502	323	66	up	up	ADP
ejde-502	323	67	to	to	ADP
ejde-502	323	68	the	the	DET
ejde-502	323	69	second	second	ADJ
ejde-502	323	70	order	order	NOUN
ejde-502	323	71	and	and	CCONJ
ejde-502	323	72	time	time	NOUN
ejde-502	323	73	derivatives	derivative	NOUN
ejde-502	323	74	of	of	ADP
ejde-502	323	75	uk	uk	PROPN
ejde-502	323	76	up	up	ADP
ejde-502	323	77	to	to	PART
ejde-502	323	78	order	order	VERB
ejde-502	323	79	one	one	PRON
ejde-502	323	80	are	be	AUX
ejde-502	323	81	uniformly	uniformly	ADV
ejde-502	323	82	bounded	bound	VERB
ejde-502	323	83	.	.	PUNCT
ejde-502	324	1	in	in	ADP
ejde-502	324	2	this	this	DET
ejde-502	324	3	case	case	NOUN
ejde-502	324	4	,	,	PUNCT
ejde-502	324	5	the	the	DET
ejde-502	324	6	previous	previous	ADJ
ejde-502	324	7	calculation	calculation	NOUN
ejde-502	324	8	applies	apply	VERB
ejde-502	324	9	rigorously	rigorously	ADV
ejde-502	324	10	for	for	ADP
ejde-502	324	11	uk	uk	PROPN
ejde-502	324	12	and	and	CCONJ
ejde-502	324	13	we	we	PRON
ejde-502	324	14	obtain	obtain	VERB
ejde-502	324	15	that	that	DET
ejde-502	324	16	∆vk(x	∆vk(x	PROPN
ejde-502	324	17	,	,	PUNCT
ejde-502	324	18	t	t	PROPN
ejde-502	324	19	)	)	PUNCT
ejde-502	324	20	≥	≥	NOUN
ejde-502	324	21	−k	−k	PROPN
ejde-502	324	22	t	t	PROPN
ejde-502	324	23	,	,	PUNCT
ejde-502	324	24	vk	vk	VERB
ejde-502	324	25	=	=	VERB
ejde-502	324	26	m	m	VERB
ejde-502	324	27	m−	m−	PROPN
ejde-502	324	28	1	1	NUM
ejde-502	325	1	um−1	um−1	PROPN
ejde-502	325	2	k	k	X
ejde-502	325	3	.	.	PUNCT
ejde-502	326	1	(	(	PUNCT
ejde-502	326	2	4.2	4.2	NUM
ejde-502	326	3	)	)	PUNCT
ejde-502	326	4	moreover	moreover	ADV
ejde-502	326	5	,	,	PUNCT
ejde-502	326	6	the	the	DET
ejde-502	326	7	comparison	comparison	NOUN
ejde-502	326	8	principle	principle	NOUN
ejde-502	326	9	for	for	ADP
ejde-502	326	10	(	(	PUNCT
ejde-502	326	11	1.5	1.5	NUM
ejde-502	326	12	)	)	PUNCT
ejde-502	326	13	(	(	PUNCT
ejde-502	326	14	see	see	VERB
ejde-502	326	15	[	[	X
ejde-502	326	16	27	27	NUM
ejde-502	326	17	,	,	PUNCT
ejde-502	326	18	theorem	theorem	VERB
ejde-502	326	19	2.1	2.1	NUM
ejde-502	326	20	]	]	PUNCT
ejde-502	326	21	)	)	PUNCT
ejde-502	326	22	entails	entail	VERB
ejde-502	326	23	that	that	SCONJ
ejde-502	326	24	solutions	solution	VERB
ejde-502	326	25	uk	uk	PROPN
ejde-502	326	26	for	for	ADP
ejde-502	326	27	k	k	PROPN
ejde-502	326	28	≥	≥	NUM
ejde-502	326	29	1	1	NUM
ejde-502	326	30	form	form	VERB
ejde-502	326	31	a	a	DET
ejde-502	326	32	non	non	ADJ
ejde-502	326	33	-	-	ADJ
ejde-502	326	34	increasing	increasing	ADJ
ejde-502	326	35	sequence	sequence	NOUN
ejde-502	326	36	of	of	ADP
ejde-502	326	37	functions	function	NOUN
ejde-502	326	38	,	,	PUNCT
ejde-502	326	39	thus	thus	ADV
ejde-502	326	40	there	there	PRON
ejde-502	326	41	exists	exist	VERB
ejde-502	326	42	a	a	DET
ejde-502	326	43	limit	limit	NOUN
ejde-502	326	44	u(x	u(x	NOUN
ejde-502	326	45	,	,	PUNCT
ejde-502	326	46	t	t	NOUN
ejde-502	326	47	)	)	PUNCT
ejde-502	327	1	=	=	PROPN
ejde-502	327	2	lim	lim	PROPN
ejde-502	327	3	k→∞	k→∞	PROPN
ejde-502	327	4	uk(x	uk(x	PROPN
ejde-502	327	5	,	,	PUNCT
ejde-502	327	6	t	t	PROPN
ejde-502	327	7	)	)	PUNCT
ejde-502	327	8	,	,	PUNCT
ejde-502	327	9	(	(	PUNCT
ejde-502	327	10	x	x	X
ejde-502	327	11	,	,	PUNCT
ejde-502	327	12	t	t	PROPN
ejde-502	327	13	)	)	PUNCT
ejde-502	327	14	∈	∈	PROPN
ejde-502	327	15	rn	rn	PROPN
ejde-502	327	16	×	×	PROPN
ejde-502	327	17	(	(	PUNCT
ejde-502	327	18	0	0	NUM
ejde-502	327	19	,	,	PUNCT
ejde-502	327	20	t	t	NOUN
ejde-502	327	21	)	)	PUNCT
ejde-502	327	22	,	,	PUNCT
ejde-502	327	23	and	and	CCONJ
ejde-502	327	24	the	the	DET
ejde-502	327	25	uniform	uniform	NOUN
ejde-502	327	26	bound	bind	VERB
ejde-502	327	27	of	of	ADP
ejde-502	327	28	uk	uk	PROPN
ejde-502	327	29	and	and	CCONJ
ejde-502	327	30	their	their	PRON
ejde-502	327	31	derivatives	derivative	NOUN
ejde-502	327	32	up	up	ADP
ejde-502	327	33	to	to	ADP
ejde-502	327	34	second	second	ADJ
ejde-502	327	35	order	order	NOUN
ejde-502	327	36	together	together	ADV
ejde-502	327	37	with	with	ADP
ejde-502	327	38	the	the	DET
ejde-502	327	39	arzelá-ascoli	arzelá-ascoli	NOUN
ejde-502	327	40	theorem	theorem	VERB
ejde-502	327	41	imply	imply	ADV
ejde-502	327	42	that	that	SCONJ
ejde-502	327	43	uk	uk	PROPN
ejde-502	327	44	→	→	SYM
ejde-502	327	45	u	u	X
ejde-502	327	46	locally	locally	ADV
ejde-502	327	47	uniformly	uniformly	ADV
ejde-502	327	48	and	and	CCONJ
ejde-502	327	49	the	the	DET
ejde-502	327	50	same	same	ADJ
ejde-502	327	51	holds	hold	VERB
ejde-502	327	52	true	true	ADJ
ejde-502	327	53	for	for	ADP
ejde-502	327	54	their	their	PRON
ejde-502	327	55	first	first	ADJ
ejde-502	327	56	order	order	NOUN
ejde-502	327	57	derivatives	derivative	NOUN
ejde-502	327	58	with	with	ADP
ejde-502	327	59	respect	respect	NOUN
ejde-502	327	60	to	to	ADP
ejde-502	327	61	the	the	DET
ejde-502	327	62	space	space	NOUN
ejde-502	327	63	variables	variable	NOUN
ejde-502	327	64	.	.	PUNCT
ejde-502	328	1	then	then	ADV
ejde-502	328	2	,	,	PUNCT
ejde-502	328	3	the	the	DET
ejde-502	328	4	uniform	uniform	NOUN
ejde-502	328	5	boundedness	boundedness	NOUN
ejde-502	328	6	of	of	ADP
ejde-502	328	7	uk	uk	PROPN
ejde-502	328	8	and	and	CCONJ
ejde-502	328	9	∂tuk	∂tuk	NOUN
ejde-502	328	10	gives	give	VERB
ejde-502	328	11	the	the	DET
ejde-502	328	12	continuity	continuity	NOUN
ejde-502	328	13	with	with	ADP
ejde-502	328	14	respect	respect	NOUN
ejde-502	328	15	to	to	ADP
ejde-502	328	16	the	the	DET
ejde-502	328	17	time	time	NOUN
ejde-502	328	18	variable	variable	ADJ
ejde-502	328	19	over	over	ADP
ejde-502	328	20	(	(	PUNCT
ejde-502	328	21	0	0	NUM
ejde-502	328	22	,	,	PUNCT
ejde-502	328	23	t	t	PROPN
ejde-502	328	24	)	)	PUNCT
ejde-502	328	25	of	of	ADP
ejde-502	328	26	the	the	DET
ejde-502	328	27	limit	limit	NOUN
ejde-502	328	28	function	function	NOUN
ejde-502	328	29	u	u	NOUN
ejde-502	328	30	,	,	PUNCT
ejde-502	328	31	while	while	SCONJ
ejde-502	328	32	the	the	DET
ejde-502	328	33	fact	fact	NOUN
ejde-502	328	34	that	that	SCONJ
ejde-502	328	35	u	u	PRON
ejde-502	328	36	belongs	belong	VERB
ejde-502	328	37	to	to	ADP
ejde-502	328	38	l∞loc	l∞loc	NOUN
ejde-502	328	39	is	be	AUX
ejde-502	328	40	obvious	obvious	ADJ
ejde-502	328	41	,	,	PUNCT
ejde-502	328	42	as	as	SCONJ
ejde-502	328	43	it	it	PRON
ejde-502	328	44	is	be	AUX
ejde-502	328	45	bounded	bound	VERB
ejde-502	328	46	from	from	ADP
ejde-502	328	47	above	above	ADV
ejde-502	328	48	by	by	ADP
ejde-502	328	49	any	any	DET
ejde-502	328	50	uk	uk	PROPN
ejde-502	328	51	.	.	PUNCT
ejde-502	329	1	we	we	PRON
ejde-502	329	2	thus	thus	ADV
ejde-502	329	3	fulfill	fulfill	VERB
ejde-502	329	4	the	the	DET
ejde-502	329	5	regularity	regularity	NOUN
ejde-502	329	6	assumption	assumption	NOUN
ejde-502	329	7	(	(	PUNCT
ejde-502	329	8	a	a	X
ejde-502	329	9	)	)	PUNCT
ejde-502	329	10	in	in	ADP
ejde-502	329	11	definition	definition	NOUN
ejde-502	329	12	1.1	1.1	NUM
ejde-502	329	13	.	.	PUNCT
ejde-502	330	1	the	the	DET
ejde-502	330	2	monotone	monotone	ADJ
ejde-502	330	3	convergence	convergence	NOUN
ejde-502	330	4	theorem	theorem	NOUN
ejde-502	330	5	then	then	ADV
ejde-502	330	6	easily	easily	ADV
ejde-502	330	7	gives	give	VERB
ejde-502	330	8	that	that	SCONJ
ejde-502	330	9	u	u	PRON
ejde-502	330	10	satisfies	satisfy	VERB
ejde-502	330	11	assumptions	assumption	NOUN
ejde-502	330	12	(	(	PUNCT
ejde-502	330	13	b	b	NOUN
ejde-502	330	14	)	)	PUNCT
ejde-502	330	15	and	and	CCONJ
ejde-502	330	16	(	(	PUNCT
ejde-502	330	17	c	c	X
ejde-502	330	18	)	)	PUNCT
ejde-502	330	19	in	in	ADP
ejde-502	330	20	definition	definition	NOUN
ejde-502	330	21	1.1	1.1	NUM
ejde-502	330	22	,	,	PUNCT
ejde-502	330	23	hence	hence	ADV
ejde-502	330	24	,	,	PUNCT
ejde-502	330	25	since	since	SCONJ
ejde-502	330	26	uk(x	uk(x	NOUN
ejde-502	330	27	,	,	PUNCT
ejde-502	330	28	0	0	NUM
ejde-502	330	29	)	)	PUNCT
ejde-502	330	30	=	=	PUNCT
ejde-502	331	1	u0,k(x)→	u0,k(x)→	PROPN
ejde-502	331	2	u0(x	u0(x	NUM
ejde-502	331	3	)	)	PUNCT
ejde-502	331	4	as	as	SCONJ
ejde-502	331	5	k	k	PROPN
ejde-502	331	6	→∞	→∞	X
ejde-502	331	7	uniformly	uniformly	ADV
ejde-502	331	8	on	on	ADP
ejde-502	331	9	rn	rn	PROPN
ejde-502	331	10	,	,	PUNCT
ejde-502	331	11	we	we	PRON
ejde-502	331	12	readily	readily	ADV
ejde-502	331	13	infer	infer	VERB
ejde-502	331	14	that	that	SCONJ
ejde-502	331	15	u	u	NOUN
ejde-502	331	16	is	be	AUX
ejde-502	331	17	a	a	DET
ejde-502	331	18	weak	weak	ADJ
ejde-502	331	19	solution	solution	NOUN
ejde-502	331	20	to	to	ADP
ejde-502	331	21	the	the	DET
ejde-502	331	22	cauchy	cauchy	ADJ
ejde-502	331	23	problem	problem	NOUN
ejde-502	331	24	(	(	PUNCT
ejde-502	331	25	1.5)-(1.2	1.5)-(1.2	NOUN
ejde-502	331	26	)	)	PUNCT
ejde-502	331	27	.	.	PUNCT
ejde-502	332	1	uniqueness	uniqueness	NOUN
ejde-502	332	2	of	of	ADP
ejde-502	332	3	solutions	solution	NOUN
ejde-502	332	4	to	to	ADP
ejde-502	332	5	the	the	DET
ejde-502	332	6	latter	latter	ADJ
ejde-502	332	7	cauchy	cauchy	PROPN
ejde-502	332	8	problem	problem	NOUN
ejde-502	332	9	,	,	PUNCT
ejde-502	332	10	established	establish	VERB
ejde-502	332	11	in	in	ADP
ejde-502	332	12	[	[	X
ejde-502	332	13	26	26	NUM
ejde-502	332	14	]	]	PUNCT
ejde-502	332	15	for	for	ADP
ejde-502	332	16	continuous	continuous	ADJ
ejde-502	332	17	and	and	CCONJ
ejde-502	332	18	bounded	bounded	ADJ
ejde-502	332	19	initial	initial	ADJ
ejde-502	332	20	conditions	condition	NOUN
ejde-502	332	21	,	,	PUNCT
ejde-502	332	22	then	then	ADV
ejde-502	332	23	proves	prove	VERB
ejde-502	332	24	that	that	SCONJ
ejde-502	332	25	u	u	NOUN
ejde-502	332	26	=	=	NOUN
ejde-502	332	27	u.	u.	PROPN
ejde-502	333	1	we	we	PRON
ejde-502	333	2	then	then	ADV
ejde-502	333	3	come	come	VERB
ejde-502	333	4	back	back	ADV
ejde-502	333	5	to	to	ADP
ejde-502	333	6	(	(	PUNCT
ejde-502	333	7	4.2	4.2	NUM
ejde-502	333	8	)	)	PUNCT
ejde-502	333	9	,	,	PUNCT
ejde-502	333	10	which	which	PRON
ejde-502	333	11	,	,	PUNCT
ejde-502	333	12	after	after	ADP
ejde-502	333	13	multiplication	multiplication	NOUN
ejde-502	333	14	by	by	ADP
ejde-502	333	15	a	a	DET
ejde-502	333	16	non	non	ADJ
ejde-502	333	17	-	-	ADJ
ejde-502	333	18	negative	negative	ADJ
ejde-502	333	19	test	test	NOUN
ejde-502	333	20	function	function	NOUN
ejde-502	333	21	and	and	CCONJ
ejde-502	333	22	integration	integration	NOUN
ejde-502	333	23	by	by	ADP
ejde-502	333	24	parts	part	NOUN
ejde-502	333	25	,	,	PUNCT
ejde-502	333	26	reads∫	reads∫	VERB
ejde-502	333	27	t	t	PROPN
ejde-502	333	28	0	0	NUM
ejde-502	333	29	∫	∫	PROPN
ejde-502	333	30	rn	rn	PROPN
ejde-502	333	31	(	(	PUNCT
ejde-502	333	32	vk(x	vk(x	PROPN
ejde-502	333	33	,	,	PUNCT
ejde-502	333	34	t)∆ϕ(x	t)∆ϕ(x	PROPN
ejde-502	333	35	,	,	PUNCT
ejde-502	333	36	t	t	PROPN
ejde-502	333	37	)	)	PUNCT
ejde-502	334	1	+	+	CCONJ
ejde-502	334	2	k	k	PROPN
ejde-502	334	3	t	t	PROPN
ejde-502	334	4	ϕ(x	ϕ(x	PROPN
ejde-502	334	5	,	,	PUNCT
ejde-502	334	6	t	t	PROPN
ejde-502	334	7	)	)	PUNCT
ejde-502	334	8	)	)	PUNCT
ejde-502	335	1	dx	dx	PROPN
ejde-502	335	2	dt	dt	X
ejde-502	335	3	≥	≥	PROPN
ejde-502	335	4	0	0	NUM
ejde-502	335	5	,	,	PUNCT
ejde-502	335	6	(	(	PUNCT
ejde-502	335	7	4.3	4.3	NUM
ejde-502	335	8	)	)	PUNCT
ejde-502	335	9	for	for	ADP
ejde-502	335	10	any	any	DET
ejde-502	335	11	ϕ	ϕ	PROPN
ejde-502	335	12	∈	∈	PROPN
ejde-502	335	13	c∞0	c∞0	X
ejde-502	335	14	(	(	PUNCT
ejde-502	335	15	rn	rn	PROPN
ejde-502	335	16	×	×	PROPN
ejde-502	335	17	(	(	PUNCT
ejde-502	335	18	0	0	NUM
ejde-502	335	19	,	,	PUNCT
ejde-502	335	20	t	t	NOUN
ejde-502	335	21	)	)	PUNCT
ejde-502	335	22	)	)	PUNCT
ejde-502	335	23	,	,	PUNCT
ejde-502	335	24	ϕ	ϕ	X
ejde-502	335	25	≥	≥	NOUN
ejde-502	335	26	0	0	NUM
ejde-502	335	27	.	.	PUNCT
ejde-502	336	1	we	we	PRON
ejde-502	336	2	pass	pass	VERB
ejde-502	336	3	to	to	ADP
ejde-502	336	4	the	the	DET
ejde-502	336	5	limit	limit	NOUN
ejde-502	336	6	in	in	ADP
ejde-502	336	7	(	(	PUNCT
ejde-502	336	8	4.3	4.3	NUM
ejde-502	336	9	)	)	PUNCT
ejde-502	336	10	as	as	ADP
ejde-502	336	11	k	k	PROPN
ejde-502	336	12	→	→	SYM
ejde-502	336	13	∞	∞	PROPN
ejde-502	336	14	,	,	PUNCT
ejde-502	336	15	taking	take	VERB
ejde-502	336	16	into	into	ADP
ejde-502	336	17	account	account	NOUN
ejde-502	336	18	that	that	SCONJ
ejde-502	336	19	vk	vk	VERB
ejde-502	336	20	→	→	SYM
ejde-502	336	21	v	v	NOUN
ejde-502	336	22	locally	locally	ADV
ejde-502	336	23	uniformly	uniformly	ADV
ejde-502	336	24	as	as	ADP
ejde-502	336	25	k	k	PROPN
ejde-502	336	26	→	→	SYM
ejde-502	336	27	∞	∞	PROPN
ejde-502	336	28	,	,	PUNCT
ejde-502	336	29	and	and	CCONJ
ejde-502	336	30	obtain	obtain	VERB
ejde-502	336	31	the	the	DET
ejde-502	336	32	claimed	claim	VERB
ejde-502	336	33	distributional	distributional	ADJ
ejde-502	336	34	form	form	NOUN
ejde-502	336	35	(	(	PUNCT
ejde-502	336	36	1.13	1.13	NUM
ejde-502	336	37	)	)	PUNCT
ejde-502	336	38	.	.	PUNCT
ejde-502	337	1	�	�	PROPN
ejde-502	337	2	we	we	PRON
ejde-502	337	3	end	end	VERB
ejde-502	337	4	this	this	DET
ejde-502	337	5	section	section	NOUN
ejde-502	337	6	with	with	ADP
ejde-502	337	7	a	a	DET
ejde-502	337	8	corollary	corollary	NOUN
ejde-502	337	9	which	which	PRON
ejde-502	337	10	will	will	AUX
ejde-502	337	11	be	be	AUX
ejde-502	337	12	used	use	VERB
ejde-502	337	13	in	in	ADP
ejde-502	337	14	the	the	DET
ejde-502	337	15	sequel	sequel	NOUN
ejde-502	337	16	.	.	PUNCT
ejde-502	338	1	corollary	corollary	ADJ
ejde-502	338	2	4.1	4.1	NUM
ejde-502	338	3	.	.	PUNCT
ejde-502	339	1	in	in	ADP
ejde-502	339	2	the	the	DET
ejde-502	339	3	same	same	ADJ
ejde-502	339	4	conditions	condition	NOUN
ejde-502	339	5	as	as	ADP
ejde-502	339	6	in	in	ADP
ejde-502	339	7	theorem	theorem	NOUN
ejde-502	339	8	1.4	1.4	NUM
ejde-502	339	9	,	,	PUNCT
ejde-502	339	10	we	we	PRON
ejde-502	339	11	have	have	VERB
ejde-502	339	12	ut	ut	PROPN
ejde-502	339	13	≥	≥	NUM
ejde-502	339	14	−	−	PROPN
ejde-502	339	15	ku	ku	PROPN
ejde-502	339	16	t	t	PROPN
ejde-502	339	17	,	,	PUNCT
ejde-502	339	18	k	k	PROPN
ejde-502	339	19	=	=	PUNCT
ejde-502	339	20	n	n	CCONJ
ejde-502	339	21	n(m−	n(m−	PROPN
ejde-502	339	22	1	1	NUM
ejde-502	339	23	)	)	PUNCT
ejde-502	339	24	+	+	NUM
ejde-502	339	25	2	2	NUM
ejde-502	339	26	,	,	PUNCT
ejde-502	339	27	in	in	ADP
ejde-502	339	28	the	the	DET
ejde-502	339	29	sense	sense	NOUN
ejde-502	339	30	of	of	ADP
ejde-502	339	31	distributions	distribution	NOUN
ejde-502	339	32	in	in	ADP
ejde-502	339	33	rn	rn	PROPN
ejde-502	339	34	.	.	PUNCT
ejde-502	340	1	proof	proof	NOUN
ejde-502	340	2	.	.	PUNCT
ejde-502	341	1	at	at	ADP
ejde-502	341	2	a	a	DET
ejde-502	341	3	formal	formal	ADJ
ejde-502	341	4	level	level	NOUN
ejde-502	341	5	,	,	PUNCT
ejde-502	341	6	we	we	PRON
ejde-502	341	7	infer	infer	VERB
ejde-502	341	8	from	from	ADP
ejde-502	341	9	(	(	PUNCT
ejde-502	341	10	4.1	4.1	NUM
ejde-502	341	11	)	)	PUNCT
ejde-502	341	12	that	that	PRON
ejde-502	341	13	vt	vt	PROPN
ejde-502	341	14	≥	≥	X
ejde-502	341	15	(	(	PUNCT
ejde-502	341	16	m−	m−	PROPN
ejde-502	341	17	1)v∆v	1)v∆v	NUM
ejde-502	341	18	,	,	PUNCT
ejde-502	341	19	hence	hence	ADV
ejde-502	341	20	(	(	PUNCT
ejde-502	341	21	m−	m−	PROPN
ejde-502	341	22	1	1	NUM
ejde-502	341	23	)	)	PUNCT
ejde-502	341	24	ut	ut	PROPN
ejde-502	341	25	u	u	PROPN
ejde-502	342	1	=	=	PROPN
ejde-502	342	2	vt	vt	PROPN
ejde-502	342	3	v	v	NUM
ejde-502	342	4	≥	≥	PROPN
ejde-502	342	5	(	(	PUNCT
ejde-502	342	6	m−	m−	PROPN
ejde-502	342	7	1)∆v	1)∆v	NUM
ejde-502	342	8	≥	≥	NOUN
ejde-502	342	9	−	−	NOUN
ejde-502	342	10	n(m−	n(m−	PROPN
ejde-502	342	11	1	1	NUM
ejde-502	342	12	)	)	PUNCT
ejde-502	342	13	(	(	PUNCT
ejde-502	342	14	n(m−	n(m−	PROPN
ejde-502	342	15	1	1	NUM
ejde-502	342	16	)	)	PUNCT
ejde-502	342	17	+	+	NUM
ejde-502	342	18	2)t	2)t	NUM
ejde-502	343	1	and	and	CCONJ
ejde-502	343	2	we	we	PRON
ejde-502	343	3	reach	reach	VERB
ejde-502	343	4	the	the	DET
ejde-502	343	5	conclusion	conclusion	NOUN
ejde-502	343	6	with	with	ADP
ejde-502	343	7	the	the	DET
ejde-502	343	8	same	same	ADJ
ejde-502	343	9	constant	constant	ADJ
ejde-502	343	10	k	k	NOUN
ejde-502	343	11	as	as	ADP
ejde-502	343	12	in	in	ADP
ejde-502	343	13	(	(	PUNCT
ejde-502	343	14	1.12	1.12	NUM
ejde-502	343	15	)	)	PUNCT
ejde-502	343	16	.	.	PUNCT
ejde-502	344	1	for	for	ADP
ejde-502	344	2	general	general	ADJ
ejde-502	344	3	weak	weak	ADJ
ejde-502	344	4	solutions	solution	NOUN
ejde-502	344	5	the	the	DET
ejde-502	344	6	estimate	estimate	NOUN
ejde-502	344	7	is	be	AUX
ejde-502	344	8	proved	prove	VERB
ejde-502	344	9	by	by	ADP
ejde-502	344	10	using	use	VERB
ejde-502	344	11	the	the	DET
ejde-502	344	12	same	same	ADJ
ejde-502	344	13	approximation	approximation	NOUN
ejde-502	344	14	as	as	ADP
ejde-502	344	15	in	in	ADP
ejde-502	344	16	the	the	DET
ejde-502	344	17	proof	proof	NOUN
ejde-502	344	18	of	of	ADP
ejde-502	344	19	theorem	theorem	ADJ
ejde-502	344	20	1.4	1.4	NUM
ejde-502	344	21	.	.	PUNCT
ejde-502	344	22	�	�	PROPN
ejde-502	344	23	remark	remark	PROPN
ejde-502	344	24	.	.	PUNCT
ejde-502	345	1	formal	formal	ADJ
ejde-502	345	2	proof	proof	NOUN
ejde-502	345	3	of	of	ADP
ejde-502	345	4	aronson	aronson	PROPN
ejde-502	345	5	-	-	PUNCT
ejde-502	345	6	bénilan	bénilan	PROPN
ejde-502	345	7	estimates	estimate	NOUN
ejde-502	345	8	for	for	ADP
ejde-502	345	9	σ	σ	PROPN
ejde-502	345	10	>	>	X
ejde-502	345	11	0	0	PROPN
ejde-502	345	12	.	.	PUNCT
ejde-502	346	1	at	at	ADP
ejde-502	346	2	a	a	DET
ejde-502	346	3	formal	formal	ADJ
ejde-502	346	4	level	level	NOUN
ejde-502	346	5	,	,	PUNCT
ejde-502	346	6	the	the	DET
ejde-502	346	7	aronson	aronson	PROPN
ejde-502	346	8	-	-	PUNCT
ejde-502	346	9	bénilan	bénilan	PROPN
ejde-502	346	10	estimates	estimate	NOUN
ejde-502	346	11	(	(	PUNCT
ejde-502	346	12	1.12	1.12	NUM
ejde-502	346	13	)	)	PUNCT
ejde-502	346	14	or	or	CCONJ
ejde-502	346	15	(	(	PUNCT
ejde-502	346	16	1.13	1.13	NUM
ejde-502	346	17	)	)	PUNCT
ejde-502	346	18	hold	hold	VERB
ejde-502	346	19	also	also	ADV
ejde-502	346	20	for	for	ADP
ejde-502	346	21	σ	σ	PROPN
ejde-502	346	22	>	>	SYM
ejde-502	346	23	0	0	PUNCT
ejde-502	346	24	and	and	CCONJ
ejde-502	346	25	n	n	PRON
ejde-502	346	26	≥	≥	NOUN
ejde-502	346	27	2	2	NUM
ejde-502	346	28	.	.	PUNCT
ejde-502	347	1	indeed	indeed	ADV
ejde-502	347	2	,	,	PUNCT
ejde-502	347	3	a	a	DET
ejde-502	347	4	slightly	slightly	ADV
ejde-502	347	5	longer	long	ADJ
ejde-502	347	6	but	but	CCONJ
ejde-502	347	7	straightforward	straightforward	ADJ
ejde-502	347	8	calculation	calculation	NOUN
ejde-502	347	9	along	along	ADP
ejde-502	347	10	the	the	DET
ejde-502	347	11	first	first	ADJ
ejde-502	347	12	few	few	ADJ
ejde-502	347	13	16	16	NUM
ejde-502	347	14	r.	r.	PROPN
ejde-502	347	15	g.	g.	PROPN
ejde-502	347	16	iagar	iagar	PROPN
ejde-502	347	17	,	,	PUNCT
ejde-502	347	18	a.	a.	NOUN
ejde-502	347	19	i.	i.	PROPN
ejde-502	347	20	muñoz	muñoz	PROPN
ejde-502	347	21	,	,	PUNCT
ejde-502	347	22	a.	a.	NOUN
ejde-502	347	23	sánchez	sánchez	PROPN
ejde-502	347	24	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	347	25	lines	line	NOUN
ejde-502	347	26	of	of	ADP
ejde-502	347	27	the	the	DET
ejde-502	347	28	proof	proof	NOUN
ejde-502	347	29	of	of	ADP
ejde-502	347	30	theorem	theorem	NOUN
ejde-502	347	31	1.4	1.4	NUM
ejde-502	347	32	by	by	ADP
ejde-502	347	33	computing	compute	VERB
ejde-502	347	34	the	the	DET
ejde-502	347	35	derivatives	derivative	NOUN
ejde-502	347	36	up	up	ADP
ejde-502	347	37	to	to	ADP
ejde-502	347	38	second	second	ADJ
ejde-502	347	39	order	order	NOUN
ejde-502	347	40	of	of	ADP
ejde-502	347	41	the	the	DET
ejde-502	347	42	reaction	reaction	NOUN
ejde-502	347	43	term	term	NOUN
ejde-502	347	44	,	,	PUNCT
ejde-502	347	45	but	but	CCONJ
ejde-502	347	46	applied	apply	VERB
ejde-502	347	47	to	to	ADP
ejde-502	347	48	(	(	PUNCT
ejde-502	347	49	1.1	1.1	NUM
ejde-502	347	50	)	)	PUNCT
ejde-502	347	51	with	with	ADP
ejde-502	347	52	σ	σ	PROPN
ejde-502	347	53	>	>	X
ejde-502	347	54	0	0	NUM
ejde-502	347	55	,	,	PUNCT
ejde-502	347	56	gives	give	VERB
ejde-502	347	57	n∑	n∑	PROPN
ejde-502	347	58	i=1	i=1	PROPN
ejde-502	347	59	∂2	∂2	PROPN
ejde-502	347	60	∂x2	∂x2	NOUN
ejde-502	347	61	i	i	PRON
ejde-502	347	62	k(m	k(m	PROPN
ejde-502	347	63	,	,	PUNCT
ejde-502	347	64	p)(1	p)(1	X
ejde-502	348	1	+	+	CCONJ
ejde-502	348	2	|x|)σv	|x|)σv	PROPN
ejde-502	348	3	m+p−2	m+p−2	PROPN
ejde-502	348	4	m−1	m−1	PROPN
ejde-502	348	5	=	=	SYM
ejde-502	348	6	k(m	k(m	PROPN
ejde-502	348	7	,	,	PUNCT
ejde-502	348	8	p	p	NOUN
ejde-502	348	9	)	)	PUNCT
ejde-502	348	10	m+	m+	NOUN
ejde-502	349	1	p−	p−	NOUN
ejde-502	349	2	2	2	NUM
ejde-502	349	3	m−	m−	PROPN
ejde-502	349	4	1	1	NUM
ejde-502	349	5	(	(	PUNCT
ejde-502	349	6	1	1	NUM
ejde-502	349	7	+	+	CCONJ
ejde-502	349	8	|x|)σv	|x|)σv	PROPN
ejde-502	349	9	p−1	p−1	PROPN
ejde-502	349	10	m−1w	m−1w	NOUN
ejde-502	349	11	+	+	PROPN
ejde-502	349	12	k(m	k(m	PROPN
ejde-502	349	13	,	,	PUNCT
ejde-502	349	14	p)(1	p)(1	X
ejde-502	349	15	+	+	CCONJ
ejde-502	349	16	|x|)σ−2v	|x|)σ−2v	PROPN
ejde-502	349	17	p−m	p−m	VERB
ejde-502	349	18	m−1	m−1	PROPN
ejde-502	349	19	[	[	PUNCT
ejde-502	349	20	(	(	PUNCT
ejde-502	349	21	2−m−	2−m−	NUM
ejde-502	349	22	p)(1−	p)(1−	ADJ
ejde-502	349	23	p	p	NOUN
ejde-502	349	24	)	)	PUNCT
ejde-502	349	25	(	(	PUNCT
ejde-502	349	26	m−	m−	PROPN
ejde-502	349	27	1)2	1)2	NUM
ejde-502	349	28	(	(	PUNCT
ejde-502	349	29	1	1	NUM
ejde-502	349	30	+	+	NUM
ejde-502	349	31	|x|)2|∇v|2	|x|)2|∇v|2	NOUN
ejde-502	350	1	−	−	NOUN
ejde-502	350	2	2σ	2σ	NOUN
ejde-502	351	1	2−m−	2−m−	NOUN
ejde-502	351	2	p	p	DET
ejde-502	351	3	m−	m−	PROPN
ejde-502	351	4	1	1	NUM
ejde-502	351	5	(	(	PUNCT
ejde-502	351	6	1	1	NUM
ejde-502	351	7	+	+	NUM
ejde-502	351	8	|x|)v	|x|)v	NOUN
ejde-502	351	9	x	x	SYM
ejde-502	351	10	|x|	|x|	PROPN
ejde-502	351	11	·	·	PUNCT
ejde-502	352	1	∇v	∇v	PROPN
ejde-502	352	2	+	+	CCONJ
ejde-502	352	3	σ	σ	PROPN
ejde-502	352	4	(	(	PUNCT
ejde-502	352	5	σ	σ	PROPN
ejde-502	352	6	−	−	PROPN
ejde-502	352	7	1	1	NUM
ejde-502	352	8	+	+	CCONJ
ejde-502	352	9	(	(	PUNCT
ejde-502	352	10	n	n	CCONJ
ejde-502	352	11	−	−	PROPN
ejde-502	352	12	1	1	NUM
ejde-502	352	13	)	)	PUNCT
ejde-502	352	14	1	1	NUM
ejde-502	353	1	+	+	CCONJ
ejde-502	353	2	|x|	|x|	PROPN
ejde-502	353	3	|x|	|x|	PROPN
ejde-502	353	4	)	)	PUNCT
ejde-502	353	5	v2	v2	NOUN
ejde-502	353	6	]	]	PUNCT
ejde-502	354	1	=	=	SYM
ejde-502	354	2	r1(x	r1(x	NOUN
ejde-502	354	3	,	,	PUNCT
ejde-502	354	4	v)w	v)w	ADJ
ejde-502	355	1	+	+	PROPN
ejde-502	355	2	r2(x	r2(x	PROPN
ejde-502	355	3	,	,	PUNCT
ejde-502	355	4	v	v	NOUN
ejde-502	355	5	)	)	PUNCT
ejde-502	355	6	,	,	PUNCT
ejde-502	356	1	where	where	SCONJ
ejde-502	356	2	r1(x	r1(x	NOUN
ejde-502	356	3	,	,	PUNCT
ejde-502	356	4	v	v	NOUN
ejde-502	356	5	)	)	PUNCT
ejde-502	356	6	=	=	SYM
ejde-502	356	7	k(m	k(m	PROPN
ejde-502	356	8	,	,	PUNCT
ejde-502	356	9	p	p	NOUN
ejde-502	356	10	)	)	PUNCT
ejde-502	356	11	m+	m+	NOUN
ejde-502	356	12	p−	p−	NOUN
ejde-502	356	13	2	2	NUM
ejde-502	356	14	m−	m−	PROPN
ejde-502	356	15	1	1	NUM
ejde-502	356	16	(	(	PUNCT
ejde-502	356	17	1	1	NUM
ejde-502	356	18	+	+	CCONJ
ejde-502	356	19	|x|)σv	|x|)σv	PROPN
ejde-502	356	20	p−1	p−1	NOUN
ejde-502	356	21	m−1	m−1	PROPN
ejde-502	356	22	<	<	X
ejde-502	356	23	0	0	PUNCT
ejde-502	356	24	and	and	CCONJ
ejde-502	356	25	r2(x	r2(x	PROPN
ejde-502	356	26	,	,	PUNCT
ejde-502	356	27	v	v	NOUN
ejde-502	356	28	)	)	PUNCT
ejde-502	356	29	gathers	gather	VERB
ejde-502	356	30	the	the	DET
ejde-502	356	31	rest	rest	NOUN
ejde-502	356	32	of	of	ADP
ejde-502	356	33	the	the	DET
ejde-502	356	34	terms	term	NOUN
ejde-502	356	35	.	.	PUNCT
ejde-502	357	1	in	in	ADP
ejde-502	357	2	order	order	NOUN
ejde-502	357	3	to	to	PART
ejde-502	357	4	proceed	proceed	VERB
ejde-502	357	5	with	with	ADP
ejde-502	357	6	the	the	DET
ejde-502	357	7	comparison	comparison	NOUN
ejde-502	357	8	principle	principle	NOUN
ejde-502	357	9	as	as	SCONJ
ejde-502	357	10	we	we	PRON
ejde-502	357	11	did	do	VERB
ejde-502	357	12	in	in	ADP
ejde-502	357	13	the	the	DET
ejde-502	357	14	body	body	NOUN
ejde-502	357	15	of	of	ADP
ejde-502	357	16	the	the	DET
ejde-502	357	17	proof	proof	NOUN
ejde-502	357	18	of	of	ADP
ejde-502	357	19	theorem	theorem	ADJ
ejde-502	357	20	1.4	1.4	NUM
ejde-502	357	21	,	,	PUNCT
ejde-502	357	22	we	we	PRON
ejde-502	357	23	still	still	ADV
ejde-502	357	24	need	need	VERB
ejde-502	357	25	to	to	PART
ejde-502	357	26	have	have	VERB
ejde-502	357	27	r2(x	r2(x	PROPN
ejde-502	357	28	,	,	PUNCT
ejde-502	357	29	v	v	NOUN
ejde-502	357	30	)	)	PUNCT
ejde-502	357	31	≥	≥	NOUN
ejde-502	357	32	0	0	NUM
ejde-502	357	33	.	.	PUNCT
ejde-502	358	1	to	to	ADP
ejde-502	358	2	this	this	DET
ejde-502	358	3	end	end	NOUN
ejde-502	358	4	,	,	PUNCT
ejde-502	358	5	we	we	PRON
ejde-502	358	6	write	write	VERB
ejde-502	358	7	r2	r2	PROPN
ejde-502	358	8	as	as	ADP
ejde-502	358	9	a	a	DET
ejde-502	358	10	square	square	NOUN
ejde-502	358	11	and	and	CCONJ
ejde-502	358	12	we	we	PRON
ejde-502	358	13	examine	examine	VERB
ejde-502	358	14	the	the	DET
ejde-502	358	15	remainders	remainder	NOUN
ejde-502	358	16	.	.	PUNCT
ejde-502	359	1	more	more	ADV
ejde-502	359	2	precisely	precisely	ADV
ejde-502	359	3	,	,	PUNCT
ejde-502	359	4	using	use	VERB
ejde-502	359	5	once	once	ADV
ejde-502	359	6	more	more	ADJ
ejde-502	359	7	a	a	DET
ejde-502	359	8	standard	standard	ADJ
ejde-502	359	9	cauchy	cauchy	NOUN
ejde-502	359	10	-	-	PUNCT
ejde-502	359	11	schwarz	schwarz	PROPN
ejde-502	359	12	inequality	inequality	NOUN
ejde-502	359	13	for	for	ADP
ejde-502	359	14	the	the	DET
ejde-502	359	15	scalar	scalar	ADJ
ejde-502	359	16	product	product	NOUN
ejde-502	359	17	∇v	∇v	PROPN
ejde-502	359	18	·	·	PUNCT
ejde-502	359	19	x/|x|	x/|x|	X
ejde-502	360	1	we	we	PRON
ejde-502	360	2	find	find	VERB
ejde-502	360	3	that	that	SCONJ
ejde-502	360	4	r2(x	r2(x	PROPN
ejde-502	360	5	,	,	PUNCT
ejde-502	360	6	v	v	NOUN
ejde-502	360	7	)	)	PUNCT
ejde-502	360	8	k(m	k(m	PROPN
ejde-502	360	9	,	,	PUNCT
ejde-502	360	10	p)(1	p)(1	X
ejde-502	360	11	+	+	CCONJ
ejde-502	360	12	|x|)σ−2v	|x|)σ−2v	PROPN
ejde-502	360	13	p−m	p−m	VERB
ejde-502	360	14	m−1	m−1	PROPN
ejde-502	360	15	≥	≥	NOUN
ejde-502	361	1	[	[	X
ejde-502	361	2	2−m−	2−m−	NOUN
ejde-502	361	3	p	p	NOUN
ejde-502	361	4	m−	m−	PROPN
ejde-502	361	5	1	1	NUM
ejde-502	361	6	(	(	PUNCT
ejde-502	361	7	1	1	NUM
ejde-502	361	8	+	+	NUM
ejde-502	361	9	|x|)|∇v|	|x|)|∇v|	NUM
ejde-502	361	10	−	−	NOUN
ejde-502	361	11	σv	σv	NOUN
ejde-502	361	12	]	]	X
ejde-502	361	13	2	2	NUM
ejde-502	361	14	+	+	CCONJ
ejde-502	361	15	2−m−	2−m−	NUM
ejde-502	361	16	p	p	NOUN
ejde-502	361	17	m−	m−	PROPN
ejde-502	361	18	1	1	NUM
ejde-502	361	19	(	(	PUNCT
ejde-502	361	20	1	1	NUM
ejde-502	361	21	+	+	NUM
ejde-502	361	22	|x|)2|∇v|2	|x|)2|∇v|2	NOUN
ejde-502	361	23	+	+	CCONJ
ejde-502	361	24	σ	σ	X
ejde-502	361	25	[	[	PUNCT
ejde-502	361	26	(	(	PUNCT
ejde-502	361	27	n	n	CCONJ
ejde-502	361	28	−	−	PROPN
ejde-502	361	29	1	1	NUM
ejde-502	361	30	)	)	PUNCT
ejde-502	361	31	1	1	NUM
ejde-502	362	1	+	+	CCONJ
ejde-502	363	1	|x|	|x|	PROPN
ejde-502	363	2	|x|	|x|	PROPN
ejde-502	363	3	−	−	NOUN
ejde-502	363	4	1	1	NUM
ejde-502	363	5	]	]	PUNCT
ejde-502	363	6	v2	v2	VERB
ejde-502	363	7	and	and	CCONJ
ejde-502	363	8	taking	take	VERB
ejde-502	363	9	into	into	ADP
ejde-502	363	10	account	account	NOUN
ejde-502	363	11	that	that	SCONJ
ejde-502	363	12	we	we	PRON
ejde-502	363	13	are	be	AUX
ejde-502	363	14	in	in	ADP
ejde-502	363	15	the	the	DET
ejde-502	363	16	range	range	NOUN
ejde-502	363	17	m	m	VERB
ejde-502	363	18	+	+	NOUN
ejde-502	364	1	p	p	X
ejde-502	364	2	−	−	PROPN
ejde-502	364	3	2	2	NUM
ejde-502	364	4	<	<	X
ejde-502	364	5	0	0	NUM
ejde-502	364	6	,	,	PUNCT
ejde-502	364	7	it	it	PRON
ejde-502	364	8	follows	follow	VERB
ejde-502	364	9	that	that	SCONJ
ejde-502	364	10	r2(x	r2(x	PROPN
ejde-502	364	11	,	,	PUNCT
ejde-502	364	12	v	v	NOUN
ejde-502	364	13	)	)	PUNCT
ejde-502	364	14	≥	≥	NOUN
ejde-502	364	15	0	0	NUM
ejde-502	364	16	provided	provide	VERB
ejde-502	364	17	that	that	SCONJ
ejde-502	364	18	σ	σ	NOUN
ejde-502	364	19	[	[	PUNCT
ejde-502	364	20	(	(	PUNCT
ejde-502	364	21	n	n	CCONJ
ejde-502	364	22	−	−	PROPN
ejde-502	364	23	1	1	NUM
ejde-502	364	24	)	)	PUNCT
ejde-502	364	25	1	1	NUM
ejde-502	364	26	+	+	CCONJ
ejde-502	364	27	|x|	|x|	PROPN
ejde-502	364	28	|x|	|x|	PROPN
ejde-502	365	1	−	−	PROPN
ejde-502	365	2	1	1	NUM
ejde-502	365	3	]	]	PUNCT
ejde-502	365	4	≥	≥	NOUN
ejde-502	365	5	0	0	NUM
ejde-502	365	6	,	,	PUNCT
ejde-502	365	7	which	which	PRON
ejde-502	365	8	holds	hold	VERB
ejde-502	365	9	for	for	ADP
ejde-502	365	10	σ	σ	NOUN
ejde-502	365	11	>	>	X
ejde-502	365	12	0	0	PUNCT
ejde-502	366	1	if	if	SCONJ
ejde-502	366	2	n	n	PRON
ejde-502	366	3	≥	≥	NOUN
ejde-502	366	4	2	2	NUM
ejde-502	366	5	.	.	PUNCT
ejde-502	366	6	we	we	PRON
ejde-502	366	7	thus	thus	ADV
ejde-502	366	8	conclude	conclude	VERB
ejde-502	366	9	,	,	PUNCT
ejde-502	366	10	at	at	ADP
ejde-502	366	11	a	a	DET
ejde-502	366	12	formal	formal	ADJ
ejde-502	366	13	level	level	NOUN
ejde-502	366	14	,	,	PUNCT
ejde-502	366	15	that	that	SCONJ
ejde-502	366	16	(	(	PUNCT
ejde-502	366	17	1.13	1.13	NUM
ejde-502	366	18	)	)	PUNCT
ejde-502	366	19	should	should	AUX
ejde-502	366	20	hold	hold	VERB
ejde-502	366	21	in	in	ADP
ejde-502	366	22	this	this	DET
ejde-502	366	23	case	case	NOUN
ejde-502	366	24	.	.	PUNCT
ejde-502	367	1	however	however	ADV
ejde-502	367	2	,	,	PUNCT
ejde-502	367	3	we	we	PRON
ejde-502	367	4	left	leave	VERB
ejde-502	367	5	this	this	DET
ejde-502	367	6	part	part	NOUN
ejde-502	367	7	out	out	ADP
ejde-502	367	8	of	of	ADP
ejde-502	367	9	the	the	DET
ejde-502	367	10	statement	statement	NOUN
ejde-502	367	11	of	of	ADP
ejde-502	367	12	theorem	theorem	ADJ
ejde-502	367	13	1.4	1.4	NUM
ejde-502	367	14	since	since	SCONJ
ejde-502	367	15	the	the	DET
ejde-502	367	16	final	final	ADJ
ejde-502	367	17	approximation	approximation	NOUN
ejde-502	367	18	argument	argument	NOUN
ejde-502	367	19	leading	lead	VERB
ejde-502	367	20	to	to	ADP
ejde-502	367	21	the	the	DET
ejde-502	367	22	rigorous	rigorous	ADJ
ejde-502	367	23	proof	proof	NOUN
ejde-502	367	24	for	for	ADP
ejde-502	367	25	σ	σ	PROPN
ejde-502	367	26	=	=	SYM
ejde-502	367	27	0	0	NUM
ejde-502	367	28	can	can	AUX
ejde-502	367	29	not	not	PART
ejde-502	367	30	be	be	AUX
ejde-502	367	31	performed	perform	VERB
ejde-502	367	32	,	,	PUNCT
ejde-502	367	33	as	as	SCONJ
ejde-502	367	34	we	we	PRON
ejde-502	367	35	are	be	AUX
ejde-502	367	36	missing	miss	VERB
ejde-502	367	37	an	an	DET
ejde-502	367	38	existence	existence	NOUN
ejde-502	367	39	and	and	CCONJ
ejde-502	367	40	uniqueness	uniqueness	NOUN
ejde-502	367	41	result	result	NOUN
ejde-502	367	42	for	for	ADP
ejde-502	367	43	solutions	solution	NOUN
ejde-502	367	44	to	to	ADP
ejde-502	367	45	the	the	DET
ejde-502	367	46	cauchy	cauchy	ADJ
ejde-502	367	47	problem	problem	NOUN
ejde-502	367	48	(	(	PUNCT
ejde-502	367	49	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	367	50	)	)	PUNCT
ejde-502	367	51	when	when	SCONJ
ejde-502	367	52	σ	σ	X
ejde-502	367	53	>	>	X
ejde-502	367	54	0	0	NUM
ejde-502	367	55	.	.	NOUN
ejde-502	368	1	5	5	NUM
ejde-502	368	2	.	.	X
ejde-502	368	3	infinite	infinite	ADJ
ejde-502	368	4	speed	speed	NOUN
ejde-502	368	5	of	of	ADP
ejde-502	368	6	propagation	propagation	NOUN
ejde-502	368	7	when	when	SCONJ
ejde-502	368	8	m+	m+	NOUN
ejde-502	368	9	p	p	X
ejde-502	368	10	<	<	X
ejde-502	368	11	2	2	NUM
ejde-502	368	12	in	in	ADP
ejde-502	368	13	this	this	DET
ejde-502	368	14	part	part	NOUN
ejde-502	368	15	we	we	PRON
ejde-502	368	16	use	use	VERB
ejde-502	368	17	the	the	DET
ejde-502	368	18	aronson	aronson	PROPN
ejde-502	368	19	-	-	PUNCT
ejde-502	368	20	bénilan	bénilan	PROPN
ejde-502	368	21	estimates	estimate	NOUN
ejde-502	368	22	in	in	ADP
ejde-502	368	23	theorem	theorem	ADJ
ejde-502	368	24	1.4	1.4	NUM
ejde-502	368	25	to	to	PART
ejde-502	368	26	establish	establish	VERB
ejde-502	368	27	the	the	DET
ejde-502	368	28	infinite	infinite	ADJ
ejde-502	368	29	speed	speed	NOUN
ejde-502	368	30	of	of	ADP
ejde-502	368	31	propagation	propagation	NOUN
ejde-502	368	32	of	of	ADP
ejde-502	368	33	the	the	DET
ejde-502	368	34	supports	support	NOUN
ejde-502	368	35	of	of	ADP
ejde-502	368	36	solutions	solution	NOUN
ejde-502	368	37	to	to	ADP
ejde-502	368	38	(	(	PUNCT
ejde-502	368	39	1.1	1.1	NUM
ejde-502	368	40	)	)	PUNCT
ejde-502	368	41	when	when	SCONJ
ejde-502	368	42	m+p−2	m+p−2	PROPN
ejde-502	368	43	<	<	X
ejde-502	368	44	0	0	PUNCT
ejde-502	368	45	and	and	CCONJ
ejde-502	368	46	thus	thus	ADV
ejde-502	368	47	complete	complete	VERB
ejde-502	368	48	the	the	DET
ejde-502	368	49	proof	proof	NOUN
ejde-502	368	50	of	of	ADP
ejde-502	368	51	theorem	theorem	ADJ
ejde-502	368	52	1.5	1.5	NUM
ejde-502	368	53	.	.	PUNCT
ejde-502	369	1	let	let	VERB
ejde-502	369	2	us	we	PRON
ejde-502	369	3	stress	stress	VERB
ejde-502	369	4	again	again	ADV
ejde-502	369	5	here	here	ADV
ejde-502	369	6	that	that	PRON
ejde-502	369	7	theorem	theorem	VERB
ejde-502	369	8	1.5	1.5	NUM
ejde-502	369	9	has	have	AUX
ejde-502	369	10	been	be	AUX
ejde-502	369	11	proved	prove	VERB
ejde-502	369	12	in	in	ADP
ejde-502	369	13	[	[	X
ejde-502	369	14	26	26	NUM
ejde-502	369	15	,	,	PUNCT
ejde-502	369	16	lemma	lemma	PROPN
ejde-502	369	17	2.4	2.4	NUM
ejde-502	369	18	]	]	PUNCT
ejde-502	369	19	for	for	ADP
ejde-502	369	20	σ	σ	PROPN
ejde-502	369	21	=	=	SYM
ejde-502	369	22	0	0	NUM
ejde-502	369	23	.	.	PUNCT
ejde-502	370	1	we	we	PRON
ejde-502	370	2	give	give	VERB
ejde-502	370	3	here	here	ADV
ejde-502	370	4	an	an	DET
ejde-502	370	5	independent	independent	ADJ
ejde-502	370	6	proof	proof	NOUN
ejde-502	370	7	,	,	PUNCT
ejde-502	370	8	based	base	VERB
ejde-502	370	9	on	on	ADP
ejde-502	370	10	a	a	DET
ejde-502	370	11	completely	completely	ADV
ejde-502	370	12	different	different	ADJ
ejde-502	370	13	argument	argument	NOUN
ejde-502	370	14	,	,	PUNCT
ejde-502	370	15	and	and	CCONJ
ejde-502	370	16	extend	extend	VERB
ejde-502	370	17	it	it	PRON
ejde-502	370	18	to	to	ADP
ejde-502	370	19	exponents	exponent	NOUN
ejde-502	370	20	σ	σ	X
ejde-502	370	21	>	>	X
ejde-502	370	22	0	0	X
ejde-502	370	23	.	.	PUNCT
ejde-502	371	1	proof	proof	NOUN
ejde-502	371	2	of	of	ADP
ejde-502	371	3	theorem	theorem	ADJ
ejde-502	371	4	1.5	1.5	NUM
ejde-502	371	5	.	.	PUNCT
ejde-502	372	1	in	in	ADP
ejde-502	372	2	a	a	DET
ejde-502	372	3	first	first	ADJ
ejde-502	372	4	step	step	NOUN
ejde-502	372	5	,	,	PUNCT
ejde-502	372	6	let	let	VERB
ejde-502	372	7	σ	σ	NOUN
ejde-502	372	8	=	=	SYM
ejde-502	372	9	0	0	PUNCT
ejde-502	372	10	and	and	CCONJ
ejde-502	372	11	assume	assume	VERB
ejde-502	372	12	for	for	ADP
ejde-502	372	13	contradiction	contradiction	NOUN
ejde-502	372	14	that	that	PRON
ejde-502	372	15	for	for	ADP
ejde-502	372	16	some	some	DET
ejde-502	372	17	compactly	compactly	ADV
ejde-502	372	18	supported	support	VERB
ejde-502	372	19	initial	initial	ADJ
ejde-502	372	20	condition	condition	NOUN
ejde-502	372	21	u0	u0	ADJ
ejde-502	372	22	as	as	ADP
ejde-502	372	23	in	in	ADP
ejde-502	372	24	(	(	PUNCT
ejde-502	372	25	1.4	1.4	NUM
ejde-502	372	26	)	)	PUNCT
ejde-502	372	27	,	,	PUNCT
ejde-502	372	28	u(t	u(t	NOUN
ejde-502	372	29	)	)	PUNCT
ejde-502	372	30	remains	remains	AUX
ejde-502	372	31	compactly	compactly	ADV
ejde-502	372	32	supported	support	VERB
ejde-502	372	33	for	for	ADP
ejde-502	372	34	t	t	PROPN
ejde-502	372	35	∈	∈	PROPN
ejde-502	372	36	(	(	PUNCT
ejde-502	372	37	0	0	NUM
ejde-502	372	38	,	,	PUNCT
ejde-502	372	39	t0	t0	PROPN
ejde-502	372	40	)	)	PUNCT
ejde-502	372	41	.	.	PUNCT
ejde-502	373	1	recalling	recall	VERB
ejde-502	373	2	the	the	DET
ejde-502	373	3	pressure	pressure	NOUN
ejde-502	373	4	equation	equation	NOUN
ejde-502	373	5	(	(	PUNCT
ejde-502	373	6	4.1	4.1	NUM
ejde-502	373	7	)	)	PUNCT
ejde-502	373	8	,	,	PUNCT
ejde-502	373	9	at	at	ADP
ejde-502	373	10	a	a	DET
ejde-502	373	11	formal	formal	ADJ
ejde-502	373	12	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	373	13	reaction	reaction	NOUN
ejde-502	373	14	-	-	PUNCT
ejde-502	373	15	diffusion	diffusion	NOUN
ejde-502	373	16	with	with	ADP
ejde-502	373	17	weighted	weight	VERB
ejde-502	373	18	strong	strong	ADJ
ejde-502	373	19	reaction	reaction	NOUN
ejde-502	373	20	17	17	NUM
ejde-502	373	21	level	level	NOUN
ejde-502	373	22	,	,	PUNCT
ejde-502	373	23	since	since	SCONJ
ejde-502	373	24	m	m	PROPN
ejde-502	373	25	+	+	NOUN
ejde-502	373	26	p	p	X
ejde-502	373	27	−	−	PROPN
ejde-502	373	28	2	2	NUM
ejde-502	373	29	<	<	X
ejde-502	373	30	0	0	NUM
ejde-502	373	31	,	,	PUNCT
ejde-502	373	32	one	one	PRON
ejde-502	373	33	reaches	reach	VERB
ejde-502	373	34	easily	easily	ADV
ejde-502	373	35	a	a	DET
ejde-502	373	36	contradiction	contradiction	NOUN
ejde-502	373	37	.	.	PUNCT
ejde-502	374	1	indeed	indeed	ADV
ejde-502	374	2	,	,	PUNCT
ejde-502	374	3	picking	pick	VERB
ejde-502	374	4	an	an	DET
ejde-502	374	5	arbitrary	arbitrary	ADJ
ejde-502	374	6	t	t	NOUN
ejde-502	374	7	∈	∈	PROPN
ejde-502	374	8	(	(	PUNCT
ejde-502	374	9	0	0	NUM
ejde-502	374	10	,	,	PUNCT
ejde-502	374	11	t0	t0	PROPN
ejde-502	374	12	)	)	PUNCT
ejde-502	374	13	,	,	PUNCT
ejde-502	374	14	since	since	SCONJ
ejde-502	374	15	∆v(x	∆v(x	PROPN
ejde-502	374	16	,	,	PUNCT
ejde-502	374	17	t	t	PROPN
ejde-502	374	18	)	)	PUNCT
ejde-502	374	19	≥	≥	NOUN
ejde-502	374	20	−k	−k	PROPN
ejde-502	374	21	/	/	SYM
ejde-502	374	22	t	t	NOUN
ejde-502	374	23	and	and	CCONJ
ejde-502	374	24	|∇v(x	|∇v(x	NUM
ejde-502	374	25	,	,	PUNCT
ejde-502	374	26	t)|2	t)|2	X
ejde-502	374	27	≥	≥	NOUN
ejde-502	374	28	0	0	NUM
ejde-502	374	29	at	at	ADP
ejde-502	374	30	any	any	DET
ejde-502	374	31	point	point	NOUN
ejde-502	374	32	x	x	X
ejde-502	374	33	∈	∈	PROPN
ejde-502	374	34	rn	rn	PROPN
ejde-502	374	35	,	,	PUNCT
ejde-502	374	36	it	it	PRON
ejde-502	374	37	follows	follow	VERB
ejde-502	374	38	that	that	SCONJ
ejde-502	374	39	at	at	ADP
ejde-502	374	40	the	the	DET
ejde-502	374	41	interface	interface	NOUN
ejde-502	374	42	point	point	NOUN
ejde-502	374	43	x	x	PUNCT
ejde-502	374	44	=	=	SYM
ejde-502	374	45	s(t	s(t	PROPN
ejde-502	374	46	)	)	PUNCT
ejde-502	374	47	we	we	PRON
ejde-502	374	48	obtain	obtain	VERB
ejde-502	374	49	vt(s(t	vt(s(t	NOUN
ejde-502	374	50	)	)	PUNCT
ejde-502	374	51	,	,	PUNCT
ejde-502	374	52	t	t	PROPN
ejde-502	374	53	)	)	PUNCT
ejde-502	374	54	=	=	PUNCT
ejde-502	375	1	+	+	PUNCT
ejde-502	375	2	∞	∞	NUM
ejde-502	375	3	in	in	ADP
ejde-502	375	4	order	order	NOUN
ejde-502	375	5	to	to	PART
ejde-502	375	6	compensate	compensate	VERB
ejde-502	375	7	the	the	DET
ejde-502	375	8	negative	negative	ADJ
ejde-502	375	9	power	power	NOUN
ejde-502	375	10	(	(	PUNCT
ejde-502	375	11	m+p−	m+p−	ADV
ejde-502	375	12	2)/(m−	2)/(m−	NUM
ejde-502	375	13	1	1	NUM
ejde-502	375	14	)	)	PUNCT
ejde-502	375	15	in	in	ADP
ejde-502	375	16	the	the	DET
ejde-502	375	17	last	last	ADJ
ejde-502	375	18	term	term	NOUN
ejde-502	375	19	of	of	ADP
ejde-502	375	20	the	the	DET
ejde-502	375	21	right	right	ADJ
ejde-502	375	22	-	-	PUNCT
ejde-502	375	23	hand	hand	NOUN
ejde-502	375	24	side	side	NOUN
ejde-502	375	25	.	.	PUNCT
ejde-502	376	1	this	this	PRON
ejde-502	376	2	is	be	AUX
ejde-502	376	3	obviously	obviously	ADV
ejde-502	376	4	equivalent	equivalent	ADJ
ejde-502	376	5	to	to	ADP
ejde-502	376	6	the	the	DET
ejde-502	376	7	infinite	infinite	ADJ
ejde-502	376	8	speed	speed	NOUN
ejde-502	376	9	of	of	ADP
ejde-502	376	10	propagation	propagation	NOUN
ejde-502	376	11	of	of	ADP
ejde-502	376	12	the	the	DET
ejde-502	376	13	supports	support	NOUN
ejde-502	376	14	.	.	PUNCT
ejde-502	377	1	more	more	ADV
ejde-502	377	2	rigorously	rigorously	ADV
ejde-502	377	3	,	,	PUNCT
ejde-502	377	4	since	since	SCONJ
ejde-502	377	5	m	m	PROPN
ejde-502	377	6	+	+	NOUN
ejde-502	377	7	p	p	X
ejde-502	377	8	<	<	X
ejde-502	377	9	2	2	NUM
ejde-502	377	10	we	we	PRON
ejde-502	377	11	multiply	multiply	VERB
ejde-502	377	12	by	by	ADP
ejde-502	377	13	v(2−m−p)/(m−1	v(2−m−p)/(m−1	PROPN
ejde-502	377	14	)	)	PUNCT
ejde-502	377	15	in	in	ADP
ejde-502	377	16	(	(	PUNCT
ejde-502	377	17	4.1	4.1	NUM
ejde-502	377	18	)	)	PUNCT
ejde-502	377	19	and	and	CCONJ
ejde-502	377	20	also	also	ADV
ejde-502	377	21	by	by	ADP
ejde-502	377	22	a	a	DET
ejde-502	377	23	test	test	NOUN
ejde-502	377	24	function	function	NOUN
ejde-502	378	1	ϕ	ϕ	PROPN
ejde-502	378	2	∈	∈	PROPN
ejde-502	378	3	c∞0	c∞0	PROPN
ejde-502	378	4	(	(	PUNCT
ejde-502	378	5	rn	rn	PROPN
ejde-502	378	6	)	)	PUNCT
ejde-502	378	7	,	,	PUNCT
ejde-502	378	8	ϕ	ϕ	X
ejde-502	378	9	≥	≥	NOUN
ejde-502	378	10	0	0	NUM
ejde-502	378	11	,	,	PUNCT
ejde-502	378	12	then	then	ADV
ejde-502	378	13	we	we	PRON
ejde-502	378	14	integrate	integrate	VERB
ejde-502	378	15	on	on	ADP
ejde-502	378	16	rn	rn	PROPN
ejde-502	378	17	and	and	CCONJ
ejde-502	378	18	on	on	ADP
ejde-502	378	19	any	any	DET
ejde-502	378	20	time	time	NOUN
ejde-502	378	21	interval	interval	NOUN
ejde-502	378	22	(	(	PUNCT
ejde-502	378	23	τ0	τ0	NOUN
ejde-502	378	24	,	,	PUNCT
ejde-502	378	25	τ1	τ1	NOUN
ejde-502	378	26	)	)	PUNCT
ejde-502	378	27	⊂	⊂	PROPN
ejde-502	378	28	(	(	PUNCT
ejde-502	378	29	0	0	NUM
ejde-502	378	30	,	,	PUNCT
ejde-502	378	31	t0	t0	PROPN
ejde-502	378	32	)	)	PUNCT
ejde-502	378	33	and	and	CCONJ
ejde-502	378	34	we	we	PRON
ejde-502	378	35	drop	drop	VERB
ejde-502	378	36	the	the	DET
ejde-502	378	37	second	second	ADJ
ejde-502	378	38	term	term	NOUN
ejde-502	378	39	in	in	ADP
ejde-502	378	40	the	the	DET
ejde-502	378	41	right	right	ADJ
ejde-502	378	42	hand	hand	NOUN
ejde-502	378	43	side	side	NOUN
ejde-502	378	44	(	(	PUNCT
ejde-502	378	45	which	which	PRON
ejde-502	378	46	is	be	AUX
ejde-502	378	47	always	always	ADV
ejde-502	378	48	positive	positive	ADJ
ejde-502	378	49	)	)	PUNCT
ejde-502	378	50	to	to	PART
ejde-502	378	51	obtain	obtain	VERB
ejde-502	378	52	m−	m−	PROPN
ejde-502	378	53	1	1	NUM
ejde-502	378	54	1−	1−	NUM
ejde-502	378	55	p	p	PRON
ejde-502	378	56	∫	∫	PROPN
ejde-502	378	57	τ1	τ1	PROPN
ejde-502	378	58	τ0	τ0	PROPN
ejde-502	378	59	∫	∫	PROPN
ejde-502	378	60	rn	rn	PROPN
ejde-502	378	61	(	(	PUNCT
ejde-502	378	62	v(1−p)/(m−1))tϕdx	v(1−p)/(m−1))tϕdx	ADP
ejde-502	378	63	dt	dt	X
ejde-502	378	64	≥	≥	X
ejde-502	378	65	(	(	PUNCT
ejde-502	378	66	m−	m−	PROPN
ejde-502	378	67	1	1	NUM
ejde-502	378	68	)	)	PUNCT
ejde-502	379	1	∫	∫	PROPN
ejde-502	380	1	τ1	τ1	NOUN
ejde-502	380	2	τ0	τ0	PROPN
ejde-502	380	3	∫	∫	PROPN
ejde-502	380	4	rn	rn	PROPN
ejde-502	380	5	v(1−p)/(m−1)∆vϕ	v(1−p)/(m−1)∆vϕ	PROPN
ejde-502	380	6	dx	dx	PROPN
ejde-502	380	7	dt+k(m	dt+k(m	PROPN
ejde-502	380	8	,	,	PUNCT
ejde-502	380	9	p	p	NOUN
ejde-502	380	10	)	)	PUNCT
ejde-502	380	11	∫	∫	PROPN
ejde-502	381	1	τ1	τ1	NOUN
ejde-502	381	2	τ0	τ0	PROPN
ejde-502	382	1	∫	∫	PROPN
ejde-502	382	2	rn	rn	PROPN
ejde-502	382	3	ϕdx	ϕdx	PROPN
ejde-502	382	4	dt	dt	PROPN
ejde-502	382	5	≥	≥	PRON
ejde-502	382	6	−	−	PROPN
ejde-502	382	7	n(m−	n(m−	PROPN
ejde-502	382	8	1	1	NUM
ejde-502	382	9	)	)	PUNCT
ejde-502	382	10	n(m−	n(m−	PROPN
ejde-502	382	11	1	1	NUM
ejde-502	382	12	)	)	PUNCT
ejde-502	382	13	+	+	CCONJ
ejde-502	382	14	2	2	NUM
ejde-502	382	15	∫	∫	NOUN
ejde-502	382	16	τ1	τ1	NOUN
ejde-502	382	17	τ0	τ0	PROPN
ejde-502	382	18	∫	∫	PROPN
ejde-502	382	19	rn	rn	PROPN
ejde-502	382	20	1	1	NUM
ejde-502	382	21	t	t	PROPN
ejde-502	382	22	v(1−p)/(m−1)ϕdx	v(1−p)/(m−1)ϕdx	PRON
ejde-502	382	23	dt	dt	X
ejde-502	383	1	+	+	PROPN
ejde-502	383	2	k(m	k(m	PROPN
ejde-502	383	3	,	,	PUNCT
ejde-502	383	4	p	p	NOUN
ejde-502	383	5	)	)	PUNCT
ejde-502	383	6	∫	∫	PROPN
ejde-502	384	1	τ1	τ1	NOUN
ejde-502	384	2	τ0	τ0	PROPN
ejde-502	384	3	∫	∫	PROPN
ejde-502	384	4	rn	rn	PROPN
ejde-502	384	5	ϕdx	ϕdx	PROPN
ejde-502	384	6	dt	dt	PROPN
ejde-502	384	7	.	.	PUNCT
ejde-502	385	1	(	(	PUNCT
ejde-502	385	2	5.1	5.1	NUM
ejde-502	385	3	)	)	PUNCT
ejde-502	385	4	we	we	PRON
ejde-502	385	5	now	now	ADV
ejde-502	385	6	consider	consider	VERB
ejde-502	385	7	a	a	DET
ejde-502	385	8	sequence	sequence	NOUN
ejde-502	385	9	of	of	ADP
ejde-502	385	10	test	test	NOUN
ejde-502	385	11	functions	function	NOUN
ejde-502	385	12	(	(	PUNCT
ejde-502	385	13	ϕn)n≥1	ϕn)n≥1	NOUN
ejde-502	385	14	defined	define	VERB
ejde-502	385	15	as	as	SCONJ
ejde-502	385	16	follows	follow	VERB
ejde-502	385	17	ϕn(x	ϕn(x	PUNCT
ejde-502	385	18	)	)	PUNCT
ejde-502	385	19	=	=	SYM
ejde-502	385	20	1	1	NUM
ejde-502	385	21	,	,	PUNCT
ejde-502	385	22	for	for	ADP
ejde-502	385	23	|x|	|x|	PROPN
ejde-502	385	24	≤	≤	NOUN
ejde-502	385	25	n	n	CCONJ
ejde-502	385	26	,	,	PUNCT
ejde-502	385	27	0	0	NUM
ejde-502	385	28	≤	≤	NOUN
ejde-502	385	29	ϕn(x	ϕn(x	X
ejde-502	385	30	)	)	PUNCT
ejde-502	385	31	≤	≤	NUM
ejde-502	385	32	1	1	NUM
ejde-502	385	33	∀x	∀x	NUM
ejde-502	385	34	∈	∈	PROPN
ejde-502	385	35	rn	rn	NOUN
ejde-502	385	36	,	,	PUNCT
ejde-502	385	37	suppϕn	suppϕn	VERB
ejde-502	385	38	⊆	⊆	X
ejde-502	385	39	b(0	b(0	NOUN
ejde-502	385	40	,	,	PUNCT
ejde-502	385	41	2n	2n	NUM
ejde-502	385	42	)	)	PUNCT
ejde-502	385	43	,	,	PUNCT
ejde-502	385	44	where	where	SCONJ
ejde-502	385	45	b(0	b(0	NOUN
ejde-502	385	46	,	,	PUNCT
ejde-502	385	47	2n	2n	NUM
ejde-502	385	48	)	)	PUNCT
ejde-502	385	49	=	=	PRON
ejde-502	385	50	{	{	PUNCT
ejde-502	385	51	x	x	PUNCT
ejde-502	385	52	∈	∈	PROPN
ejde-502	385	53	rn	rn	NOUN
ejde-502	385	54	:	:	PUNCT
ejde-502	385	55	|x|	|x|	PROPN
ejde-502	385	56	≤	≤	NUM
ejde-502	385	57	2n	2n	NUM
ejde-502	385	58	}	}	PUNCT
ejde-502	385	59	,	,	PUNCT
ejde-502	385	60	and	and	CCONJ
ejde-502	385	61	let	let	VERB
ejde-502	385	62	ϕ	ϕ	NOUN
ejde-502	385	63	=	=	VERB
ejde-502	385	64	ϕn	ϕn	X
ejde-502	385	65	in	in	ADP
ejde-502	385	66	(	(	PUNCT
ejde-502	385	67	5.1	5.1	NUM
ejde-502	385	68	)	)	PUNCT
ejde-502	385	69	for	for	ADP
ejde-502	385	70	any	any	DET
ejde-502	385	71	positive	positive	ADJ
ejde-502	385	72	integer	integer	NOUN
ejde-502	385	73	n	n	PRON
ejde-502	385	74	≥	≥	NOUN
ejde-502	385	75	1	1	NUM
ejde-502	385	76	.	.	PUNCT
ejde-502	386	1	since	since	SCONJ
ejde-502	386	2	the	the	DET
ejde-502	386	3	support	support	NOUN
ejde-502	386	4	of	of	ADP
ejde-502	386	5	v	v	NOUN
ejde-502	386	6	is	be	AUX
ejde-502	386	7	uniformly	uniformly	ADV
ejde-502	386	8	localized	localize	VERB
ejde-502	386	9	for	for	ADP
ejde-502	386	10	t	t	PROPN
ejde-502	386	11	∈	∈	PROPN
ejde-502	387	1	[	[	X
ejde-502	387	2	τ0	τ0	NOUN
ejde-502	387	3	,	,	PUNCT
ejde-502	387	4	τ1	τ1	NOUN
ejde-502	387	5	]	]	X
ejde-502	387	6	(	(	PUNCT
ejde-502	387	7	as	as	ADP
ejde-502	387	8	τ1	τ1	PROPN
ejde-502	387	9	<	<	X
ejde-502	387	10	t0	t0	PROPN
ejde-502	387	11	)	)	PUNCT
ejde-502	387	12	,	,	PUNCT
ejde-502	387	13	it	it	PRON
ejde-502	387	14	follows	follow	VERB
ejde-502	387	15	that	that	SCONJ
ejde-502	387	16	the	the	DET
ejde-502	387	17	right	right	ADJ
ejde-502	387	18	-	-	PUNCT
ejde-502	387	19	hand	hand	NOUN
ejde-502	387	20	side	side	NOUN
ejde-502	387	21	of	of	ADP
ejde-502	387	22	(	(	PUNCT
ejde-502	387	23	5.1	5.1	NUM
ejde-502	387	24	)	)	PUNCT
ejde-502	387	25	tends	tend	VERB
ejde-502	387	26	to	to	ADP
ejde-502	387	27	+	+	NOUN
ejde-502	387	28	∞	∞	PROPN
ejde-502	387	29	as	as	ADP
ejde-502	387	30	n	n	PROPN
ejde-502	387	31	→	→	SYM
ejde-502	387	32	∞	∞	PROPN
ejde-502	387	33	due	due	ADP
ejde-502	387	34	to	to	ADP
ejde-502	387	35	the	the	DET
ejde-502	387	36	last	last	ADJ
ejde-502	387	37	integral	integral	NOUN
ejde-502	387	38	only	only	ADV
ejde-502	387	39	of	of	ADP
ejde-502	387	40	ϕn	ϕn	INTJ
ejde-502	387	41	.	.	PUNCT
ejde-502	388	1	we	we	PRON
ejde-502	388	2	thus	thus	ADV
ejde-502	388	3	infer	infer	VERB
ejde-502	388	4	that	that	SCONJ
ejde-502	388	5	lim	lim	PROPN
ejde-502	388	6	n→∞	n→∞	PRON
ejde-502	388	7	∫	∫	PROPN
ejde-502	388	8	τ1	τ1	NOUN
ejde-502	388	9	τ0	τ0	PROPN
ejde-502	388	10	∫	∫	PROPN
ejde-502	388	11	rn	rn	PROPN
ejde-502	388	12	(	(	PUNCT
ejde-502	388	13	v(1−p)/(m−1))tϕn	v(1−p)/(m−1))tϕn	PROPN
ejde-502	388	14	dx	dx	PROPN
ejde-502	388	15	dt	dt	PROPN
ejde-502	389	1	=	=	PUNCT
ejde-502	390	1	+	+	NOUN
ejde-502	390	2	∞	∞	NUM
ejde-502	390	3	or	or	CCONJ
ejde-502	390	4	equivalently	equivalently	ADV
ejde-502	390	5	lim	lim	PROPN
ejde-502	390	6	n→∞	n→∞	NUM
ejde-502	390	7	∫	∫	PROPN
ejde-502	390	8	rn	rn	PROPN
ejde-502	390	9	[	[	PUNCT
ejde-502	390	10	v(1−p)/(m−1)(τ1)−	v(1−p)/(m−1)(τ1)−	PROPN
ejde-502	390	11	v(1−p)/(m−1)(τ0	v(1−p)/(m−1)(τ0	PROPN
ejde-502	390	12	)	)	PUNCT
ejde-502	390	13	]	]	PUNCT
ejde-502	390	14	ϕn	ϕn	ADP
ejde-502	390	15	dx	dx	PROPN
ejde-502	391	1	=	=	PUNCT
ejde-502	392	1	+	+	NUM
ejde-502	392	2	∞	∞	PROPN
ejde-502	392	3	,	,	PUNCT
ejde-502	392	4	which	which	PRON
ejde-502	392	5	is	be	AUX
ejde-502	392	6	a	a	DET
ejde-502	392	7	contradiction	contradiction	NOUN
ejde-502	392	8	with	with	ADP
ejde-502	392	9	the	the	DET
ejde-502	392	10	localization	localization	NOUN
ejde-502	392	11	of	of	ADP
ejde-502	392	12	the	the	DET
ejde-502	392	13	supports	support	NOUN
ejde-502	392	14	of	of	ADP
ejde-502	392	15	v(t	v(t	NOUN
ejde-502	392	16	)	)	PUNCT
ejde-502	392	17	for	for	ADP
ejde-502	392	18	t	t	PROPN
ejde-502	392	19	∈	∈	PROPN
ejde-502	393	1	[	[	X
ejde-502	393	2	τ0	τ0	NOUN
ejde-502	393	3	,	,	PUNCT
ejde-502	393	4	τ1	τ1	NOUN
ejde-502	393	5	]	]	PUNCT
ejde-502	393	6	.	.	PUNCT
ejde-502	394	1	let	let	VERB
ejde-502	394	2	us	we	PRON
ejde-502	394	3	notice	notice	VERB
ejde-502	394	4	here	here	ADV
ejde-502	394	5	that	that	SCONJ
ejde-502	394	6	the	the	DET
ejde-502	394	7	chosen	choose	VERB
ejde-502	394	8	range	range	NOUN
ejde-502	394	9	of	of	ADP
ejde-502	394	10	exponents	exponent	NOUN
ejde-502	394	11	p	p	X
ejde-502	394	12	∈	∈	PROPN
ejde-502	394	13	(	(	PUNCT
ejde-502	394	14	0	0	NUM
ejde-502	394	15	,	,	PUNCT
ejde-502	394	16	1	1	NUM
ejde-502	394	17	)	)	PUNCT
ejde-502	394	18	and	and	CCONJ
ejde-502	394	19	m+p	m+p	NOUN
ejde-502	394	20	<	<	X
ejde-502	394	21	2	2	NUM
ejde-502	394	22	were	be	AUX
ejde-502	394	23	decisive	decisive	ADJ
ejde-502	394	24	,	,	PUNCT
ejde-502	394	25	as	as	ADP
ejde-502	394	26	after	after	ADP
ejde-502	394	27	multiplication	multiplication	NOUN
ejde-502	394	28	by	by	ADP
ejde-502	394	29	a	a	DET
ejde-502	394	30	positive	positive	ADJ
ejde-502	394	31	power	power	NOUN
ejde-502	394	32	v(2−m−p)/(m−1	v(2−m−p)/(m−1	PROPN
ejde-502	394	33	)	)	PUNCT
ejde-502	394	34	,	,	PUNCT
ejde-502	394	35	we	we	PRON
ejde-502	394	36	got	get	VERB
ejde-502	394	37	in	in	ADP
ejde-502	394	38	the	the	DET
ejde-502	394	39	left	left	ADJ
ejde-502	394	40	-	-	PUNCT
ejde-502	394	41	hand	hand	NOUN
ejde-502	394	42	side	side	NOUN
ejde-502	394	43	also	also	ADV
ejde-502	394	44	a	a	DET
ejde-502	394	45	positive	positive	ADJ
ejde-502	394	46	power	power	NOUN
ejde-502	394	47	v(1−p)/(m−1	v(1−p)/(m−1	NUM
ejde-502	394	48	)	)	PUNCT
ejde-502	394	49	of	of	ADP
ejde-502	394	50	v	v	NOUN
ejde-502	394	51	,	,	PUNCT
ejde-502	394	52	thus	thus	ADV
ejde-502	394	53	we	we	PRON
ejde-502	394	54	do	do	AUX
ejde-502	394	55	not	not	PART
ejde-502	394	56	create	create	VERB
ejde-502	394	57	new	new	ADJ
ejde-502	394	58	singularities	singularity	NOUN
ejde-502	394	59	at	at	ADP
ejde-502	394	60	the	the	DET
ejde-502	394	61	edges	edge	NOUN
ejde-502	394	62	of	of	ADP
ejde-502	394	63	the	the	DET
ejde-502	394	64	supports	support	NOUN
ejde-502	394	65	.	.	PUNCT
ejde-502	395	1	we	we	PRON
ejde-502	395	2	pass	pass	VERB
ejde-502	395	3	now	now	ADV
ejde-502	395	4	to	to	ADP
ejde-502	395	5	σ	σ	PROPN
ejde-502	395	6	>	>	X
ejde-502	395	7	0	0	X
ejde-502	395	8	.	.	PUNCT
ejde-502	396	1	assume	assume	VERB
ejde-502	396	2	that	that	SCONJ
ejde-502	396	3	there	there	PRON
ejde-502	396	4	is	be	VERB
ejde-502	396	5	a	a	DET
ejde-502	396	6	weak	weak	ADJ
ejde-502	396	7	solution	solution	NOUN
ejde-502	396	8	u	u	NOUN
ejde-502	396	9	to	to	ADP
ejde-502	396	10	the	the	DET
ejde-502	396	11	cauchy	cauchy	PROPN
ejde-502	396	12	problem	problem	NOUN
ejde-502	396	13	eq	eq	ADJ
ejde-502	396	14	.	.	PUNCT
ejde-502	397	1	(	(	PUNCT
ejde-502	397	2	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	397	3	)	)	PUNCT
ejde-502	397	4	defined	define	VERB
ejde-502	397	5	for	for	ADP
ejde-502	397	6	t	t	PROPN
ejde-502	397	7	∈	∈	PROPN
ejde-502	397	8	(	(	PUNCT
ejde-502	397	9	0	0	NUM
ejde-502	397	10	,	,	PUNCT
ejde-502	397	11	t	t	NOUN
ejde-502	397	12	)	)	PUNCT
ejde-502	397	13	with	with	ADP
ejde-502	397	14	some	some	DET
ejde-502	397	15	t	t	PROPN
ejde-502	397	16	>	>	X
ejde-502	397	17	0	0	X
ejde-502	397	18	.	.	PUNCT
ejde-502	398	1	since	since	SCONJ
ejde-502	398	2	(	(	PUNCT
ejde-502	398	3	1	1	NUM
ejde-502	398	4	+	+	NUM
ejde-502	398	5	|x|)σ	|x|)σ	X
ejde-502	398	6	≥	≥	NUM
ejde-502	398	7	1	1	NUM
ejde-502	398	8	for	for	ADP
ejde-502	398	9	any	any	DET
ejde-502	398	10	x	x	SYM
ejde-502	398	11	∈	∈	PROPN
ejde-502	398	12	rn	rn	NOUN
ejde-502	398	13	we	we	PRON
ejde-502	398	14	deduce	deduce	VERB
ejde-502	398	15	that	that	SCONJ
ejde-502	398	16	u	u	NOUN
ejde-502	398	17	is	be	AUX
ejde-502	398	18	a	a	DET
ejde-502	398	19	supersolution	supersolution	NOUN
ejde-502	398	20	to	to	ADP
ejde-502	398	21	the	the	DET
ejde-502	398	22	cauchy	cauchy	ADJ
ejde-502	398	23	problem	problem	NOUN
ejde-502	398	24	(	(	PUNCT
ejde-502	398	25	1.5)(1.2	1.5)(1.2	NOUN
ejde-502	398	26	)	)	PUNCT
ejde-502	398	27	.	.	PUNCT
ejde-502	399	1	we	we	PRON
ejde-502	399	2	infer	infer	VERB
ejde-502	399	3	from	from	ADP
ejde-502	399	4	the	the	DET
ejde-502	399	5	comparison	comparison	NOUN
ejde-502	399	6	principle	principle	NOUN
ejde-502	399	7	(	(	PUNCT
ejde-502	399	8	which	which	PRON
ejde-502	399	9	holds	hold	VERB
ejde-502	399	10	true	true	ADJ
ejde-502	399	11	for	for	ADP
ejde-502	399	12	(	(	PUNCT
ejde-502	399	13	1.5	1.5	NUM
ejde-502	399	14	)	)	PUNCT
ejde-502	399	15	and	and	CCONJ
ejde-502	399	16	non	non	ADJ
ejde-502	399	17	-	-	ADJ
ejde-502	399	18	trivial	trivial	ADJ
ejde-502	399	19	initial	initial	ADJ
ejde-502	399	20	data	datum	NOUN
ejde-502	399	21	in	in	ADP
ejde-502	399	22	the	the	DET
ejde-502	399	23	range	range	NOUN
ejde-502	399	24	m	m	VERB
ejde-502	399	25	+	+	X
ejde-502	399	26	p	p	X
ejde-502	399	27	<	<	X
ejde-502	399	28	2	2	NUM
ejde-502	399	29	,	,	PUNCT
ejde-502	399	30	[	[	X
ejde-502	399	31	27	27	NUM
ejde-502	399	32	]	]	PUNCT
ejde-502	399	33	)	)	PUNCT
ejde-502	399	34	that	that	PRON
ejde-502	399	35	u(x	u(x	VERB
ejde-502	399	36	,	,	PUNCT
ejde-502	399	37	t	t	PROPN
ejde-502	399	38	)	)	PUNCT
ejde-502	399	39	>	>	X
ejde-502	399	40	0	0	PUNCT
ejde-502	399	41	for	for	ADP
ejde-502	399	42	any	any	DET
ejde-502	399	43	(	(	PUNCT
ejde-502	399	44	x	x	NOUN
ejde-502	399	45	,	,	PUNCT
ejde-502	399	46	t	t	PROPN
ejde-502	399	47	)	)	PUNCT
ejde-502	399	48	∈	∈	PROPN
ejde-502	399	49	rn	rn	PROPN
ejde-502	399	50	×	×	PROPN
ejde-502	399	51	(	(	PUNCT
ejde-502	399	52	0	0	NUM
ejde-502	399	53	,	,	PUNCT
ejde-502	399	54	t	t	NOUN
ejde-502	399	55	)	)	PUNCT
ejde-502	399	56	.	.	PUNCT
ejde-502	399	57	�	�	PROPN
ejde-502	399	58	extensions	extension	NOUN
ejde-502	399	59	and	and	CCONJ
ejde-502	399	60	open	open	ADJ
ejde-502	399	61	problems	problem	NOUN
ejde-502	399	62	we	we	PRON
ejde-502	399	63	gather	gather	VERB
ejde-502	399	64	here	here	ADV
ejde-502	399	65	some	some	DET
ejde-502	399	66	extensions	extension	NOUN
ejde-502	399	67	related	relate	VERB
ejde-502	399	68	to	to	ADP
ejde-502	399	69	the	the	DET
ejde-502	399	70	previous	previous	ADJ
ejde-502	399	71	results	result	NOUN
ejde-502	399	72	,	,	PUNCT
ejde-502	399	73	that	that	SCONJ
ejde-502	399	74	we	we	PRON
ejde-502	399	75	consider	consider	VERB
ejde-502	399	76	interesting	interesting	ADJ
ejde-502	399	77	.	.	PUNCT
ejde-502	400	1	18	18	NUM
ejde-502	400	2	r.	r.	PROPN
ejde-502	400	3	g.	g.	PROPN
ejde-502	400	4	iagar	iagar	PROPN
ejde-502	400	5	,	,	PUNCT
ejde-502	400	6	a.	a.	NOUN
ejde-502	400	7	i.	i.	PROPN
ejde-502	400	8	muñoz	muñoz	PROPN
ejde-502	400	9	,	,	PUNCT
ejde-502	400	10	a.	a.	NOUN
ejde-502	400	11	sánchez	sánchez	PROPN
ejde-502	400	12	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	400	13	1	1	NUM
ejde-502	400	14	.	.	PUNCT
ejde-502	400	15	finite	finite	PROPN
ejde-502	400	16	time	time	NOUN
ejde-502	400	17	blow	blow	NOUN
ejde-502	400	18	-	-	PUNCT
ejde-502	400	19	up	up	NOUN
ejde-502	400	20	.	.	PUNCT
ejde-502	401	1	a	a	DET
ejde-502	401	2	natural	natural	ADJ
ejde-502	401	3	question	question	NOUN
ejde-502	401	4	is	be	AUX
ejde-502	401	5	whether	whether	SCONJ
ejde-502	401	6	any	any	DET
ejde-502	401	7	solution	solution	NOUN
ejde-502	401	8	to	to	ADP
ejde-502	401	9	(	(	PUNCT
ejde-502	401	10	1.1	1.1	NUM
ejde-502	401	11	)	)	PUNCT
ejde-502	401	12	blows	blow	VERB
ejde-502	401	13	up	up	ADP
ejde-502	401	14	in	in	ADP
ejde-502	401	15	finite	finite	ADJ
ejde-502	401	16	time	time	NOUN
ejde-502	401	17	or	or	CCONJ
ejde-502	401	18	there	there	PRON
ejde-502	401	19	are	be	VERB
ejde-502	401	20	some	some	DET
ejde-502	401	21	initial	initial	ADJ
ejde-502	401	22	conditions	condition	NOUN
ejde-502	401	23	u0	u0	ADJ
ejde-502	401	24	producing	produce	VERB
ejde-502	401	25	(	(	PUNCT
ejde-502	401	26	minimal	minimal	ADJ
ejde-502	401	27	)	)	PUNCT
ejde-502	401	28	solutions	solution	NOUN
ejde-502	401	29	that	that	PRON
ejde-502	401	30	are	be	AUX
ejde-502	401	31	global	global	ADJ
ejde-502	401	32	in	in	ADP
ejde-502	401	33	time	time	NOUN
ejde-502	401	34	.	.	PUNCT
ejde-502	402	1	our	our	PRON
ejde-502	402	2	conjecture	conjecture	NOUN
ejde-502	402	3	is	be	AUX
ejde-502	402	4	that	that	SCONJ
ejde-502	402	5	,	,	PUNCT
ejde-502	402	6	if	if	SCONJ
ejde-502	402	7	l	l	PROPN
ejde-502	402	8	>	>	X
ejde-502	402	9	0	0	PROPN
ejde-502	402	10	,	,	PUNCT
ejde-502	402	11	each	each	DET
ejde-502	402	12	non	non	ADJ
ejde-502	402	13	-	-	ADJ
ejde-502	402	14	trivial	trivial	ADJ
ejde-502	402	15	solution	solution	NOUN
ejde-502	402	16	is	be	AUX
ejde-502	402	17	expected	expect	VERB
ejde-502	402	18	to	to	PART
ejde-502	402	19	blow	blow	VERB
ejde-502	402	20	up	up	ADP
ejde-502	402	21	in	in	ADP
ejde-502	402	22	finite	finite	ADJ
ejde-502	402	23	time	time	NOUN
ejde-502	402	24	,	,	PUNCT
ejde-502	402	25	while	while	SCONJ
ejde-502	402	26	if	if	SCONJ
ejde-502	402	27	l	l	PROPN
ejde-502	402	28	≤	≤	NUM
ejde-502	402	29	0	0	NUM
ejde-502	402	30	,	,	PUNCT
ejde-502	402	31	there	there	PRON
ejde-502	402	32	are	be	VERB
ejde-502	402	33	initial	initial	ADJ
ejde-502	402	34	conditions	condition	NOUN
ejde-502	402	35	producing	produce	VERB
ejde-502	402	36	solutions	solution	NOUN
ejde-502	402	37	that	that	PRON
ejde-502	402	38	are	be	AUX
ejde-502	402	39	global	global	ADJ
ejde-502	402	40	in	in	ADP
ejde-502	402	41	time	time	NOUN
ejde-502	402	42	(	(	PUNCT
ejde-502	402	43	we	we	PRON
ejde-502	402	44	recall	recall	VERB
ejde-502	402	45	that	that	SCONJ
ejde-502	402	46	l	l	NOUN
ejde-502	402	47	is	be	AUX
ejde-502	402	48	defined	define	VERB
ejde-502	402	49	in	in	ADP
ejde-502	402	50	(	(	PUNCT
ejde-502	402	51	1.8	1.8	NUM
ejde-502	402	52	)	)	PUNCT
ejde-502	402	53	)	)	PUNCT
ejde-502	402	54	.	.	PUNCT
ejde-502	403	1	a	a	DET
ejde-502	403	2	formal	formal	ADJ
ejde-502	403	3	argument	argument	NOUN
ejde-502	403	4	about	about	ADP
ejde-502	403	5	general	general	ADJ
ejde-502	403	6	finite	finite	ADJ
ejde-502	403	7	time	time	NOUN
ejde-502	403	8	blow	blow	VERB
ejde-502	403	9	-	-	PUNCT
ejde-502	403	10	up	up	NOUN
ejde-502	403	11	if	if	SCONJ
ejde-502	403	12	l	l	NOUN
ejde-502	403	13	>	>	X
ejde-502	403	14	0	0	PUNCT
ejde-502	403	15	is	be	AUX
ejde-502	403	16	based	base	VERB
ejde-502	403	17	on	on	ADP
ejde-502	403	18	comparison	comparison	NOUN
ejde-502	403	19	with	with	ADP
ejde-502	403	20	subsolutions	subsolution	NOUN
ejde-502	403	21	in	in	ADP
ejde-502	403	22	self	self	NOUN
ejde-502	403	23	-	-	PUNCT
ejde-502	403	24	similar	similar	ADJ
ejde-502	403	25	form	form	NOUN
ejde-502	403	26	obtained	obtain	VERB
ejde-502	403	27	in	in	ADP
ejde-502	403	28	our	our	PRON
ejde-502	403	29	recent	recent	ADJ
ejde-502	403	30	papers	paper	NOUN
ejde-502	403	31	[	[	X
ejde-502	403	32	19	19	NUM
ejde-502	403	33	,	,	PUNCT
ejde-502	403	34	21	21	NUM
ejde-502	403	35	,	,	PUNCT
ejde-502	403	36	16	16	NUM
ejde-502	403	37	]	]	PUNCT
ejde-502	403	38	.	.	PUNCT
ejde-502	404	1	more	more	ADV
ejde-502	404	2	precisely	precisely	ADV
ejde-502	404	3	,	,	PUNCT
ejde-502	404	4	it	it	PRON
ejde-502	404	5	goes	go	VERB
ejde-502	404	6	by	by	ADP
ejde-502	404	7	contradiction	contradiction	NOUN
ejde-502	404	8	as	as	SCONJ
ejde-502	404	9	follows	follow	VERB
ejde-502	404	10	:	:	PUNCT
ejde-502	404	11	assume	assume	VERB
ejde-502	404	12	that	that	SCONJ
ejde-502	404	13	there	there	PRON
ejde-502	404	14	exists	exist	VERB
ejde-502	404	15	u0	u0	ADJ
ejde-502	404	16	∈	∈	PROPN
ejde-502	404	17	c0(rn	c0(rn	PROPN
ejde-502	404	18	)	)	PUNCT
ejde-502	404	19	as	as	ADP
ejde-502	404	20	in	in	ADP
ejde-502	404	21	(	(	PUNCT
ejde-502	404	22	1.4	1.4	NUM
ejde-502	404	23	)	)	PUNCT
ejde-502	404	24	such	such	ADJ
ejde-502	404	25	that	that	SCONJ
ejde-502	404	26	the	the	DET
ejde-502	404	27	minimal	minimal	ADJ
ejde-502	404	28	solution	solution	NOUN
ejde-502	404	29	m(u0	m(u0	NOUN
ejde-502	404	30	)	)	PUNCT
ejde-502	404	31	is	be	AUX
ejde-502	404	32	defined	define	VERB
ejde-502	404	33	for	for	ADP
ejde-502	404	34	t	t	PROPN
ejde-502	404	35	∈	∈	PROPN
ejde-502	404	36	(	(	PUNCT
ejde-502	404	37	0,∞	0,∞	NUM
ejde-502	404	38	)	)	PUNCT
ejde-502	404	39	.	.	PUNCT
ejde-502	405	1	we	we	PRON
ejde-502	405	2	infer	infer	VERB
ejde-502	405	3	by	by	ADP
ejde-502	405	4	comparison	comparison	NOUN
ejde-502	405	5	with	with	ADP
ejde-502	405	6	the	the	DET
ejde-502	405	7	absolute	absolute	ADJ
ejde-502	405	8	minimal	minimal	ADJ
ejde-502	405	9	solution	solution	NOUN
ejde-502	405	10	e	e	NOUN
ejde-502	405	11	defined	define	VERB
ejde-502	405	12	in	in	ADP
ejde-502	405	13	[	[	X
ejde-502	405	14	26	26	NUM
ejde-502	405	15	]	]	PUNCT
ejde-502	405	16	and	and	CCONJ
ejde-502	405	17	(	(	PUNCT
ejde-502	405	18	3.1	3.1	NUM
ejde-502	405	19	)	)	PUNCT
ejde-502	405	20	that	that	SCONJ
ejde-502	405	21	for	for	ADP
ejde-502	405	22	large	large	ADJ
ejde-502	405	23	t	t	PROPN
ejde-502	405	24	,	,	PUNCT
ejde-502	405	25	solution	solution	NOUN
ejde-502	405	26	m(u0)(t	m(u0)(t	NOUN
ejde-502	405	27	)	)	PUNCT
ejde-502	405	28	is	be	AUX
ejde-502	405	29	as	as	ADV
ejde-502	405	30	large	large	ADJ
ejde-502	405	31	as	as	SCONJ
ejde-502	405	32	we	we	PRON
ejde-502	405	33	want	want	VERB
ejde-502	405	34	both	both	PRON
ejde-502	405	35	in	in	ADP
ejde-502	405	36	amplitude	amplitude	NOUN
ejde-502	405	37	and	and	CCONJ
ejde-502	405	38	support	support	NOUN
ejde-502	405	39	.	.	PUNCT
ejde-502	406	1	we	we	PRON
ejde-502	406	2	can	can	AUX
ejde-502	406	3	thus	thus	ADV
ejde-502	406	4	find	find	VERB
ejde-502	406	5	a	a	DET
ejde-502	406	6	blow	blow	NOUN
ejde-502	406	7	-	-	PUNCT
ejde-502	406	8	up	up	ADP
ejde-502	406	9	self	self	NOUN
ejde-502	406	10	-	-	PUNCT
ejde-502	406	11	similar	similar	ADJ
ejde-502	406	12	solution	solution	NOUN
ejde-502	406	13	to	to	ADP
ejde-502	406	14	(	(	PUNCT
ejde-502	406	15	1.7	1.7	NUM
ejde-502	406	16	)	)	PUNCT
ejde-502	406	17	as	as	ADP
ejde-502	406	18	in	in	ADP
ejde-502	406	19	[	[	X
ejde-502	406	20	19	19	NUM
ejde-502	406	21	,	,	PUNCT
ejde-502	406	22	21	21	NUM
ejde-502	406	23	,	,	PUNCT
ejde-502	406	24	16	16	NUM
ejde-502	406	25	]	]	PUNCT
ejde-502	406	26	(	(	PUNCT
ejde-502	406	27	which	which	PRON
ejde-502	406	28	is	be	AUX
ejde-502	406	29	a	a	DET
ejde-502	406	30	subsolution	subsolution	NOUN
ejde-502	406	31	to	to	ADP
ejde-502	406	32	(	(	PUNCT
ejde-502	406	33	1.1	1.1	NUM
ejde-502	406	34	)	)	PUNCT
ejde-502	406	35	)	)	PUNCT
ejde-502	406	36	below	below	ADP
ejde-502	406	37	it	it	PRON
ejde-502	406	38	.	.	PUNCT
ejde-502	407	1	if	if	SCONJ
ejde-502	407	2	comparison	comparison	NOUN
ejde-502	407	3	would	would	AUX
ejde-502	407	4	be	be	AUX
ejde-502	407	5	allowed	allow	VERB
ejde-502	407	6	,	,	PUNCT
ejde-502	407	7	then	then	ADV
ejde-502	407	8	we	we	PRON
ejde-502	407	9	would	would	AUX
ejde-502	407	10	get	get	VERB
ejde-502	407	11	an	an	DET
ejde-502	407	12	easy	easy	ADJ
ejde-502	407	13	contradiction	contradiction	NOUN
ejde-502	407	14	proving	prove	VERB
ejde-502	407	15	that	that	SCONJ
ejde-502	407	16	any	any	DET
ejde-502	407	17	minimal	minimal	ADJ
ejde-502	407	18	solution	solution	NOUN
ejde-502	407	19	(	(	PUNCT
ejde-502	407	20	and	and	CCONJ
ejde-502	407	21	then	then	ADV
ejde-502	407	22	any	any	DET
ejde-502	407	23	other	other	ADJ
ejde-502	407	24	solution	solution	NOUN
ejde-502	407	25	)	)	PUNCT
ejde-502	407	26	blows	blow	VERB
ejde-502	407	27	up	up	ADP
ejde-502	407	28	in	in	ADP
ejde-502	407	29	finite	finite	ADJ
ejde-502	407	30	time	time	NOUN
ejde-502	407	31	.	.	PUNCT
ejde-502	408	1	but	but	CCONJ
ejde-502	408	2	as	as	SCONJ
ejde-502	408	3	we	we	PRON
ejde-502	408	4	see	see	VERB
ejde-502	408	5	by	by	ADP
ejde-502	408	6	considering	consider	VERB
ejde-502	408	7	the	the	DET
ejde-502	408	8	subsolution	subsolution	NOUN
ejde-502	408	9	u	u	NOUN
ejde-502	408	10	with	with	ADP
ejde-502	408	11	initial	initial	ADJ
ejde-502	408	12	condition	condition	NOUN
ejde-502	408	13	u(x	u(x	NOUN
ejde-502	408	14	,	,	PUNCT
ejde-502	408	15	0	0	NUM
ejde-502	408	16	)	)	PUNCT
ejde-502	408	17	=	=	SYM
ejde-502	408	18	0	0	NUM
ejde-502	409	1	for	for	ADP
ejde-502	409	2	any	any	DET
ejde-502	409	3	x	x	SYM
ejde-502	409	4	∈	∈	PROPN
ejde-502	409	5	rn	rn	PROPN
ejde-502	409	6	,	,	PUNCT
ejde-502	409	7	which	which	PRON
ejde-502	409	8	is	be	AUX
ejde-502	409	9	defined	define	VERB
ejde-502	409	10	explicitly	explicitly	ADV
ejde-502	409	11	by	by	ADP
ejde-502	409	12	u(x	u(x	NOUN
ejde-502	409	13	,	,	PUNCT
ejde-502	409	14	t	t	PROPN
ejde-502	409	15	)	)	PUNCT
ejde-502	409	16	=	=	PUNCT
ejde-502	409	17	(	(	PUNCT
ejde-502	409	18	1	1	NUM
ejde-502	409	19	1−	1−	NUM
ejde-502	409	20	p	p	NOUN
ejde-502	409	21	)	)	PUNCT
ejde-502	409	22	1/(p−1	1/(p−1	NUM
ejde-502	409	23	)	)	PUNCT
ejde-502	409	24	t1/(1−p)(1	t1/(1−p)(1	PROPN
ejde-502	409	25	+	+	CCONJ
ejde-502	409	26	|x|)σ/(1−p	|x|)σ/(1−p	NOUN
ejde-502	409	27	)	)	PUNCT
ejde-502	409	28	,	,	PUNCT
ejde-502	409	29	(	(	PUNCT
ejde-502	409	30	5.2	5.2	NUM
ejde-502	409	31	)	)	PUNCT
ejde-502	409	32	comparison	comparison	NOUN
ejde-502	409	33	does	do	AUX
ejde-502	409	34	not	not	PART
ejde-502	409	35	hold	hold	VERB
ejde-502	409	36	in	in	ADP
ejde-502	409	37	general	general	ADJ
ejde-502	409	38	,	,	PUNCT
ejde-502	409	39	as	as	SCONJ
ejde-502	409	40	otherwise	otherwise	ADV
ejde-502	409	41	any	any	DET
ejde-502	409	42	solution	solution	NOUN
ejde-502	409	43	(	(	PUNCT
ejde-502	409	44	even	even	ADV
ejde-502	409	45	the	the	DET
ejde-502	409	46	ones	one	NOUN
ejde-502	409	47	with	with	ADP
ejde-502	409	48	compact	compact	ADJ
ejde-502	409	49	support	support	NOUN
ejde-502	409	50	)	)	PUNCT
ejde-502	409	51	could	could	AUX
ejde-502	409	52	have	have	AUX
ejde-502	409	53	been	be	AUX
ejde-502	409	54	compared	compare	VERB
ejde-502	409	55	to	to	ADP
ejde-502	409	56	u	u	NOUN
ejde-502	409	57	in	in	ADP
ejde-502	409	58	order	order	NOUN
ejde-502	409	59	to	to	PART
ejde-502	409	60	force	force	VERB
ejde-502	409	61	it	it	PRON
ejde-502	409	62	to	to	PART
ejde-502	409	63	become	become	VERB
ejde-502	409	64	positive	positive	ADJ
ejde-502	409	65	everywhere	everywhere	ADV
ejde-502	409	66	,	,	PUNCT
ejde-502	409	67	contradicting	contradict	VERB
ejde-502	409	68	the	the	DET
ejde-502	409	69	finite	finite	ADJ
ejde-502	409	70	speed	speed	NOUN
ejde-502	409	71	of	of	ADP
ejde-502	409	72	propagation	propagation	NOUN
ejde-502	409	73	at	at	ADP
ejde-502	409	74	least	least	ADJ
ejde-502	409	75	in	in	ADP
ejde-502	409	76	the	the	DET
ejde-502	409	77	range	range	NOUN
ejde-502	409	78	m+	m+	NOUN
ejde-502	409	79	p	p	NOUN
ejde-502	409	80	≥	≥	NUM
ejde-502	409	81	2	2	NUM
ejde-502	409	82	.	.	PUNCT
ejde-502	410	1	this	this	PRON
ejde-502	410	2	opens	open	VERB
ejde-502	410	3	the	the	DET
ejde-502	410	4	question	question	NOUN
ejde-502	410	5	on	on	ADP
ejde-502	410	6	whether	whether	SCONJ
ejde-502	410	7	the	the	DET
ejde-502	410	8	self	self	NOUN
ejde-502	410	9	-	-	PUNCT
ejde-502	410	10	similar	similar	ADJ
ejde-502	410	11	solutions	solution	NOUN
ejde-502	410	12	to	to	ADP
ejde-502	410	13	(	(	PUNCT
ejde-502	410	14	1.1	1.1	NUM
ejde-502	410	15	)	)	PUNCT
ejde-502	410	16	are	be	AUX
ejde-502	410	17	minimal	minimal	ADJ
ejde-502	410	18	solutions	solution	NOUN
ejde-502	410	19	in	in	ADP
ejde-502	410	20	the	the	DET
ejde-502	410	21	sense	sense	NOUN
ejde-502	410	22	of	of	ADP
ejde-502	410	23	the	the	DET
ejde-502	410	24	construction	construction	NOUN
ejde-502	410	25	performed	perform	VERB
ejde-502	410	26	in	in	ADP
ejde-502	410	27	proposition	proposition	NOUN
ejde-502	410	28	2.1	2.1	NUM
ejde-502	410	29	.	.	PUNCT
ejde-502	411	1	2	2	NUM
ejde-502	411	2	.	.	X
ejde-502	411	3	non	non	ADJ
ejde-502	411	4	-	-	NOUN
ejde-502	411	5	existence	existence	NOUN
ejde-502	411	6	of	of	ADP
ejde-502	411	7	positive	positive	ADJ
ejde-502	411	8	solutions	solution	NOUN
ejde-502	411	9	if	if	SCONJ
ejde-502	411	10	σ	σ	PROPN
ejde-502	411	11	>	>	X
ejde-502	411	12	2(1−	2(1−	NUM
ejde-502	411	13	p)/(m−	p)/(m−	PROPN
ejde-502	411	14	1	1	NUM
ejde-502	411	15	)	)	PUNCT
ejde-502	411	16	.	.	PUNCT
ejde-502	412	1	strongly	strongly	ADV
ejde-502	412	2	connected	connect	VERB
ejde-502	412	3	to	to	ADP
ejde-502	412	4	the	the	DET
ejde-502	412	5	first	first	ADJ
ejde-502	412	6	comment	comment	NOUN
ejde-502	412	7	,	,	PUNCT
ejde-502	412	8	and	and	CCONJ
ejde-502	412	9	expecting	expect	VERB
ejde-502	412	10	(	(	PUNCT
ejde-502	412	11	at	at	ADP
ejde-502	412	12	a	a	DET
ejde-502	412	13	formal	formal	ADJ
ejde-502	412	14	level	level	NOUN
ejde-502	412	15	)	)	PUNCT
ejde-502	412	16	that	that	SCONJ
ejde-502	412	17	we	we	PRON
ejde-502	412	18	might	might	AUX
ejde-502	412	19	compare	compare	VERB
ejde-502	412	20	a	a	DET
ejde-502	412	21	strictly	strictly	ADV
ejde-502	412	22	positive	positive	ADJ
ejde-502	412	23	solution	solution	NOUN
ejde-502	412	24	to	to	ADP
ejde-502	412	25	(	(	PUNCT
ejde-502	412	26	1.1	1.1	NUM
ejde-502	412	27	)	)	PUNCT
ejde-502	412	28	with	with	ADP
ejde-502	412	29	the	the	DET
ejde-502	412	30	subsolution	subsolution	NOUN
ejde-502	412	31	u	u	PROPN
ejde-502	412	32	introduced	introduce	VERB
ejde-502	412	33	in	in	ADP
ejde-502	412	34	(	(	PUNCT
ejde-502	412	35	5.2	5.2	NUM
ejde-502	412	36	)	)	PUNCT
ejde-502	412	37	,	,	PUNCT
ejde-502	412	38	we	we	PRON
ejde-502	412	39	conjecture	conjecture	VERB
ejde-502	412	40	that	that	SCONJ
ejde-502	412	41	,	,	PUNCT
ejde-502	412	42	if	if	SCONJ
ejde-502	412	43	l	l	PROPN
ejde-502	412	44	>	>	X
ejde-502	412	45	0	0	PUNCT
ejde-502	412	46	(	(	PUNCT
ejde-502	412	47	that	that	PRON
ejde-502	412	48	is	is	ADV
ejde-502	412	49	,	,	PUNCT
ejde-502	412	50	σ	σ	X
ejde-502	412	51	>	>	X
ejde-502	412	52	2(1	2(1	NUM
ejde-502	412	53	−	−	PUNCT
ejde-502	413	1	p)/(m	p)/(m	INTJ
ejde-502	413	2	−	−	NOUN
ejde-502	413	3	1	1	NUM
ejde-502	413	4	)	)	PUNCT
ejde-502	413	5	)	)	PUNCT
ejde-502	413	6	there	there	PRON
ejde-502	413	7	are	be	VERB
ejde-502	413	8	no	no	DET
ejde-502	413	9	solutions	solution	NOUN
ejde-502	413	10	to	to	ADP
ejde-502	413	11	(	(	PUNCT
ejde-502	413	12	1.1	1.1	NUM
ejde-502	413	13	)	)	PUNCT
ejde-502	413	14	such	such	ADJ
ejde-502	413	15	that	that	SCONJ
ejde-502	413	16	u(x	u(x	NOUN
ejde-502	413	17	,	,	PUNCT
ejde-502	413	18	t	t	PROPN
ejde-502	413	19	)	)	PUNCT
ejde-502	413	20	>	>	X
ejde-502	413	21	0	0	PUNCT
ejde-502	414	1	for	for	ADP
ejde-502	414	2	any	any	DET
ejde-502	414	3	x	x	SYM
ejde-502	414	4	∈	∈	PROPN
ejde-502	414	5	rn	rn	PROPN
ejde-502	414	6	.	.	PUNCT
ejde-502	415	1	indeed	indeed	ADV
ejde-502	415	2	,	,	PUNCT
ejde-502	415	3	assuming	assume	VERB
ejde-502	415	4	that	that	SCONJ
ejde-502	415	5	the	the	DET
ejde-502	415	6	comparison	comparison	NOUN
ejde-502	415	7	can	can	AUX
ejde-502	415	8	be	be	AUX
ejde-502	415	9	performed	perform	VERB
ejde-502	415	10	rigorously	rigorously	ADV
ejde-502	415	11	if	if	SCONJ
ejde-502	415	12	u(x	u(x	NOUN
ejde-502	415	13	,	,	PUNCT
ejde-502	415	14	t	t	PROPN
ejde-502	415	15	)	)	PUNCT
ejde-502	415	16	>	>	X
ejde-502	415	17	0	0	PUNCT
ejde-502	415	18	for	for	ADP
ejde-502	415	19	any	any	DET
ejde-502	415	20	x	x	SYM
ejde-502	415	21	∈	∈	PROPN
ejde-502	415	22	rn	rn	PROPN
ejde-502	415	23	,	,	PUNCT
ejde-502	415	24	we	we	PRON
ejde-502	415	25	would	would	AUX
ejde-502	415	26	get	get	VERB
ejde-502	415	27	a	a	DET
ejde-502	415	28	solution	solution	NOUN
ejde-502	415	29	with	with	ADP
ejde-502	415	30	local	local	ADJ
ejde-502	415	31	behavior	behavior	NOUN
ejde-502	415	32	u(x	u(x	NOUN
ejde-502	415	33	,	,	PUNCT
ejde-502	415	34	t	t	PROPN
ejde-502	415	35	)	)	PUNCT
ejde-502	416	1	≥	≥	NOUN
ejde-502	416	2	c(1	c(1	PROPN
ejde-502	416	3	+	+	CCONJ
ejde-502	416	4	|x|)σ/(1−p	|x|)σ/(1−p	NOUN
ejde-502	416	5	)	)	PUNCT
ejde-502	416	6	,	,	PUNCT
ejde-502	416	7	as	as	ADP
ejde-502	416	8	|x|	|x|	PROPN
ejde-502	416	9	→	→	SYM
ejde-502	416	10	∞	∞	PROPN
ejde-502	416	11	,	,	PUNCT
ejde-502	416	12	for	for	ADP
ejde-502	416	13	all	all	DET
ejde-502	416	14	t	t	PROPN
ejde-502	416	15	>	>	X
ejde-502	416	16	0	0	NUM
ejde-502	416	17	.	.	PUNCT
ejde-502	417	1	but	but	CCONJ
ejde-502	417	2	any	any	DET
ejde-502	417	3	solution	solution	NOUN
ejde-502	417	4	to	to	ADP
ejde-502	417	5	(	(	PUNCT
ejde-502	417	6	1.1	1.1	NUM
ejde-502	417	7	)	)	PUNCT
ejde-502	417	8	is	be	AUX
ejde-502	417	9	a	a	DET
ejde-502	417	10	supersolution	supersolution	NOUN
ejde-502	417	11	to	to	ADP
ejde-502	417	12	the	the	DET
ejde-502	417	13	standard	standard	ADJ
ejde-502	417	14	porous	porous	ADJ
ejde-502	417	15	medium	medium	ADJ
ejde-502	417	16	equation	equation	NOUN
ejde-502	417	17	and	and	CCONJ
ejde-502	417	18	classical	classical	ADJ
ejde-502	417	19	results	result	NOUN
ejde-502	417	20	on	on	ADP
ejde-502	417	21	the	the	DET
ejde-502	417	22	porous	porous	ADJ
ejde-502	417	23	medium	medium	ADJ
ejde-502	417	24	equation	equation	NOUN
ejde-502	417	25	(	(	PUNCT
ejde-502	417	26	see	see	VERB
ejde-502	417	27	for	for	ADP
ejde-502	417	28	example	example	NOUN
ejde-502	417	29	[	[	X
ejde-502	417	30	4	4	NUM
ejde-502	417	31	,	,	PUNCT
ejde-502	417	32	7	7	NUM
ejde-502	417	33	,	,	PUNCT
ejde-502	417	34	8	8	NUM
ejde-502	417	35	]	]	PUNCT
ejde-502	417	36	)	)	PUNCT
ejde-502	417	37	state	state	NOUN
ejde-502	417	38	that	that	SCONJ
ejde-502	417	39	there	there	PRON
ejde-502	417	40	are	be	VERB
ejde-502	417	41	no	no	DET
ejde-502	417	42	solutions	solution	NOUN
ejde-502	417	43	(	(	PUNCT
ejde-502	417	44	and	and	CCONJ
ejde-502	417	45	it	it	PRON
ejde-502	417	46	seems	seem	VERB
ejde-502	417	47	to	to	ADP
ejde-502	417	48	us	we	PRON
ejde-502	417	49	that	that	SCONJ
ejde-502	417	50	the	the	DET
ejde-502	417	51	proofs	proof	NOUN
ejde-502	417	52	can	can	AUX
ejde-502	417	53	be	be	AUX
ejde-502	417	54	extended	extend	VERB
ejde-502	417	55	to	to	ADP
ejde-502	417	56	supersolutions	supersolution	NOUN
ejde-502	417	57	)	)	PUNCT
ejde-502	417	58	to	to	ADP
ejde-502	417	59	it	it	PRON
ejde-502	417	60	increasing	increase	VERB
ejde-502	417	61	at	at	ADP
ejde-502	417	62	infinity	infinity	NOUN
ejde-502	417	63	faster	fast	ADV
ejde-502	417	64	than	than	ADP
ejde-502	417	65	|x|2/(m−1	|x|2/(m−1	PROPN
ejde-502	417	66	)	)	PUNCT
ejde-502	417	67	.	.	PUNCT
ejde-502	418	1	since	since	SCONJ
ejde-502	418	2	σ/(1−	σ/(1−	PROPN
ejde-502	418	3	p	p	PROPN
ejde-502	418	4	)	)	PUNCT
ejde-502	418	5	>	>	X
ejde-502	418	6	2/(m	2/(m	NUM
ejde-502	418	7	−	−	NOUN
ejde-502	418	8	1	1	X
ejde-502	418	9	)	)	PUNCT
ejde-502	418	10	if	if	SCONJ
ejde-502	418	11	l	l	PROPN
ejde-502	418	12	>	>	X
ejde-502	418	13	0	0	NUM
ejde-502	418	14	,	,	PUNCT
ejde-502	418	15	we	we	PRON
ejde-502	418	16	would	would	AUX
ejde-502	418	17	be	be	AUX
ejde-502	418	18	in	in	ADP
ejde-502	418	19	this	this	DET
ejde-502	418	20	case	case	NOUN
ejde-502	418	21	.	.	PUNCT
ejde-502	419	1	in	in	ADP
ejde-502	419	2	particular	particular	ADJ
ejde-502	419	3	,	,	PUNCT
ejde-502	419	4	since	since	SCONJ
ejde-502	419	5	infinite	infinite	ADJ
ejde-502	419	6	speed	speed	NOUN
ejde-502	419	7	of	of	ADP
ejde-502	419	8	propagation	propagation	NOUN
ejde-502	419	9	is	be	AUX
ejde-502	419	10	in	in	ADP
ejde-502	419	11	force	force	NOUN
ejde-502	419	12	for	for	ADP
ejde-502	419	13	m	m	PROPN
ejde-502	419	14	+	+	X
ejde-502	419	15	p	p	X
ejde-502	419	16	<	<	X
ejde-502	419	17	2	2	NUM
ejde-502	419	18	by	by	ADP
ejde-502	419	19	theorem	theorem	ADJ
ejde-502	419	20	1.5	1.5	NUM
ejde-502	419	21	,	,	PUNCT
ejde-502	419	22	we	we	PRON
ejde-502	419	23	expect	expect	VERB
ejde-502	419	24	complete	complete	ADJ
ejde-502	419	25	non	non	ADJ
ejde-502	419	26	-	-	NOUN
ejde-502	419	27	existence	existence	NOUN
ejde-502	419	28	of	of	ADP
ejde-502	419	29	non	non	ADJ
ejde-502	419	30	-	-	ADJ
ejde-502	419	31	trivial	trivial	ADJ
ejde-502	419	32	solutions	solution	NOUN
ejde-502	419	33	if	if	SCONJ
ejde-502	419	34	l	l	PROPN
ejde-502	419	35	>	>	X
ejde-502	419	36	0	0	PUNCT
ejde-502	420	1	and	and	CCONJ
ejde-502	420	2	m+	m+	NUM
ejde-502	420	3	p	p	X
ejde-502	420	4	<	<	X
ejde-502	420	5	2	2	NUM
ejde-502	420	6	.	.	NOUN
ejde-502	420	7	3	3	NUM
ejde-502	420	8	.	.	X
ejde-502	421	1	establishing	establish	VERB
ejde-502	421	2	which	which	DET
ejde-502	421	3	self	self	NOUN
ejde-502	421	4	-	-	PUNCT
ejde-502	421	5	similar	similar	ADJ
ejde-502	421	6	solutions	solution	NOUN
ejde-502	421	7	are	be	AUX
ejde-502	421	8	minimal	minimal	ADJ
ejde-502	421	9	.	.	PUNCT
ejde-502	422	1	an	an	DET
ejde-502	422	2	immediate	immediate	ADJ
ejde-502	422	3	adaptation	adaptation	NOUN
ejde-502	422	4	of	of	ADP
ejde-502	422	5	the	the	DET
ejde-502	422	6	result	result	NOUN
ejde-502	422	7	in	in	ADP
ejde-502	422	8	[	[	X
ejde-502	422	9	19	19	NUM
ejde-502	422	10	]	]	PUNCT
ejde-502	422	11	proves	prove	VERB
ejde-502	422	12	that	that	SCONJ
ejde-502	422	13	(	(	PUNCT
ejde-502	422	14	1.1	1.1	NUM
ejde-502	422	15	)	)	PUNCT
ejde-502	422	16	with	with	ADP
ejde-502	422	17	m+	m+	NOUN
ejde-502	422	18	p	p	X
ejde-502	422	19	>	>	SYM
ejde-502	422	20	2	2	NUM
ejde-502	422	21	presents	present	VERB
ejde-502	422	22	two	two	NUM
ejde-502	422	23	types	type	NOUN
ejde-502	422	24	of	of	ADP
ejde-502	422	25	blow	blow	NOUN
ejde-502	422	26	-	-	PUNCT
ejde-502	422	27	up	up	ADP
ejde-502	422	28	self	self	NOUN
ejde-502	422	29	-	-	PUNCT
ejde-502	422	30	similar	similar	ADJ
ejde-502	422	31	solutions	solution	NOUN
ejde-502	422	32	u(x	u(x	NOUN
ejde-502	422	33	,	,	PUNCT
ejde-502	422	34	t	t	NOUN
ejde-502	422	35	)	)	PUNCT
ejde-502	422	36	=	=	PUNCT
ejde-502	423	1	(	(	PUNCT
ejde-502	423	2	t	t	PROPN
ejde-502	423	3	−	−	PROPN
ejde-502	423	4	t)−αf((1	t)−αf((1	PROPN
ejde-502	423	5	+	+	CCONJ
ejde-502	423	6	|x|)(t	|x|)(t	ADJ
ejde-502	423	7	−	−	NOUN
ejde-502	423	8	t)β	t)β	NOUN
ejde-502	423	9	)	)	PUNCT
ejde-502	423	10	,	,	PUNCT
ejde-502	423	11	α	α	X
ejde-502	423	12	=	=	PUNCT
ejde-502	423	13	σ	σ	PROPN
ejde-502	424	1	+	+	NUM
ejde-502	424	2	2	2	NUM
ejde-502	424	3	l	l	NOUN
ejde-502	424	4	,	,	PUNCT
ejde-502	424	5	β	β	X
ejde-502	424	6	=	=	PUNCT
ejde-502	424	7	m−	m−	PROPN
ejde-502	424	8	p	p	PROPN
ejde-502	424	9	l	l	NOUN
ejde-502	424	10	,	,	PUNCT
ejde-502	424	11	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	424	12	reaction	reaction	NOUN
ejde-502	424	13	-	-	PUNCT
ejde-502	424	14	diffusion	diffusion	NOUN
ejde-502	424	15	with	with	ADP
ejde-502	424	16	weighted	weight	VERB
ejde-502	424	17	strong	strong	ADJ
ejde-502	424	18	reaction	reaction	NOUN
ejde-502	424	19	19	19	NUM
ejde-502	424	20	which	which	PRON
ejde-502	424	21	differ	differ	VERB
ejde-502	424	22	with	with	ADP
ejde-502	424	23	respect	respect	NOUN
ejde-502	424	24	to	to	ADP
ejde-502	424	25	the	the	DET
ejde-502	424	26	local	local	ADJ
ejde-502	424	27	behavior	behavior	NOUN
ejde-502	424	28	of	of	ADP
ejde-502	424	29	the	the	DET
ejde-502	424	30	profile	profile	NOUN
ejde-502	424	31	f(ξ	f(ξ	NOUN
ejde-502	424	32	)	)	PUNCT
ejde-502	424	33	near	near	ADP
ejde-502	424	34	the	the	DET
ejde-502	424	35	interface	interface	NOUN
ejde-502	424	36	point	point	NOUN
ejde-502	424	37	ξ0	ξ0	PROPN
ejde-502	424	38	∈	∈	PROPN
ejde-502	424	39	(	(	PUNCT
ejde-502	424	40	0,∞	0,∞	NOUN
ejde-502	424	41	)	)	PUNCT
ejde-502	424	42	,	,	PUNCT
ejde-502	424	43	namely	namely	ADV
ejde-502	424	44	f(ξ	f(ξ	NOUN
ejde-502	424	45	)	)	PUNCT
ejde-502	424	46	∼	∼	NOUN
ejde-502	424	47	(	(	PUNCT
ejde-502	424	48	ξ0	ξ0	PROPN
ejde-502	424	49	−	−	PROPN
ejde-502	424	50	ξ)1/(m−1	ξ)1/(m−1	PROPN
ejde-502	424	51	)	)	PUNCT
ejde-502	424	52	(	(	PUNCT
ejde-502	424	53	type	type	NOUN
ejde-502	424	54	i	i	NOUN
ejde-502	424	55	)	)	PUNCT
ejde-502	424	56	or	or	CCONJ
ejde-502	424	57	f(ξ	f(ξ	X
ejde-502	424	58	)	)	PUNCT
ejde-502	424	59	∼	∼	NOUN
ejde-502	424	60	(	(	PUNCT
ejde-502	424	61	ξ0	ξ0	NOUN
ejde-502	424	62	−	−	PROPN
ejde-502	424	63	ξ)1/(1−p	ξ)1/(1−p	PROPN
ejde-502	424	64	)	)	PUNCT
ejde-502	424	65	(	(	PUNCT
ejde-502	424	66	type	type	NOUN
ejde-502	424	67	ii	ii	PROPN
ejde-502	424	68	)	)	PUNCT
ejde-502	424	69	,	,	PUNCT
ejde-502	424	70	both	both	PRON
ejde-502	424	71	taken	take	VERB
ejde-502	424	72	in	in	ADP
ejde-502	424	73	the	the	DET
ejde-502	424	74	limit	limit	NOUN
ejde-502	424	75	ξ	ξ	X
ejde-502	424	76	→	→	SYM
ejde-502	424	77	ξ0	ξ0	PROPN
ejde-502	424	78	,	,	PUNCT
ejde-502	424	79	ξ	ξ	X
ejde-502	424	80	<	<	X
ejde-502	424	81	ξ0	ξ0	PROPN
ejde-502	424	82	.	.	PUNCT
ejde-502	425	1	it	it	PRON
ejde-502	425	2	is	be	AUX
ejde-502	425	3	also	also	ADV
ejde-502	425	4	shown	show	VERB
ejde-502	425	5	at	at	ADP
ejde-502	425	6	least	least	ADJ
ejde-502	425	7	at	at	ADP
ejde-502	425	8	a	a	DET
ejde-502	425	9	formal	formal	ADJ
ejde-502	425	10	level	level	NOUN
ejde-502	425	11	that	that	PRON
ejde-502	425	12	these	these	DET
ejde-502	425	13	solutions	solution	NOUN
ejde-502	425	14	satisfy	satisfy	VERB
ejde-502	425	15	two	two	NUM
ejde-502	425	16	different	different	ADJ
ejde-502	425	17	interface	interface	NOUN
ejde-502	425	18	equations	equation	NOUN
ejde-502	425	19	.	.	PUNCT
ejde-502	426	1	thus	thus	ADV
ejde-502	426	2	,	,	PUNCT
ejde-502	426	3	a	a	DET
ejde-502	426	4	natural	natural	ADJ
ejde-502	426	5	question	question	NOUN
ejde-502	426	6	would	would	AUX
ejde-502	426	7	be	be	AUX
ejde-502	426	8	whether	whether	SCONJ
ejde-502	426	9	any	any	PRON
ejde-502	426	10	of	of	ADP
ejde-502	426	11	these	these	DET
ejde-502	426	12	self	self	NOUN
ejde-502	426	13	-	-	PUNCT
ejde-502	426	14	similar	similar	ADJ
ejde-502	426	15	solutions	solution	NOUN
ejde-502	426	16	is	be	AUX
ejde-502	426	17	minimal	minimal	ADJ
ejde-502	426	18	in	in	ADP
ejde-502	426	19	the	the	DET
ejde-502	426	20	sense	sense	NOUN
ejde-502	426	21	of	of	ADP
ejde-502	426	22	the	the	DET
ejde-502	426	23	construction	construction	NOUN
ejde-502	426	24	in	in	ADP
ejde-502	426	25	proposition	proposition	NOUN
ejde-502	426	26	2.1	2.1	NUM
ejde-502	426	27	,	,	PUNCT
ejde-502	426	28	which	which	PRON
ejde-502	426	29	would	would	AUX
ejde-502	426	30	allow	allow	VERB
ejde-502	426	31	us	we	PRON
ejde-502	426	32	to	to	PART
ejde-502	426	33	compare	compare	VERB
ejde-502	426	34	and	and	CCONJ
ejde-502	426	35	conclude	conclude	VERB
ejde-502	426	36	on	on	ADP
ejde-502	426	37	the	the	DET
ejde-502	426	38	finite	finite	ADJ
ejde-502	426	39	time	time	NOUN
ejde-502	426	40	blow	blow	NOUN
ejde-502	426	41	-	-	PUNCT
ejde-502	426	42	up	up	NOUN
ejde-502	426	43	.	.	PUNCT
ejde-502	427	1	this	this	PRON
ejde-502	427	2	is	be	AUX
ejde-502	427	3	not	not	PART
ejde-502	427	4	an	an	DET
ejde-502	427	5	easy	easy	ADJ
ejde-502	427	6	question	question	NOUN
ejde-502	427	7	and	and	CCONJ
ejde-502	427	8	our	our	PRON
ejde-502	427	9	intuition	intuition	NOUN
ejde-502	427	10	suggests	suggest	VERB
ejde-502	427	11	that	that	SCONJ
ejde-502	427	12	minimality	minimality	NOUN
ejde-502	427	13	has	have	VERB
ejde-502	427	14	to	to	PART
ejde-502	427	15	do	do	VERB
ejde-502	427	16	with	with	ADP
ejde-502	427	17	the	the	DET
ejde-502	427	18	interface	interface	NOUN
ejde-502	427	19	equation	equation	NOUN
ejde-502	427	20	:	:	PUNCT
ejde-502	427	21	we	we	PRON
ejde-502	427	22	might	might	AUX
ejde-502	427	23	expect	expect	VERB
ejde-502	427	24	that	that	SCONJ
ejde-502	427	25	the	the	DET
ejde-502	427	26	solutions	solution	NOUN
ejde-502	427	27	with	with	ADP
ejde-502	427	28	interface	interface	NOUN
ejde-502	427	29	of	of	ADP
ejde-502	427	30	type	type	NOUN
ejde-502	427	31	i	i	PRON
ejde-502	427	32	are	be	AUX
ejde-502	427	33	minimal	minimal	ADJ
ejde-502	427	34	,	,	PUNCT
ejde-502	427	35	while	while	SCONJ
ejde-502	427	36	the	the	DET
ejde-502	427	37	other	other	ADJ
ejde-502	427	38	ones	one	NOUN
ejde-502	427	39	are	be	AUX
ejde-502	427	40	not	not	PART
ejde-502	427	41	.	.	PUNCT
ejde-502	428	1	this	this	DET
ejde-502	428	2	conjecture	conjecture	NOUN
ejde-502	428	3	is	be	AUX
ejde-502	428	4	supported	support	VERB
ejde-502	428	5	by	by	ADP
ejde-502	428	6	the	the	DET
ejde-502	428	7	analogy	analogy	NOUN
ejde-502	428	8	with	with	ADP
ejde-502	428	9	the	the	DET
ejde-502	428	10	minimality	minimality	NOUN
ejde-502	428	11	of	of	ADP
ejde-502	428	12	the	the	DET
ejde-502	428	13	traveling	travel	VERB
ejde-502	428	14	wave	wave	NOUN
ejde-502	428	15	solutions	solution	NOUN
ejde-502	428	16	to	to	ADP
ejde-502	428	17	(	(	PUNCT
ejde-502	428	18	1.5	1.5	NUM
ejde-502	428	19	)	)	PUNCT
ejde-502	428	20	,	,	PUNCT
ejde-502	428	21	see	see	VERB
ejde-502	428	22	for	for	ADP
ejde-502	428	23	example	example	NOUN
ejde-502	429	1	[	[	X
ejde-502	429	2	27	27	NUM
ejde-502	429	3	,	,	PUNCT
ejde-502	429	4	theorem	theorem	VERB
ejde-502	429	5	4.1	4.1	NUM
ejde-502	429	6	]	]	PUNCT
ejde-502	429	7	.	.	PUNCT
ejde-502	430	1	4	4	X
ejde-502	430	2	.	.	X
ejde-502	430	3	connection	connection	NOUN
ejde-502	430	4	between	between	ADP
ejde-502	430	5	non	non	ADJ
ejde-502	430	6	-	-	ADJ
ejde-502	430	7	uniqueness	uniqueness	ADJ
ejde-502	430	8	and	and	CCONJ
ejde-502	430	9	blow	blow	NOUN
ejde-502	430	10	-	-	PUNCT
ejde-502	430	11	up	up	ADP
ejde-502	430	12	time	time	NOUN
ejde-502	430	13	.	.	PUNCT
ejde-502	431	1	a	a	DET
ejde-502	431	2	much	much	ADV
ejde-502	431	3	deeper	deep	ADJ
ejde-502	431	4	open	open	ADJ
ejde-502	431	5	question	question	NOUN
ejde-502	431	6	is	be	AUX
ejde-502	431	7	related	relate	VERB
ejde-502	431	8	to	to	ADP
ejde-502	431	9	whether	whether	SCONJ
ejde-502	431	10	,	,	PUNCT
ejde-502	431	11	in	in	ADP
ejde-502	431	12	the	the	DET
ejde-502	431	13	range	range	NOUN
ejde-502	431	14	σ	σ	PROPN
ejde-502	431	15	>	>	X
ejde-502	431	16	2(1−p)/(m−1	2(1−p)/(m−1	NUM
ejde-502	431	17	)	)	PUNCT
ejde-502	431	18	,	,	PUNCT
ejde-502	431	19	prescribing	prescribe	VERB
ejde-502	431	20	a	a	DET
ejde-502	431	21	blowup	blowup	ADJ
ejde-502	431	22	time	time	NOUN
ejde-502	431	23	t	t	PROPN
ejde-502	431	24	and	and	CCONJ
ejde-502	431	25	a	a	DET
ejde-502	431	26	function	function	NOUN
ejde-502	431	27	u0	u0	ADJ
ejde-502	431	28	,	,	PUNCT
ejde-502	431	29	there	there	PRON
ejde-502	431	30	exists	exist	VERB
ejde-502	431	31	a	a	DET
ejde-502	431	32	unique	unique	ADJ
ejde-502	431	33	solution	solution	NOUN
ejde-502	431	34	to	to	ADP
ejde-502	431	35	the	the	DET
ejde-502	431	36	cauchy	cauchy	ADJ
ejde-502	431	37	problem	problem	NOUN
ejde-502	431	38	(	(	PUNCT
ejde-502	431	39	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	431	40	)	)	PUNCT
ejde-502	431	41	with	with	ADP
ejde-502	431	42	condition	condition	NOUN
ejde-502	431	43	u0	u0	NOUN
ejde-502	431	44	blowing	blow	VERB
ejde-502	431	45	up	up	ADP
ejde-502	431	46	in	in	ADP
ejde-502	431	47	finite	finite	ADJ
ejde-502	431	48	time	time	NOUN
ejde-502	431	49	exactly	exactly	ADV
ejde-502	431	50	at	at	ADP
ejde-502	431	51	the	the	DET
ejde-502	431	52	given	give	VERB
ejde-502	431	53	time	time	NOUN
ejde-502	431	54	t	t	PROPN
ejde-502	431	55	.	.	PUNCT
ejde-502	432	1	more	more	ADV
ejde-502	432	2	precisely	precisely	ADV
ejde-502	432	3	,	,	PUNCT
ejde-502	432	4	taking	take	VERB
ejde-502	432	5	u0	u0	PROPN
ejde-502	432	6	∈	∈	PROPN
ejde-502	432	7	c0(r	c0(r	NOUN
ejde-502	432	8	)	)	PUNCT
ejde-502	432	9	satisfying	satisfying	NOUN
ejde-502	432	10	(	(	PUNCT
ejde-502	432	11	1.4	1.4	NUM
ejde-502	432	12	)	)	PUNCT
ejde-502	432	13	,	,	PUNCT
ejde-502	432	14	there	there	PRON
ejde-502	432	15	exists	exist	VERB
ejde-502	432	16	a	a	DET
ejde-502	432	17	minimal	minimal	ADJ
ejde-502	432	18	solution	solution	NOUN
ejde-502	432	19	m(u0	m(u0	NOUN
ejde-502	432	20	)	)	PUNCT
ejde-502	432	21	which	which	PRON
ejde-502	432	22	(	(	PUNCT
ejde-502	432	23	assuming	assume	VERB
ejde-502	432	24	that	that	DET
ejde-502	432	25	point	point	NOUN
ejde-502	432	26	1	1	NUM
ejde-502	432	27	in	in	ADP
ejde-502	432	28	this	this	DET
ejde-502	432	29	enumeration	enumeration	NOUN
ejde-502	432	30	of	of	ADP
ejde-502	432	31	open	open	ADJ
ejde-502	432	32	problems	problem	NOUN
ejde-502	432	33	holds	hold	VERB
ejde-502	432	34	true	true	ADJ
ejde-502	432	35	,	,	PUNCT
ejde-502	432	36	as	as	SCONJ
ejde-502	432	37	we	we	PRON
ejde-502	432	38	strongly	strongly	ADV
ejde-502	432	39	expect	expect	VERB
ejde-502	432	40	)	)	PUNCT
ejde-502	432	41	comes	come	VERB
ejde-502	432	42	with	with	ADP
ejde-502	432	43	a	a	DET
ejde-502	432	44	finite	finite	ADJ
ejde-502	432	45	blow	blow	NOUN
ejde-502	432	46	-	-	PUNCT
ejde-502	432	47	up	up	ADP
ejde-502	432	48	time	time	NOUN
ejde-502	432	49	t0	t0	PROPN
ejde-502	432	50	∈	∈	PROPN
ejde-502	432	51	(	(	PUNCT
ejde-502	432	52	0,∞	0,∞	NOUN
ejde-502	432	53	)	)	PUNCT
ejde-502	432	54	.	.	PUNCT
ejde-502	433	1	we	we	PRON
ejde-502	433	2	have	have	AUX
ejde-502	433	3	also	also	ADV
ejde-502	433	4	proved	prove	VERB
ejde-502	433	5	in	in	ADP
ejde-502	433	6	section	section	NOUN
ejde-502	433	7	3	3	NUM
ejde-502	433	8	that	that	SCONJ
ejde-502	433	9	the	the	DET
ejde-502	433	10	cauchy	cauchy	PROPN
ejde-502	433	11	problem	problem	NOUN
ejde-502	433	12	(	(	PUNCT
ejde-502	433	13	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	433	14	)	)	PUNCT
ejde-502	433	15	has	have	VERB
ejde-502	433	16	an	an	DET
ejde-502	433	17	infinite	infinite	ADJ
ejde-502	433	18	number	number	NOUN
ejde-502	433	19	of	of	ADP
ejde-502	433	20	compactly	compactly	ADV
ejde-502	433	21	supported	support	VERB
ejde-502	433	22	solutions	solution	NOUN
ejde-502	433	23	with	with	ADP
ejde-502	433	24	interfaces	interface	NOUN
ejde-502	433	25	advancing	advance	VERB
ejde-502	433	26	faster	fast	ADV
ejde-502	433	27	than	than	ADP
ejde-502	433	28	the	the	DET
ejde-502	433	29	minimal	minimal	ADJ
ejde-502	433	30	one	one	NUM
ejde-502	433	31	,	,	PUNCT
ejde-502	433	32	and	and	CCONJ
ejde-502	433	33	estimate	estimate	VERB
ejde-502	433	34	(	(	PUNCT
ejde-502	433	35	3.4	3.4	NUM
ejde-502	433	36	)	)	PUNCT
ejde-502	433	37	shows	show	VERB
ejde-502	433	38	that	that	SCONJ
ejde-502	433	39	faster	fast	ADV
ejde-502	433	40	advancing	advance	VERB
ejde-502	433	41	speed	speed	NOUN
ejde-502	433	42	of	of	ADP
ejde-502	433	43	the	the	DET
ejde-502	433	44	interface	interface	NOUN
ejde-502	433	45	implies	imply	VERB
ejde-502	433	46	shorter	short	ADJ
ejde-502	433	47	lifetime	lifetime	NOUN
ejde-502	433	48	before	before	ADP
ejde-502	433	49	blow	blow	NOUN
ejde-502	433	50	-	-	PUNCT
ejde-502	433	51	up	up	NOUN
ejde-502	433	52	.	.	PUNCT
ejde-502	434	1	thus	thus	ADV
ejde-502	434	2	one	one	PRON
ejde-502	434	3	can	can	AUX
ejde-502	434	4	wonder	wonder	VERB
ejde-502	434	5	naturally	naturally	ADV
ejde-502	434	6	whether	whether	SCONJ
ejde-502	434	7	,	,	PUNCT
ejde-502	434	8	given	give	VERB
ejde-502	434	9	t	t	PROPN
ejde-502	434	10	∈	∈	PROPN
ejde-502	434	11	(	(	PUNCT
ejde-502	434	12	0	0	NUM
ejde-502	434	13	,	,	PUNCT
ejde-502	434	14	t0	t0	PROPN
ejde-502	434	15	)	)	PUNCT
ejde-502	434	16	,	,	PUNCT
ejde-502	434	17	there	there	PRON
ejde-502	434	18	exists	exist	VERB
ejde-502	434	19	one	one	NUM
ejde-502	434	20	solution	solution	NOUN
ejde-502	434	21	to	to	ADP
ejde-502	434	22	the	the	DET
ejde-502	434	23	cauchy	cauchy	ADJ
ejde-502	434	24	problem	problem	NOUN
ejde-502	434	25	(	(	PUNCT
ejde-502	434	26	1.1)-(1.2	1.1)-(1.2	NUM
ejde-502	434	27	)	)	PUNCT
ejde-502	434	28	blowing	blow	VERB
ejde-502	434	29	up	up	ADP
ejde-502	434	30	exactly	exactly	ADV
ejde-502	434	31	at	at	ADP
ejde-502	434	32	this	this	DET
ejde-502	434	33	time	time	NOUN
ejde-502	434	34	t	t	NOUN
ejde-502	434	35	.	.	PUNCT
ejde-502	435	1	we	we	PRON
ejde-502	435	2	do	do	AUX
ejde-502	435	3	not	not	PART
ejde-502	435	4	have	have	VERB
ejde-502	435	5	by	by	ADP
ejde-502	435	6	now	now	ADV
ejde-502	435	7	a	a	DET
ejde-502	435	8	suggestion	suggestion	NOUN
ejde-502	435	9	of	of	ADP
ejde-502	435	10	how	how	SCONJ
ejde-502	435	11	to	to	PART
ejde-502	435	12	approach	approach	VERB
ejde-502	435	13	this	this	DET
ejde-502	435	14	problem	problem	NOUN
ejde-502	435	15	,	,	PUNCT
ejde-502	435	16	but	but	CCONJ
ejde-502	435	17	it	it	PRON
ejde-502	435	18	is	be	AUX
ejde-502	435	19	in	in	ADP
ejde-502	435	20	our	our	PRON
ejde-502	435	21	opinion	opinion	NOUN
ejde-502	435	22	an	an	DET
ejde-502	435	23	interesting	interesting	ADJ
ejde-502	435	24	and	and	CCONJ
ejde-502	435	25	deep	deep	ADJ
ejde-502	435	26	open	open	ADJ
ejde-502	435	27	question	question	NOUN
ejde-502	435	28	.	.	PUNCT
ejde-502	436	1	acknowledgments	acknowledgment	NOUN
ejde-502	436	2	.	.	PUNCT
ejde-502	437	1	r.	r.	PROPN
ejde-502	437	2	g.	g.	PROPN
ejde-502	437	3	i.	i.	PROPN
ejde-502	437	4	and	and	CCONJ
ejde-502	437	5	a.	a.	PROPN
ejde-502	437	6	s.	s.	PROPN
ejde-502	437	7	were	be	AUX
ejde-502	437	8	partially	partially	ADV
ejde-502	437	9	supported	support	VERB
ejde-502	437	10	by	by	ADP
ejde-502	437	11	the	the	DET
ejde-502	437	12	project	project	NOUN
ejde-502	437	13	pid2020	pid2020	ADV
ejde-502	437	14	-	-	PUNCT
ejde-502	437	15	115273gb	115273gb	NOUN
ejde-502	437	16	-	-	PUNCT
ejde-502	437	17	i00	i00	NOUN
ejde-502	437	18	and	and	CCONJ
ejde-502	437	19	by	by	ADP
ejde-502	437	20	the	the	DET
ejde-502	437	21	grant	grant	PROPN
ejde-502	437	22	red2022	red2022	NOUN
ejde-502	437	23	-	-	PUNCT
ejde-502	437	24	134301	134301	NUM
ejde-502	437	25	-	-	PUNCT
ejde-502	437	26	t	t	PROPN
ejde-502	437	27	(	(	PUNCT
ejde-502	437	28	spain	spain	PROPN
ejde-502	437	29	)	)	PUNCT
ejde-502	437	30	.	.	PUNCT
ejde-502	438	1	references	reference	NOUN
ejde-502	438	2	[	[	X
ejde-502	438	3	1	1	NUM
ejde-502	438	4	]	]	PUNCT
ejde-502	438	5	d.	d.	PROPN
ejde-502	438	6	andreucci	andreucci	PROPN
ejde-502	438	7	,	,	PUNCT
ejde-502	438	8	e.	e.	PROPN
ejde-502	438	9	dibenedetto	dibenedetto	PROPN
ejde-502	438	10	;	;	PUNCT
ejde-502	438	11	on	on	ADP
ejde-502	438	12	the	the	DET
ejde-502	438	13	cauchy	cauchy	ADJ
ejde-502	438	14	problem	problem	NOUN
ejde-502	438	15	and	and	CCONJ
ejde-502	438	16	initial	initial	ADJ
ejde-502	438	17	traces	trace	NOUN
ejde-502	438	18	for	for	ADP
ejde-502	438	19	a	a	DET
ejde-502	438	20	class	class	NOUN
ejde-502	438	21	of	of	ADP
ejde-502	438	22	evolution	evolution	NOUN
ejde-502	438	23	equations	equation	NOUN
ejde-502	438	24	with	with	ADP
ejde-502	438	25	strongly	strongly	ADV
ejde-502	438	26	nonlinear	nonlinear	ADJ
ejde-502	438	27	sources	source	NOUN
ejde-502	438	28	,	,	PUNCT
ejde-502	438	29	ann	ann	PROPN
ejde-502	438	30	.	.	PROPN
ejde-502	438	31	scuola	scuola	PROPN
ejde-502	438	32	norm	norm	NOUN
ejde-502	438	33	.	.	PUNCT
ejde-502	439	1	sup	sup	NOUN
ejde-502	439	2	.	.	PUNCT
ejde-502	440	1	pisa	pisa	PROPN
ejde-502	440	2	,	,	PUNCT
ejde-502	440	3	18	18	NUM
ejde-502	440	4	(	(	PUNCT
ejde-502	440	5	1991	1991	NUM
ejde-502	440	6	)	)	PUNCT
ejde-502	440	7	,	,	PUNCT
ejde-502	440	8	363	363	NUM
ejde-502	440	9	-	-	SYM
ejde-502	440	10	441	441	NUM
ejde-502	440	11	.	.	PUNCT
ejde-502	441	1	[	[	X
ejde-502	441	2	2	2	NUM
ejde-502	441	3	]	]	PUNCT
ejde-502	441	4	d.	d.	PROPN
ejde-502	441	5	andreucci	andreucci	PROPN
ejde-502	441	6	,	,	PUNCT
ejde-502	441	7	a.	a.	PROPN
ejde-502	441	8	f.	f.	PROPN
ejde-502	441	9	tedeev	tedeev	PROPN
ejde-502	441	10	;	;	PUNCT
ejde-502	441	11	universal	universal	ADJ
ejde-502	441	12	bounds	bound	NOUN
ejde-502	441	13	at	at	ADP
ejde-502	441	14	the	the	DET
ejde-502	441	15	blow	blow	NOUN
ejde-502	441	16	-	-	PUNCT
ejde-502	441	17	up	up	ADP
ejde-502	441	18	time	time	NOUN
ejde-502	441	19	for	for	ADP
ejde-502	441	20	nonlinear	nonlinear	ADJ
ejde-502	441	21	parabolic	parabolic	PROPN
ejde-502	441	22	equations	equation	NOUN
ejde-502	441	23	,	,	PUNCT
ejde-502	441	24	adv	adv	PROPN
ejde-502	441	25	.	.	PUNCT
ejde-502	441	26	differential	differential	PROPN
ejde-502	441	27	equations	equation	NOUN
ejde-502	441	28	,	,	PUNCT
ejde-502	441	29	10	10	NUM
ejde-502	441	30	(	(	PUNCT
ejde-502	441	31	2005	2005	NUM
ejde-502	441	32	)	)	PUNCT
ejde-502	441	33	,	,	PUNCT
ejde-502	441	34	no	no	INTJ
ejde-502	441	35	.	.	NOUN
ejde-502	441	36	1	1	NUM
ejde-502	441	37	,	,	PUNCT
ejde-502	441	38	89	89	NUM
ejde-502	441	39	-	-	SYM
ejde-502	441	40	120	120	NUM
ejde-502	441	41	.	.	PUNCT
ejde-502	442	1	[	[	X
ejde-502	442	2	3	3	X
ejde-502	442	3	]	]	X
ejde-502	442	4	d.	d.	PROPN
ejde-502	442	5	g.	g.	PROPN
ejde-502	442	6	aronson	aronson	PROPN
ejde-502	442	7	,	,	PUNCT
ejde-502	442	8	ph	ph	PROPN
ejde-502	442	9	.	.	PROPN
ejde-502	442	10	bénilan	bénilan	PROPN
ejde-502	442	11	;	;	PUNCT
ejde-502	442	12	régularité	régularité	SYM
ejde-502	442	13	des	des	X
ejde-502	442	14	solutions	solutions	X
ejde-502	442	15	de	de	X
ejde-502	442	16	l’équation	l’équation	PROPN
ejde-502	442	17	des	des	PROPN
ejde-502	442	18	milieux	milieux	PROPN
ejde-502	442	19	poreux	poreux	PROPN
ejde-502	442	20	dans	dans	PROPN
ejde-502	442	21	rn	rn	PROPN
ejde-502	442	22	(	(	PUNCT
ejde-502	442	23	french	french	PROPN
ejde-502	442	24	)	)	PUNCT
ejde-502	442	25	,	,	PUNCT
ejde-502	442	26	cr	cr	PROPN
ejde-502	442	27	acad	acad	PROPN
ejde-502	442	28	.	.	PUNCT
ejde-502	443	1	sci	sci	PROPN
ejde-502	443	2	.	.	PROPN
ejde-502	444	1	paris	paris	PROPN
ejde-502	444	2	sér	sér	PROPN
ejde-502	444	3	a	a	PRON
ejde-502	444	4	,	,	PUNCT
ejde-502	444	5	288	288	NUM
ejde-502	444	6	(	(	PUNCT
ejde-502	444	7	1979	1979	NUM
ejde-502	444	8	)	)	PUNCT
ejde-502	444	9	,	,	PUNCT
ejde-502	444	10	103	103	NUM
ejde-502	444	11	-	-	SYM
ejde-502	444	12	105	105	NUM
ejde-502	444	13	.	.	PUNCT
ejde-502	445	1	[	[	X
ejde-502	445	2	4	4	X
ejde-502	445	3	]	]	X
ejde-502	445	4	d.	d.	PROPN
ejde-502	445	5	g.	g.	PROPN
ejde-502	445	6	aronson	aronson	PROPN
ejde-502	445	7	,	,	PUNCT
ejde-502	445	8	l.	l.	PROPN
ejde-502	445	9	a.	a.	PROPN
ejde-502	445	10	caffarelli	caffarelli	PROPN
ejde-502	445	11	;	;	PUNCT
ejde-502	445	12	the	the	DET
ejde-502	445	13	initial	initial	ADJ
ejde-502	445	14	trace	trace	NOUN
ejde-502	445	15	of	of	ADP
ejde-502	445	16	a	a	DET
ejde-502	445	17	solution	solution	NOUN
ejde-502	445	18	of	of	ADP
ejde-502	445	19	the	the	DET
ejde-502	445	20	porous	porous	ADJ
ejde-502	445	21	medium	medium	NOUN
ejde-502	445	22	equation	equation	NOUN
ejde-502	445	23	,	,	PUNCT
ejde-502	445	24	trans	trans	PROPN
ejde-502	445	25	.	.	PROPN
ejde-502	445	26	amer	amer	PROPN
ejde-502	445	27	.	.	PUNCT
ejde-502	445	28	math	math	PROPN
ejde-502	445	29	.	.	PUNCT
ejde-502	446	1	soc	soc	PROPN
ejde-502	446	2	.	.	PROPN
ejde-502	446	3	,	,	PUNCT
ejde-502	446	4	280	280	NUM
ejde-502	446	5	(	(	PUNCT
ejde-502	446	6	1983	1983	NUM
ejde-502	446	7	)	)	PUNCT
ejde-502	446	8	,	,	PUNCT
ejde-502	446	9	no	no	INTJ
ejde-502	446	10	.	.	NOUN
ejde-502	446	11	1	1	NUM
ejde-502	446	12	,	,	PUNCT
ejde-502	446	13	351	351	NUM
ejde-502	446	14	-	-	SYM
ejde-502	446	15	366	366	NUM
ejde-502	446	16	.	.	PUNCT
ejde-502	447	1	[	[	X
ejde-502	447	2	5	5	NUM
ejde-502	447	3	]	]	PUNCT
ejde-502	447	4	c.	c.	NOUN
ejde-502	447	5	bandle	bandle	PROPN
ejde-502	447	6	,	,	PUNCT
ejde-502	447	7	h.	h.	PROPN
ejde-502	447	8	levine	levine	PROPN
ejde-502	447	9	;	;	PUNCT
ejde-502	447	10	on	on	ADP
ejde-502	447	11	the	the	DET
ejde-502	447	12	existence	existence	NOUN
ejde-502	447	13	and	and	CCONJ
ejde-502	447	14	nonexistence	nonexistence	NOUN
ejde-502	447	15	of	of	ADP
ejde-502	447	16	global	global	ADJ
ejde-502	447	17	solutions	solution	NOUN
ejde-502	447	18	of	of	ADP
ejde-502	447	19	reactiondiffusion	reactiondiffusion	NOUN
ejde-502	447	20	equations	equation	NOUN
ejde-502	447	21	in	in	ADP
ejde-502	447	22	sectorial	sectorial	ADJ
ejde-502	447	23	domains	domain	NOUN
ejde-502	447	24	,	,	PUNCT
ejde-502	447	25	trans	trans	PROPN
ejde-502	447	26	.	.	PROPN
ejde-502	448	1	amer	amer	PROPN
ejde-502	448	2	.	.	PUNCT
ejde-502	448	3	math	math	PROPN
ejde-502	448	4	.	.	PUNCT
ejde-502	449	1	soc	soc	PROPN
ejde-502	449	2	.	.	PUNCT
ejde-502	449	3	,	,	PUNCT
ejde-502	449	4	316	316	NUM
ejde-502	449	5	(	(	PUNCT
ejde-502	449	6	1989	1989	NUM
ejde-502	449	7	)	)	PUNCT
ejde-502	449	8	,	,	PUNCT
ejde-502	449	9	595	595	NUM
ejde-502	449	10	-	-	SYM
ejde-502	449	11	622	622	NUM
ejde-502	449	12	.	.	PUNCT
ejde-502	450	1	[	[	X
ejde-502	450	2	6	6	NUM
ejde-502	450	3	]	]	X
ejde-502	450	4	p.	p.	PROPN
ejde-502	450	5	baras	baras	PROPN
ejde-502	450	6	,	,	PUNCT
ejde-502	450	7	r.	r.	PROPN
ejde-502	450	8	kersner	kersner	PROPN
ejde-502	450	9	;	;	PUNCT
ejde-502	450	10	local	local	ADJ
ejde-502	450	11	and	and	CCONJ
ejde-502	450	12	global	global	ADJ
ejde-502	450	13	solvability	solvability	NOUN
ejde-502	450	14	of	of	ADP
ejde-502	450	15	a	a	DET
ejde-502	450	16	class	class	NOUN
ejde-502	450	17	of	of	ADP
ejde-502	450	18	semilinear	semilinear	PROPN
ejde-502	450	19	parabolic	parabolic	PROPN
ejde-502	450	20	equations	equation	NOUN
ejde-502	450	21	,	,	PUNCT
ejde-502	450	22	j.	j.	PROPN
ejde-502	450	23	differential	differential	PROPN
ejde-502	450	24	equations	equation	NOUN
ejde-502	450	25	,	,	PUNCT
ejde-502	450	26	68	68	NUM
ejde-502	450	27	(	(	PUNCT
ejde-502	450	28	1987	1987	NUM
ejde-502	450	29	)	)	PUNCT
ejde-502	450	30	,	,	PUNCT
ejde-502	450	31	238	238	NUM
ejde-502	450	32	-	-	SYM
ejde-502	450	33	252	252	NUM
ejde-502	450	34	.	.	PUNCT
ejde-502	451	1	[	[	X
ejde-502	451	2	7	7	X
ejde-502	451	3	]	]	X
ejde-502	451	4	p.	p.	PROPN
ejde-502	451	5	bénilan	bénilan	PROPN
ejde-502	451	6	,	,	PUNCT
ejde-502	451	7	m.	m.	NOUN
ejde-502	451	8	g.	g.	PROPN
ejde-502	451	9	crandall	crandall	PROPN
ejde-502	451	10	,	,	PUNCT
ejde-502	451	11	m.	m.	NOUN
ejde-502	451	12	pierre	pierre	NOUN
ejde-502	451	13	;	;	PUNCT
ejde-502	451	14	solutions	solution	NOUN
ejde-502	451	15	of	of	ADP
ejde-502	451	16	the	the	DET
ejde-502	451	17	porous	porous	ADJ
ejde-502	451	18	medium	medium	NOUN
ejde-502	451	19	in	in	ADP
ejde-502	451	20	rn	rn	PROPN
ejde-502	451	21	under	under	ADP
ejde-502	451	22	optimal	optimal	ADJ
ejde-502	451	23	conditions	condition	NOUN
ejde-502	451	24	on	on	ADP
ejde-502	451	25	the	the	DET
ejde-502	451	26	initial	initial	ADJ
ejde-502	451	27	values	value	NOUN
ejde-502	451	28	,	,	PUNCT
ejde-502	451	29	indiana	indiana	PROPN
ejde-502	451	30	univ	univ	PROPN
ejde-502	451	31	.	.	PUNCT
ejde-502	452	1	math	math	PROPN
ejde-502	452	2	.	.	PUNCT
ejde-502	453	1	jour	jour	PROPN
ejde-502	453	2	.	.	PROPN
ejde-502	453	3	,	,	PUNCT
ejde-502	453	4	33	33	NUM
ejde-502	453	5	(	(	PUNCT
ejde-502	453	6	1984	1984	NUM
ejde-502	453	7	)	)	PUNCT
ejde-502	453	8	,	,	PUNCT
ejde-502	453	9	no	no	INTJ
ejde-502	453	10	.	.	NOUN
ejde-502	453	11	1	1	NUM
ejde-502	453	12	,	,	PUNCT
ejde-502	453	13	51	51	NUM
ejde-502	453	14	-	-	SYM
ejde-502	453	15	87	87	NUM
ejde-502	453	16	.	.	PUNCT
ejde-502	454	1	[	[	X
ejde-502	454	2	8	8	NUM
ejde-502	454	3	]	]	X
ejde-502	454	4	l.	l.	PROPN
ejde-502	454	5	a.	a.	PROPN
ejde-502	454	6	caffarelli	caffarelli	PROPN
ejde-502	454	7	,	,	PUNCT
ejde-502	454	8	j.	j.	PROPN
ejde-502	454	9	l.	l.	PROPN
ejde-502	454	10	vázquez	vázquez	PROPN
ejde-502	454	11	,	,	PUNCT
ejde-502	454	12	n.	n.	PROPN
ejde-502	454	13	i.	i.	PROPN
ejde-502	454	14	wolanski	wolanski	PROPN
ejde-502	454	15	;	;	PUNCT
ejde-502	454	16	lipschitz	lipschitz	VERB
ejde-502	454	17	continuity	continuity	NOUN
ejde-502	454	18	of	of	ADP
ejde-502	454	19	solutions	solution	NOUN
ejde-502	454	20	and	and	CCONJ
ejde-502	454	21	interfaces	interface	NOUN
ejde-502	454	22	of	of	ADP
ejde-502	454	23	the	the	DET
ejde-502	454	24	n	n	ADV
ejde-502	454	25	-dimensional	-dimensional	ADJ
ejde-502	454	26	porous	porous	ADJ
ejde-502	454	27	medium	medium	NOUN
ejde-502	454	28	equation	equation	NOUN
ejde-502	454	29	,	,	PUNCT
ejde-502	454	30	indiana	indiana	PROPN
ejde-502	454	31	univ	univ	PROPN
ejde-502	454	32	.	.	PUNCT
ejde-502	455	1	math	math	PROPN
ejde-502	455	2	.	.	PUNCT
ejde-502	456	1	j.	j.	PROPN
ejde-502	456	2	,	,	PUNCT
ejde-502	456	3	36	36	NUM
ejde-502	456	4	(	(	PUNCT
ejde-502	456	5	1987	1987	NUM
ejde-502	456	6	)	)	PUNCT
ejde-502	456	7	,	,	PUNCT
ejde-502	456	8	no	no	INTJ
ejde-502	456	9	.	.	NOUN
ejde-502	456	10	2	2	NUM
ejde-502	456	11	,	,	PUNCT
ejde-502	456	12	373	373	NUM
ejde-502	456	13	-	-	SYM
ejde-502	456	14	401	401	NUM
ejde-502	456	15	.	.	PUNCT
ejde-502	457	1	[	[	X
ejde-502	457	2	9	9	NUM
ejde-502	457	3	]	]	X
ejde-502	457	4	e.	e.	PROPN
ejde-502	457	5	dibenedetto	dibenedetto	PROPN
ejde-502	457	6	;	;	PUNCT
ejde-502	457	7	continuity	continuity	NOUN
ejde-502	457	8	of	of	ADP
ejde-502	457	9	weak	weak	ADJ
ejde-502	457	10	solutions	solution	NOUN
ejde-502	457	11	to	to	ADP
ejde-502	457	12	a	a	DET
ejde-502	457	13	general	general	ADJ
ejde-502	457	14	porous	porous	ADJ
ejde-502	457	15	medium	medium	NOUN
ejde-502	457	16	equation	equation	NOUN
ejde-502	457	17	,	,	PUNCT
ejde-502	457	18	indiana	indiana	PROPN
ejde-502	457	19	univ	univ	PROPN
ejde-502	457	20	.	.	PUNCT
ejde-502	458	1	math	math	PROPN
ejde-502	458	2	.	.	PUNCT
ejde-502	459	1	j.	j.	PROPN
ejde-502	459	2	,	,	PUNCT
ejde-502	459	3	32	32	NUM
ejde-502	459	4	(	(	PUNCT
ejde-502	459	5	1983	1983	NUM
ejde-502	459	6	)	)	PUNCT
ejde-502	459	7	,	,	PUNCT
ejde-502	459	8	no	no	INTJ
ejde-502	459	9	.	.	NOUN
ejde-502	459	10	1	1	NUM
ejde-502	459	11	,	,	PUNCT
ejde-502	459	12	83	83	NUM
ejde-502	459	13	-	-	SYM
ejde-502	459	14	118	118	NUM
ejde-502	459	15	.	.	NOUN
ejde-502	459	16	20	20	NUM
ejde-502	459	17	r.	r.	PROPN
ejde-502	459	18	g.	g.	PROPN
ejde-502	459	19	iagar	iagar	PROPN
ejde-502	459	20	,	,	PUNCT
ejde-502	459	21	a.	a.	NOUN
ejde-502	459	22	i.	i.	PROPN
ejde-502	459	23	muñoz	muñoz	PROPN
ejde-502	459	24	,	,	PUNCT
ejde-502	459	25	a.	a.	NOUN
ejde-502	459	26	sánchez	sánchez	PROPN
ejde-502	459	27	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	459	28	[	[	X
ejde-502	459	29	10	10	NUM
ejde-502	459	30	]	]	PUNCT
ejde-502	459	31	s.	s.	PROPN
ejde-502	459	32	filippas	filippas	PROPN
ejde-502	459	33	,	,	PUNCT
ejde-502	459	34	a.	a.	NOUN
ejde-502	459	35	tertikas	tertikas	ADV
ejde-502	459	36	;	;	PUNCT
ejde-502	459	37	on	on	ADP
ejde-502	459	38	similarity	similarity	NOUN
ejde-502	459	39	solutions	solution	NOUN
ejde-502	459	40	of	of	ADP
ejde-502	459	41	a	a	DET
ejde-502	459	42	heat	heat	NOUN
ejde-502	459	43	equation	equation	NOUN
ejde-502	459	44	with	with	ADP
ejde-502	459	45	a	a	DET
ejde-502	459	46	nonhomogeneous	nonhomogeneous	ADJ
ejde-502	459	47	nonlinearity	nonlinearity	NOUN
ejde-502	459	48	,	,	PUNCT
ejde-502	459	49	j.	j.	PROPN
ejde-502	459	50	differential	differential	PROPN
ejde-502	459	51	equations	equations	PROPN
ejde-502	459	52	,	,	PUNCT
ejde-502	459	53	165	165	NUM
ejde-502	459	54	(	(	PUNCT
ejde-502	459	55	2000	2000	NUM
ejde-502	459	56	)	)	PUNCT
ejde-502	459	57	,	,	PUNCT
ejde-502	459	58	no	no	INTJ
ejde-502	459	59	.	.	NOUN
ejde-502	459	60	2	2	NUM
ejde-502	459	61	,	,	PUNCT
ejde-502	459	62	468	468	NUM
ejde-502	459	63	-	-	SYM
ejde-502	459	64	492	492	NUM
ejde-502	459	65	.	.	PUNCT
ejde-502	460	1	[	[	X
ejde-502	460	2	11	11	NUM
ejde-502	460	3	]	]	X
ejde-502	460	4	j.-s	j.-	NOUN
ejde-502	460	5	.	.	PUNCT
ejde-502	461	1	guo	guo	PROPN
ejde-502	461	2	,	,	PUNCT
ejde-502	461	3	c.-s	c.-	NOUN
ejde-502	461	4	.	.	PUNCT
ejde-502	462	1	lin	lin	PROPN
ejde-502	462	2	,	,	PUNCT
ejde-502	462	3	m.	m.	NOUN
ejde-502	462	4	shimojo	shimojo	PROPN
ejde-502	462	5	;	;	PUNCT
ejde-502	462	6	blow	blow	NOUN
ejde-502	462	7	-	-	PUNCT
ejde-502	462	8	up	up	ADP
ejde-502	462	9	behavior	behavior	NOUN
ejde-502	462	10	for	for	ADP
ejde-502	462	11	a	a	DET
ejde-502	462	12	parabolic	parabolic	ADJ
ejde-502	462	13	equation	equation	NOUN
ejde-502	462	14	with	with	ADP
ejde-502	462	15	spatially	spatially	ADV
ejde-502	462	16	dependent	dependent	ADJ
ejde-502	462	17	coefficient	coefficient	NOUN
ejde-502	462	18	,	,	PUNCT
ejde-502	462	19	dynam	dynam	PROPN
ejde-502	462	20	.	.	PUNCT
ejde-502	463	1	systems	system	NOUN
ejde-502	463	2	appl	appl	PROPN
ejde-502	463	3	.	.	PROPN
ejde-502	463	4	,	,	PUNCT
ejde-502	463	5	19	19	NUM
ejde-502	463	6	(	(	PUNCT
ejde-502	463	7	2010	2010	NUM
ejde-502	463	8	)	)	PUNCT
ejde-502	463	9	,	,	PUNCT
ejde-502	464	1	no	no	INTJ
ejde-502	464	2	.	.	NOUN
ejde-502	465	1	3	3	NUM
ejde-502	465	2	-	-	SYM
ejde-502	465	3	4	4	NUM
ejde-502	465	4	,	,	PUNCT
ejde-502	465	5	415	415	NUM
ejde-502	465	6	-	-	SYM
ejde-502	465	7	433	433	NUM
ejde-502	465	8	.	.	PUNCT
ejde-502	466	1	[	[	X
ejde-502	466	2	12	12	NUM
ejde-502	466	3	]	]	X
ejde-502	466	4	j.-s	j.-	NOUN
ejde-502	466	5	.	.	PUNCT
ejde-502	467	1	guo	guo	PROPN
ejde-502	467	2	,	,	PUNCT
ejde-502	467	3	c.-s	c.-	NOUN
ejde-502	467	4	.	.	PUNCT
ejde-502	468	1	lin	lin	PROPN
ejde-502	468	2	,	,	PUNCT
ejde-502	468	3	m.	m.	NOUN
ejde-502	468	4	shimojo	shimojo	PROPN
ejde-502	468	5	;	;	PUNCT
ejde-502	468	6	blow	blow	NOUN
ejde-502	468	7	-	-	PUNCT
ejde-502	468	8	up	up	NOUN
ejde-502	468	9	for	for	ADP
ejde-502	468	10	a	a	DET
ejde-502	468	11	reaction	reaction	NOUN
ejde-502	468	12	-	-	PUNCT
ejde-502	468	13	diffusion	diffusion	NOUN
ejde-502	468	14	equation	equation	NOUN
ejde-502	468	15	with	with	ADP
ejde-502	468	16	variable	variable	ADJ
ejde-502	468	17	coefficient	coefficient	NOUN
ejde-502	468	18	,	,	PUNCT
ejde-502	468	19	appl	appl	PROPN
ejde-502	468	20	.	.	PROPN
ejde-502	468	21	math	math	PROPN
ejde-502	468	22	.	.	PUNCT
ejde-502	469	1	lett	lett	PROPN
ejde-502	469	2	.	.	PROPN
ejde-502	470	1	,	,	PUNCT
ejde-502	470	2	26	26	NUM
ejde-502	470	3	(	(	PUNCT
ejde-502	470	4	2013	2013	NUM
ejde-502	470	5	)	)	PUNCT
ejde-502	470	6	,	,	PUNCT
ejde-502	471	1	no	no	INTJ
ejde-502	471	2	.	.	NOUN
ejde-502	471	3	1	1	NUM
ejde-502	471	4	,	,	PUNCT
ejde-502	471	5	150	150	NUM
ejde-502	471	6	-	-	SYM
ejde-502	471	7	153	153	NUM
ejde-502	471	8	.	.	PUNCT
ejde-502	472	1	[	[	X
ejde-502	472	2	13	13	NUM
ejde-502	472	3	]	]	X
ejde-502	472	4	j.-s	j.-	NOUN
ejde-502	472	5	.	.	PUNCT
ejde-502	473	1	guo	guo	PROPN
ejde-502	473	2	,	,	PUNCT
ejde-502	473	3	m.	m.	NOUN
ejde-502	473	4	shimojo	shimojo	NOUN
ejde-502	473	5	;	;	PUNCT
ejde-502	473	6	blowing	blow	VERB
ejde-502	473	7	up	up	ADP
ejde-502	473	8	at	at	ADP
ejde-502	473	9	zero	zero	NUM
ejde-502	473	10	points	point	NOUN
ejde-502	473	11	of	of	ADP
ejde-502	473	12	potential	potential	NOUN
ejde-502	473	13	for	for	ADP
ejde-502	473	14	an	an	DET
ejde-502	473	15	initial	initial	ADJ
ejde-502	473	16	boundary	boundary	ADJ
ejde-502	473	17	value	value	NOUN
ejde-502	473	18	problem	problem	NOUN
ejde-502	473	19	,	,	PUNCT
ejde-502	473	20	commun	commun	PROPN
ejde-502	473	21	.	.	PUNCT
ejde-502	474	1	pure	pure	ADJ
ejde-502	474	2	appl	appl	PROPN
ejde-502	474	3	.	.	PUNCT
ejde-502	475	1	anal	anal	PROPN
ejde-502	475	2	.	.	PROPN
ejde-502	475	3	,	,	PUNCT
ejde-502	475	4	10	10	NUM
ejde-502	475	5	(	(	PUNCT
ejde-502	475	6	2011	2011	NUM
ejde-502	475	7	)	)	PUNCT
ejde-502	475	8	,	,	PUNCT
ejde-502	476	1	no	no	INTJ
ejde-502	476	2	.	.	NOUN
ejde-502	476	3	1	1	NUM
ejde-502	476	4	,	,	PUNCT
ejde-502	476	5	161	161	NUM
ejde-502	476	6	-	-	SYM
ejde-502	476	7	177	177	NUM
ejde-502	476	8	.	.	PUNCT
ejde-502	477	1	[	[	X
ejde-502	477	2	14	14	NUM
ejde-502	477	3	]	]	X
ejde-502	477	4	j.-s	j.-	NOUN
ejde-502	477	5	.	.	PUNCT
ejde-502	478	1	guo	guo	PROPN
ejde-502	478	2	,	,	PUNCT
ejde-502	478	3	p.	p.	NOUN
ejde-502	478	4	souplet	souplet	NOUN
ejde-502	478	5	;	;	PUNCT
ejde-502	478	6	excluding	exclude	VERB
ejde-502	478	7	blowup	blowup	NOUN
ejde-502	478	8	at	at	ADP
ejde-502	478	9	zero	zero	NUM
ejde-502	478	10	points	point	NOUN
ejde-502	478	11	of	of	ADP
ejde-502	478	12	the	the	DET
ejde-502	478	13	potential	potential	NOUN
ejde-502	478	14	by	by	ADP
ejde-502	478	15	means	mean	NOUN
ejde-502	478	16	of	of	ADP
ejde-502	478	17	liouvilletype	liouvilletype	NOUN
ejde-502	478	18	theorems	theorem	NOUN
ejde-502	478	19	,	,	PUNCT
ejde-502	478	20	j.	j.	PROPN
ejde-502	478	21	differential	differential	PROPN
ejde-502	478	22	equations	equation	NOUN
ejde-502	478	23	,	,	PUNCT
ejde-502	478	24	265	265	NUM
ejde-502	478	25	(	(	PUNCT
ejde-502	478	26	2018	2018	NUM
ejde-502	478	27	)	)	PUNCT
ejde-502	478	28	,	,	PUNCT
ejde-502	478	29	no	no	INTJ
ejde-502	478	30	.	.	NOUN
ejde-502	478	31	10	10	NUM
ejde-502	478	32	,	,	PUNCT
ejde-502	478	33	4942	4942	NUM
ejde-502	478	34	-	-	SYM
ejde-502	478	35	4964	4964	NUM
ejde-502	478	36	.	.	PUNCT
ejde-502	479	1	[	[	X
ejde-502	479	2	15	15	NUM
ejde-502	479	3	]	]	X
ejde-502	479	4	r.	r.	PROPN
ejde-502	479	5	g.	g.	PROPN
ejde-502	479	6	iagar	iagar	PROPN
ejde-502	479	7	,	,	PUNCT
ejde-502	479	8	m.	m.	NOUN
ejde-502	479	9	latorre	latorre	NOUN
ejde-502	479	10	,	,	PUNCT
ejde-502	479	11	a.	a.	NOUN
ejde-502	479	12	sánchez	sánchez	PROPN
ejde-502	479	13	;	;	PUNCT
ejde-502	479	14	blow	blow	NOUN
ejde-502	479	15	-	-	PUNCT
ejde-502	479	16	up	up	ADP
ejde-502	479	17	patterns	pattern	NOUN
ejde-502	479	18	for	for	ADP
ejde-502	479	19	a	a	DET
ejde-502	479	20	reaction	reaction	NOUN
ejde-502	479	21	-	-	PUNCT
ejde-502	479	22	diffusion	diffusion	NOUN
ejde-502	479	23	equation	equation	NOUN
ejde-502	479	24	with	with	ADP
ejde-502	479	25	weighted	weighted	ADJ
ejde-502	479	26	reaction	reaction	NOUN
ejde-502	479	27	in	in	ADP
ejde-502	479	28	general	general	ADJ
ejde-502	479	29	dimension	dimension	NOUN
ejde-502	479	30	,	,	PUNCT
ejde-502	479	31	adv	adv	PROPN
ejde-502	479	32	.	.	PUNCT
ejde-502	479	33	differential	differential	PROPN
ejde-502	479	34	equations	equation	NOUN
ejde-502	479	35	(	(	PUNCT
ejde-502	479	36	accepted	accept	VERB
ejde-502	479	37	november	november	PROPN
ejde-502	479	38	2022	2022	NUM
ejde-502	479	39	)	)	PUNCT
ejde-502	479	40	,	,	PUNCT
ejde-502	479	41	preprint	preprint	VERB
ejde-502	479	42	arxiv	arxiv	PROPN
ejde-502	479	43	no	no	INTJ
ejde-502	479	44	.	.	PUNCT
ejde-502	480	1	2205.09407	2205.09407	X
ejde-502	480	2	.	.	PUNCT
ejde-502	481	1	[	[	X
ejde-502	481	2	16	16	NUM
ejde-502	481	3	]	]	PUNCT
ejde-502	481	4	r.	r.	PROPN
ejde-502	481	5	g.	g.	PROPN
ejde-502	481	6	iagar	iagar	PROPN
ejde-502	481	7	,	,	PUNCT
ejde-502	481	8	a.	a.	NOUN
ejde-502	481	9	i.	i.	PROPN
ejde-502	481	10	muñoz	muñoz	PROPN
ejde-502	481	11	,	,	PUNCT
ejde-502	481	12	a.	a.	NOUN
ejde-502	481	13	sánchez	sánchez	PROPN
ejde-502	481	14	;	;	PUNCT
ejde-502	481	15	self	self	NOUN
ejde-502	481	16	-	-	PUNCT
ejde-502	481	17	similar	similar	ADJ
ejde-502	481	18	blow	blow	NOUN
ejde-502	481	19	-	-	PUNCT
ejde-502	481	20	up	up	ADP
ejde-502	481	21	patterns	pattern	NOUN
ejde-502	481	22	for	for	ADP
ejde-502	481	23	a	a	DET
ejde-502	481	24	reaction	reaction	NOUN
ejde-502	481	25	-	-	PUNCT
ejde-502	481	26	diffusion	diffusion	NOUN
ejde-502	481	27	equation	equation	NOUN
ejde-502	481	28	with	with	ADP
ejde-502	481	29	weighted	weighted	ADJ
ejde-502	481	30	reaction	reaction	NOUN
ejde-502	481	31	in	in	ADP
ejde-502	481	32	general	general	ADJ
ejde-502	481	33	dimension	dimension	NOUN
ejde-502	481	34	,	,	PUNCT
ejde-502	481	35	commun	commun	PROPN
ejde-502	481	36	.	.	PUNCT
ejde-502	482	1	pure	pure	ADJ
ejde-502	482	2	appl	appl	PROPN
ejde-502	482	3	.	.	PUNCT
ejde-502	483	1	analysis	analysis	NOUN
ejde-502	483	2	,	,	PUNCT
ejde-502	483	3	21	21	NUM
ejde-502	483	4	(	(	PUNCT
ejde-502	483	5	2022	2022	NUM
ejde-502	483	6	)	)	PUNCT
ejde-502	483	7	,	,	PUNCT
ejde-502	483	8	no	no	INTJ
ejde-502	483	9	.	.	NOUN
ejde-502	483	10	3	3	NUM
ejde-502	483	11	,	,	PUNCT
ejde-502	483	12	891	891	NUM
ejde-502	483	13	-	-	SYM
ejde-502	483	14	925	925	NUM
ejde-502	483	15	.	.	PUNCT
ejde-502	484	1	[	[	X
ejde-502	484	2	17	17	NUM
ejde-502	484	3	]	]	PUNCT
ejde-502	484	4	r.	r.	PROPN
ejde-502	484	5	g.	g.	PROPN
ejde-502	484	6	iagar	iagar	PROPN
ejde-502	484	7	,	,	PUNCT
ejde-502	484	8	a.	a.	NOUN
ejde-502	484	9	sánchez	sánchez	PROPN
ejde-502	484	10	;	;	PUNCT
ejde-502	484	11	blow	blow	VERB
ejde-502	484	12	up	up	ADP
ejde-502	484	13	profiles	profile	NOUN
ejde-502	484	14	for	for	ADP
ejde-502	484	15	a	a	DET
ejde-502	484	16	quasilinear	quasilinear	NOUN
ejde-502	484	17	reaction	reaction	NOUN
ejde-502	484	18	-	-	PUNCT
ejde-502	484	19	diffusion	diffusion	NOUN
ejde-502	484	20	equation	equation	NOUN
ejde-502	484	21	with	with	ADP
ejde-502	484	22	weighted	weighted	ADJ
ejde-502	484	23	reaction	reaction	NOUN
ejde-502	484	24	with	with	ADP
ejde-502	484	25	linear	linear	ADJ
ejde-502	484	26	growth	growth	NOUN
ejde-502	484	27	,	,	PUNCT
ejde-502	484	28	j.	j.	PROPN
ejde-502	484	29	dynam	dynam	PROPN
ejde-502	484	30	.	.	PUNCT
ejde-502	485	1	differential	differential	ADJ
ejde-502	485	2	equations	equation	NOUN
ejde-502	485	3	,	,	PUNCT
ejde-502	485	4	31	31	NUM
ejde-502	485	5	(	(	PUNCT
ejde-502	485	6	2019	2019	NUM
ejde-502	485	7	)	)	PUNCT
ejde-502	485	8	,	,	PUNCT
ejde-502	485	9	no	no	INTJ
ejde-502	485	10	.	.	NOUN
ejde-502	485	11	4	4	NUM
ejde-502	485	12	,	,	PUNCT
ejde-502	485	13	2061	2061	NUM
ejde-502	485	14	-	-	SYM
ejde-502	485	15	2094	2094	NUM
ejde-502	485	16	.	.	PUNCT
ejde-502	486	1	[	[	X
ejde-502	486	2	18	18	NUM
ejde-502	486	3	]	]	PUNCT
ejde-502	486	4	r.	r.	PROPN
ejde-502	486	5	g.	g.	PROPN
ejde-502	486	6	iagar	iagar	PROPN
ejde-502	486	7	,	,	PUNCT
ejde-502	486	8	a.	a.	NOUN
ejde-502	486	9	sánchez	sánchez	PROPN
ejde-502	486	10	;	;	PUNCT
ejde-502	486	11	blow	blow	VERB
ejde-502	486	12	up	up	ADP
ejde-502	486	13	profiles	profile	NOUN
ejde-502	486	14	for	for	ADP
ejde-502	486	15	a	a	DET
ejde-502	486	16	reaction	reaction	NOUN
ejde-502	486	17	-	-	PUNCT
ejde-502	486	18	diffusion	diffusion	NOUN
ejde-502	486	19	equation	equation	NOUN
ejde-502	486	20	with	with	ADP
ejde-502	486	21	critical	critical	ADJ
ejde-502	486	22	weighted	weight	VERB
ejde-502	486	23	reaction	reaction	NOUN
ejde-502	486	24	,	,	PUNCT
ejde-502	486	25	nonlinear	nonlinear	ADJ
ejde-502	486	26	anal	anal	NOUN
ejde-502	486	27	.	.	PUNCT
ejde-502	486	28	,	,	PUNCT
ejde-502	486	29	191	191	NUM
ejde-502	486	30	(	(	PUNCT
ejde-502	486	31	2020	2020	NUM
ejde-502	486	32	)	)	PUNCT
ejde-502	486	33	,	,	PUNCT
ejde-502	486	34	paper	paper	NOUN
ejde-502	486	35	no	no	NOUN
ejde-502	486	36	.	.	PROPN
ejde-502	486	37	111628	111628	NUM
ejde-502	486	38	,	,	PUNCT
ejde-502	486	39	24	24	NUM
ejde-502	486	40	pages	page	NOUN
ejde-502	486	41	.	.	PUNCT
ejde-502	487	1	[	[	X
ejde-502	487	2	19	19	NUM
ejde-502	487	3	]	]	PUNCT
ejde-502	487	4	r.	r.	PROPN
ejde-502	487	5	g.	g.	PROPN
ejde-502	487	6	iagar	iagar	PROPN
ejde-502	487	7	,	,	PUNCT
ejde-502	487	8	a.	a.	NOUN
ejde-502	487	9	sánchez	sánchez	PROPN
ejde-502	487	10	;	;	PUNCT
ejde-502	487	11	self	self	NOUN
ejde-502	487	12	-	-	PUNCT
ejde-502	487	13	similar	similar	ADJ
ejde-502	487	14	blow	blow	NOUN
ejde-502	487	15	-	-	PUNCT
ejde-502	487	16	up	up	ADP
ejde-502	487	17	profiles	profile	NOUN
ejde-502	487	18	for	for	ADP
ejde-502	487	19	a	a	DET
ejde-502	487	20	reaction	reaction	NOUN
ejde-502	487	21	-	-	PUNCT
ejde-502	487	22	diffusion	diffusion	NOUN
ejde-502	487	23	equation	equation	NOUN
ejde-502	487	24	with	with	ADP
ejde-502	487	25	strong	strong	ADJ
ejde-502	487	26	weighted	weighted	ADJ
ejde-502	487	27	reaction	reaction	NOUN
ejde-502	487	28	,	,	PUNCT
ejde-502	487	29	adv	adv	PROPN
ejde-502	487	30	.	.	PUNCT
ejde-502	487	31	nonl	nonl	PROPN
ejde-502	487	32	.	.	PUNCT
ejde-502	488	1	studies	study	NOUN
ejde-502	488	2	,	,	PUNCT
ejde-502	488	3	20	20	NUM
ejde-502	488	4	(	(	PUNCT
ejde-502	488	5	2020	2020	NUM
ejde-502	488	6	)	)	PUNCT
ejde-502	488	7	,	,	PUNCT
ejde-502	488	8	no	no	INTJ
ejde-502	488	9	.	.	NOUN
ejde-502	488	10	4	4	NUM
ejde-502	488	11	,	,	PUNCT
ejde-502	488	12	867	867	NUM
ejde-502	488	13	-	-	SYM
ejde-502	488	14	894	894	NUM
ejde-502	488	15	.	.	PUNCT
ejde-502	489	1	[	[	X
ejde-502	489	2	20	20	NUM
ejde-502	489	3	]	]	PUNCT
ejde-502	489	4	r.	r.	PROPN
ejde-502	489	5	g.	g.	PROPN
ejde-502	489	6	iagar	iagar	PROPN
ejde-502	489	7	,	,	PUNCT
ejde-502	489	8	a.	a.	NOUN
ejde-502	489	9	sánchez	sánchez	PROPN
ejde-502	489	10	;	;	PUNCT
ejde-502	489	11	blow	blow	VERB
ejde-502	489	12	up	up	ADP
ejde-502	489	13	profiles	profile	NOUN
ejde-502	489	14	for	for	ADP
ejde-502	489	15	a	a	DET
ejde-502	489	16	quasilinear	quasilinear	NOUN
ejde-502	489	17	reaction	reaction	NOUN
ejde-502	489	18	-	-	PUNCT
ejde-502	489	19	diffusion	diffusion	NOUN
ejde-502	489	20	equation	equation	NOUN
ejde-502	489	21	with	with	ADP
ejde-502	489	22	weighted	weight	VERB
ejde-502	489	23	reaction	reaction	NOUN
ejde-502	489	24	,	,	PUNCT
ejde-502	489	25	j.	j.	PROPN
ejde-502	489	26	differential	differential	PROPN
ejde-502	489	27	equations	equations	PROPN
ejde-502	489	28	,	,	PUNCT
ejde-502	489	29	272	272	NUM
ejde-502	489	30	(	(	PUNCT
ejde-502	489	31	2021	2021	NUM
ejde-502	489	32	)	)	PUNCT
ejde-502	489	33	,	,	PUNCT
ejde-502	489	34	560	560	NUM
ejde-502	489	35	-	-	SYM
ejde-502	489	36	605	605	NUM
ejde-502	489	37	.	.	PUNCT
ejde-502	490	1	[	[	X
ejde-502	490	2	21	21	NUM
ejde-502	490	3	]	]	X
ejde-502	490	4	r.	r.	PROPN
ejde-502	490	5	g.	g.	PROPN
ejde-502	490	6	iagar	iagar	PROPN
ejde-502	490	7	,	,	PUNCT
ejde-502	490	8	a.	a.	NOUN
ejde-502	490	9	sánchez	sánchez	PROPN
ejde-502	490	10	;	;	PUNCT
ejde-502	490	11	self	self	NOUN
ejde-502	490	12	-	-	PUNCT
ejde-502	490	13	similar	similar	ADJ
ejde-502	490	14	blow	blow	NOUN
ejde-502	490	15	-	-	PUNCT
ejde-502	490	16	up	up	ADP
ejde-502	490	17	profiles	profile	NOUN
ejde-502	490	18	for	for	ADP
ejde-502	490	19	a	a	DET
ejde-502	490	20	reaction	reaction	NOUN
ejde-502	490	21	-	-	PUNCT
ejde-502	490	22	diffusion	diffusion	NOUN
ejde-502	490	23	equation	equation	NOUN
ejde-502	490	24	with	with	ADP
ejde-502	490	25	critically	critically	ADV
ejde-502	490	26	strong	strong	ADJ
ejde-502	490	27	weighted	weighted	ADJ
ejde-502	490	28	reaction	reaction	NOUN
ejde-502	490	29	,	,	PUNCT
ejde-502	490	30	j.	j.	PROPN
ejde-502	490	31	dynam	dynam	PROPN
ejde-502	490	32	.	.	PUNCT
ejde-502	491	1	differential	differential	ADJ
ejde-502	491	2	equations	equation	NOUN
ejde-502	491	3	,	,	PUNCT
ejde-502	491	4	34	34	NUM
ejde-502	491	5	(	(	PUNCT
ejde-502	491	6	2022	2022	NUM
ejde-502	491	7	)	)	PUNCT
ejde-502	491	8	,	,	PUNCT
ejde-502	491	9	no	no	INTJ
ejde-502	491	10	.	.	NOUN
ejde-502	491	11	2	2	NUM
ejde-502	491	12	,	,	PUNCT
ejde-502	491	13	11391172	11391172	NUM
ejde-502	491	14	.	.	PUNCT
ejde-502	492	1	[	[	X
ejde-502	492	2	22	22	NUM
ejde-502	492	3	]	]	X
ejde-502	492	4	r.	r.	PROPN
ejde-502	492	5	iagar	iagar	PROPN
ejde-502	492	6	,	,	PUNCT
ejde-502	492	7	a.	a.	NOUN
ejde-502	492	8	sánchez	sánchez	PROPN
ejde-502	492	9	;	;	PUNCT
ejde-502	492	10	separate	separate	ADJ
ejde-502	492	11	variable	variable	ADJ
ejde-502	492	12	blow	blow	NOUN
ejde-502	492	13	-	-	PUNCT
ejde-502	492	14	up	up	ADP
ejde-502	492	15	patterns	pattern	NOUN
ejde-502	492	16	for	for	ADP
ejde-502	492	17	a	a	DET
ejde-502	492	18	reaction	reaction	NOUN
ejde-502	492	19	-	-	PUNCT
ejde-502	492	20	diffusion	diffusion	NOUN
ejde-502	492	21	equation	equation	NOUN
ejde-502	492	22	with	with	ADP
ejde-502	492	23	critical	critical	ADJ
ejde-502	492	24	weighted	weight	VERB
ejde-502	492	25	reaction	reaction	NOUN
ejde-502	492	26	,	,	PUNCT
ejde-502	492	27	nonlinear	nonlinear	ADJ
ejde-502	492	28	anal	anal	NOUN
ejde-502	492	29	.	.	PUNCT
ejde-502	492	30	,	,	PUNCT
ejde-502	492	31	217	217	NUM
ejde-502	492	32	(	(	PUNCT
ejde-502	492	33	2022	2022	NUM
ejde-502	492	34	)	)	PUNCT
ejde-502	492	35	,	,	PUNCT
ejde-502	492	36	article	article	NOUN
ejde-502	492	37	i	i	PROPN
ejde-502	492	38	d	d	PROPN
ejde-502	492	39	112740	112740	NUM
ejde-502	492	40	,	,	PUNCT
ejde-502	492	41	33	33	NUM
ejde-502	492	42	pages	page	NOUN
ejde-502	492	43	.	.	PUNCT
ejde-502	493	1	[	[	X
ejde-502	493	2	23	23	NUM
ejde-502	493	3	]	]	X
ejde-502	493	4	o.	o.	PROPN
ejde-502	493	5	a.	a.	PROPN
ejde-502	493	6	ladyzhenskaya	ladyzhenskaya	PROPN
ejde-502	493	7	,	,	PUNCT
ejde-502	493	8	m.	m.	NOUN
ejde-502	493	9	a.	a.	NOUN
ejde-502	493	10	solonnikov	solonnikov	PROPN
ejde-502	493	11	,	,	PUNCT
ejde-502	493	12	n.	n.	PROPN
ejde-502	493	13	n.	n.	PROPN
ejde-502	493	14	uraltseva	uraltseva	PROPN
ejde-502	493	15	;	;	PUNCT
ejde-502	493	16	linear	linear	ADJ
ejde-502	493	17	and	and	CCONJ
ejde-502	493	18	quasilinear	quasilinear	ADJ
ejde-502	493	19	equations	equation	NOUN
ejde-502	493	20	of	of	ADP
ejde-502	493	21	parabolic	parabolic	ADJ
ejde-502	493	22	type	type	NOUN
ejde-502	493	23	,	,	PUNCT
ejde-502	493	24	transl	transl	PROPN
ejde-502	493	25	.	.	PUNCT
ejde-502	494	1	math	math	NOUN
ejde-502	494	2	.	.	PUNCT
ejde-502	495	1	monographs	monograph	NOUN
ejde-502	495	2	,	,	PUNCT
ejde-502	495	3	23	23	NUM
ejde-502	495	4	,	,	PUNCT
ejde-502	495	5	amer	amer	PROPN
ejde-502	495	6	.	.	PUNCT
ejde-502	495	7	math	math	PROPN
ejde-502	495	8	.	.	PUNCT
ejde-502	496	1	society	society	NOUN
ejde-502	496	2	,	,	PUNCT
ejde-502	496	3	providence	providence	NOUN
ejde-502	496	4	,	,	PUNCT
ejde-502	496	5	1968	1968	NUM
ejde-502	496	6	.	.	PUNCT
ejde-502	497	1	[	[	X
ejde-502	497	2	24	24	NUM
ejde-502	497	3	]	]	PUNCT
ejde-502	497	4	a.	a.	NOUN
ejde-502	497	5	mukai	mukai	PROPN
ejde-502	497	6	,	,	PUNCT
ejde-502	497	7	y.	y.	PROPN
ejde-502	497	8	seki	seki	PROPN
ejde-502	497	9	;	;	PUNCT
ejde-502	497	10	refined	refined	ADJ
ejde-502	497	11	construction	construction	NOUN
ejde-502	497	12	of	of	ADP
ejde-502	497	13	type	type	NOUN
ejde-502	497	14	ii	ii	PROPN
ejde-502	497	15	blow	blow	NOUN
ejde-502	497	16	-	-	PUNCT
ejde-502	497	17	up	up	ADP
ejde-502	497	18	solutions	solution	NOUN
ejde-502	497	19	for	for	ADP
ejde-502	497	20	semilinear	semilinear	ADJ
ejde-502	497	21	heat	heat	NOUN
ejde-502	497	22	equations	equation	NOUN
ejde-502	497	23	with	with	ADP
ejde-502	497	24	joseph	joseph	PROPN
ejde-502	497	25	-	-	PUNCT
ejde-502	497	26	lundgren	lundgren	PROPN
ejde-502	497	27	supercritical	supercritical	ADJ
ejde-502	497	28	nonlinearity	nonlinearity	NOUN
ejde-502	497	29	,	,	PUNCT
ejde-502	497	30	discrete	discrete	ADJ
ejde-502	497	31	cont	cont	NOUN
ejde-502	497	32	.	.	PUNCT
ejde-502	498	1	dynamical	dynamical	ADJ
ejde-502	498	2	systems	system	NOUN
ejde-502	498	3	,	,	PUNCT
ejde-502	498	4	41	41	NUM
ejde-502	498	5	(	(	PUNCT
ejde-502	498	6	2021	2021	NUM
ejde-502	498	7	)	)	PUNCT
ejde-502	498	8	,	,	PUNCT
ejde-502	498	9	no	no	INTJ
ejde-502	498	10	.	.	NOUN
ejde-502	498	11	10	10	NUM
ejde-502	498	12	,	,	PUNCT
ejde-502	498	13	4847	4847	NUM
ejde-502	498	14	-	-	SYM
ejde-502	498	15	4885	4885	NUM
ejde-502	498	16	.	.	PUNCT
ejde-502	499	1	[	[	X
ejde-502	499	2	25	25	NUM
ejde-502	499	3	]	]	PUNCT
ejde-502	499	4	a.	a.	NOUN
ejde-502	499	5	de	de	X
ejde-502	499	6	pablo	pablo	PROPN
ejde-502	499	7	;	;	PUNCT
ejde-502	499	8	large	large	ADJ
ejde-502	499	9	-	-	PUNCT
ejde-502	499	10	time	time	NOUN
ejde-502	499	11	behaviour	behaviour	NOUN
ejde-502	499	12	of	of	ADP
ejde-502	499	13	solutions	solution	NOUN
ejde-502	499	14	of	of	ADP
ejde-502	499	15	a	a	DET
ejde-502	499	16	reaction	reaction	NOUN
ejde-502	499	17	-	-	PUNCT
ejde-502	499	18	diffusion	diffusion	NOUN
ejde-502	499	19	equation	equation	NOUN
ejde-502	499	20	,	,	PUNCT
ejde-502	499	21	proc	proc	NOUN
ejde-502	499	22	.	.	PUNCT
ejde-502	500	1	roy	roy	PROPN
ejde-502	500	2	.	.	PROPN
ejde-502	500	3	soc	soc	PROPN
ejde-502	500	4	.	.	PUNCT
ejde-502	501	1	edinburgh	edinburgh	PROPN
ejde-502	501	2	sect	sect	PROPN
ejde-502	501	3	.	.	PUNCT
ejde-502	502	1	a	a	PRON
ejde-502	502	2	,	,	PUNCT
ejde-502	502	3	124	124	NUM
ejde-502	502	4	(	(	PUNCT
ejde-502	502	5	1994	1994	NUM
ejde-502	502	6	)	)	PUNCT
ejde-502	502	7	,	,	PUNCT
ejde-502	502	8	no	no	INTJ
ejde-502	502	9	.	.	NOUN
ejde-502	502	10	2	2	NUM
ejde-502	502	11	,	,	PUNCT
ejde-502	502	12	389	389	NUM
ejde-502	502	13	-	-	SYM
ejde-502	502	14	398	398	NUM
ejde-502	502	15	.	.	PUNCT
ejde-502	503	1	[	[	X
ejde-502	503	2	26	26	NUM
ejde-502	503	3	]	]	PUNCT
ejde-502	503	4	a.	a.	NOUN
ejde-502	503	5	de	de	PROPN
ejde-502	503	6	pablo	pablo	PROPN
ejde-502	503	7	,	,	PUNCT
ejde-502	503	8	j.	j.	PROPN
ejde-502	503	9	l.	l.	PROPN
ejde-502	503	10	vázquez	vázquez	PROPN
ejde-502	503	11	;	;	PUNCT
ejde-502	503	12	the	the	DET
ejde-502	503	13	balance	balance	NOUN
ejde-502	503	14	between	between	ADP
ejde-502	503	15	strong	strong	ADJ
ejde-502	503	16	reaction	reaction	NOUN
ejde-502	503	17	and	and	CCONJ
ejde-502	503	18	slow	slow	ADJ
ejde-502	503	19	diffusion	diffusion	NOUN
ejde-502	503	20	,	,	PUNCT
ejde-502	503	21	comm	comm	NOUN
ejde-502	503	22	.	.	PUNCT
ejde-502	504	1	partial	partial	ADJ
ejde-502	504	2	differential	differential	NOUN
ejde-502	504	3	equations	equation	NOUN
ejde-502	504	4	,	,	PUNCT
ejde-502	504	5	15	15	NUM
ejde-502	504	6	(	(	PUNCT
ejde-502	504	7	1990	1990	NUM
ejde-502	504	8	)	)	PUNCT
ejde-502	504	9	,	,	PUNCT
ejde-502	504	10	no	no	INTJ
ejde-502	504	11	.	.	NOUN
ejde-502	504	12	2	2	NUM
ejde-502	504	13	,	,	PUNCT
ejde-502	504	14	159	159	NUM
ejde-502	504	15	-	-	SYM
ejde-502	504	16	183	183	NUM
ejde-502	504	17	.	.	PUNCT
ejde-502	505	1	[	[	X
ejde-502	505	2	27	27	NUM
ejde-502	505	3	]	]	PUNCT
ejde-502	505	4	a.	a.	NOUN
ejde-502	505	5	de	de	PROPN
ejde-502	505	6	pablo	pablo	PROPN
ejde-502	505	7	,	,	PUNCT
ejde-502	505	8	j.	j.	PROPN
ejde-502	505	9	l.	l.	PROPN
ejde-502	505	10	vázquez	vázquez	PROPN
ejde-502	505	11	;	;	PUNCT
ejde-502	505	12	travelling	travel	VERB
ejde-502	505	13	waves	wave	NOUN
ejde-502	505	14	and	and	CCONJ
ejde-502	505	15	finite	finite	ADJ
ejde-502	505	16	propagation	propagation	NOUN
ejde-502	505	17	in	in	ADP
ejde-502	505	18	a	a	DET
ejde-502	505	19	reaction	reaction	NOUN
ejde-502	505	20	-	-	PUNCT
ejde-502	505	21	diffusion	diffusion	NOUN
ejde-502	505	22	equation	equation	NOUN
ejde-502	505	23	,	,	PUNCT
ejde-502	505	24	j.	j.	PROPN
ejde-502	505	25	differential	differential	PROPN
ejde-502	505	26	equations	equations	PROPN
ejde-502	505	27	,	,	PUNCT
ejde-502	505	28	93	93	NUM
ejde-502	505	29	(	(	PUNCT
ejde-502	505	30	1991	1991	NUM
ejde-502	505	31	)	)	PUNCT
ejde-502	505	32	,	,	PUNCT
ejde-502	505	33	no	no	INTJ
ejde-502	505	34	.	.	NOUN
ejde-502	505	35	1	1	NUM
ejde-502	505	36	,	,	PUNCT
ejde-502	505	37	19	19	NUM
ejde-502	505	38	-	-	SYM
ejde-502	505	39	61	61	NUM
ejde-502	505	40	.	.	PUNCT
ejde-502	506	1	[	[	X
ejde-502	506	2	28	28	NUM
ejde-502	506	3	]	]	X
ejde-502	506	4	a.	a.	NOUN
ejde-502	506	5	de	de	PROPN
ejde-502	506	6	pablo	pablo	PROPN
ejde-502	506	7	,	,	PUNCT
ejde-502	506	8	j.	j.	PROPN
ejde-502	506	9	l.	l.	PROPN
ejde-502	506	10	vázquez	vázquez	PROPN
ejde-502	506	11	;	;	PUNCT
ejde-502	506	12	an	an	DET
ejde-502	506	13	overdetermined	overdetermined	ADJ
ejde-502	506	14	initial	initial	ADJ
ejde-502	506	15	and	and	CCONJ
ejde-502	506	16	boundary	boundary	ADJ
ejde-502	506	17	-	-	PUNCT
ejde-502	506	18	value	value	NOUN
ejde-502	506	19	problem	problem	NOUN
ejde-502	506	20	for	for	ADP
ejde-502	506	21	a	a	DET
ejde-502	506	22	reaction	reaction	NOUN
ejde-502	506	23	-	-	PUNCT
ejde-502	506	24	diffusion	diffusion	NOUN
ejde-502	506	25	equation	equation	NOUN
ejde-502	506	26	,	,	PUNCT
ejde-502	506	27	nonlinear	nonlinear	ADJ
ejde-502	506	28	anal	anal	NOUN
ejde-502	506	29	.	.	PUNCT
ejde-502	506	30	,	,	PUNCT
ejde-502	506	31	19	19	NUM
ejde-502	506	32	(	(	PUNCT
ejde-502	506	33	1992	1992	NUM
ejde-502	506	34	)	)	PUNCT
ejde-502	506	35	,	,	PUNCT
ejde-502	506	36	no	no	INTJ
ejde-502	506	37	.	.	NOUN
ejde-502	506	38	3	3	NUM
ejde-502	506	39	,	,	PUNCT
ejde-502	506	40	259	259	NUM
ejde-502	506	41	-	-	SYM
ejde-502	506	42	269	269	NUM
ejde-502	506	43	.	.	PUNCT
ejde-502	507	1	[	[	X
ejde-502	507	2	29	29	NUM
ejde-502	507	3	]	]	X
ejde-502	507	4	r.	r.	PROPN
ejde-502	507	5	g.	g.	PROPN
ejde-502	507	6	pinsky	pinsky	PROPN
ejde-502	507	7	;	;	PUNCT
ejde-502	507	8	existence	existence	NOUN
ejde-502	507	9	and	and	CCONJ
ejde-502	507	10	nonexistence	nonexistence	NOUN
ejde-502	507	11	of	of	ADP
ejde-502	507	12	global	global	ADJ
ejde-502	507	13	solutions	solution	NOUN
ejde-502	507	14	for	for	ADP
ejde-502	507	15	ut	ut	PROPN
ejde-502	507	16	=	=	PROPN
ejde-502	507	17	∆u+	∆u+	PROPN
ejde-502	507	18	a(x)up	a(x)up	VERB
ejde-502	507	19	in	in	ADP
ejde-502	507	20	rd	rd	PROPN
ejde-502	507	21	,	,	PUNCT
ejde-502	507	22	j.	j.	PROPN
ejde-502	507	23	differential	differential	PROPN
ejde-502	507	24	equations	equations	PROPN
ejde-502	507	25	,	,	PUNCT
ejde-502	507	26	133	133	NUM
ejde-502	507	27	(	(	PUNCT
ejde-502	507	28	1997	1997	NUM
ejde-502	507	29	)	)	PUNCT
ejde-502	507	30	,	,	PUNCT
ejde-502	507	31	no	no	INTJ
ejde-502	507	32	.	.	NOUN
ejde-502	507	33	1	1	NUM
ejde-502	507	34	,	,	PUNCT
ejde-502	507	35	152	152	NUM
ejde-502	507	36	-	-	SYM
ejde-502	507	37	177	177	NUM
ejde-502	507	38	.	.	PUNCT
ejde-502	508	1	[	[	X
ejde-502	508	2	30	30	NUM
ejde-502	508	3	]	]	X
ejde-502	508	4	r.	r.	PROPN
ejde-502	508	5	g.	g.	PROPN
ejde-502	508	6	pinsky	pinsky	PROPN
ejde-502	508	7	;	;	PUNCT
ejde-502	508	8	the	the	DET
ejde-502	508	9	behavior	behavior	NOUN
ejde-502	508	10	of	of	ADP
ejde-502	508	11	the	the	DET
ejde-502	508	12	life	life	NOUN
ejde-502	508	13	span	span	NOUN
ejde-502	508	14	for	for	ADP
ejde-502	508	15	solutions	solution	NOUN
ejde-502	508	16	to	to	PART
ejde-502	508	17	ut	ut	PROPN
ejde-502	508	18	=	=	PUNCT
ejde-502	509	1	∆u	∆u	PROPN
ejde-502	509	2	+	+	CCONJ
ejde-502	509	3	a(x)up	a(x)up	VERB
ejde-502	509	4	in	in	ADP
ejde-502	509	5	rd	rd	PROPN
ejde-502	509	6	,	,	PUNCT
ejde-502	509	7	j.	j.	PROPN
ejde-502	509	8	differential	differential	PROPN
ejde-502	509	9	equations	equation	NOUN
ejde-502	509	10	,	,	PUNCT
ejde-502	509	11	147	147	NUM
ejde-502	509	12	(	(	PUNCT
ejde-502	509	13	1998	1998	NUM
ejde-502	509	14	)	)	PUNCT
ejde-502	509	15	,	,	PUNCT
ejde-502	509	16	no	no	INTJ
ejde-502	509	17	.	.	NOUN
ejde-502	509	18	1	1	NUM
ejde-502	509	19	,	,	PUNCT
ejde-502	509	20	30	30	NUM
ejde-502	509	21	-	-	SYM
ejde-502	509	22	57	57	NUM
ejde-502	509	23	.	.	PUNCT
ejde-502	510	1	[	[	X
ejde-502	510	2	31	31	NUM
ejde-502	510	3	]	]	PUNCT
ejde-502	510	4	y.-w	y.-w	PROPN
ejde-502	510	5	.	.	PUNCT
ejde-502	511	1	qi	qi	PROPN
ejde-502	511	2	;	;	PUNCT
ejde-502	511	3	the	the	DET
ejde-502	511	4	critical	critical	ADJ
ejde-502	511	5	exponents	exponent	NOUN
ejde-502	511	6	of	of	ADP
ejde-502	511	7	parabolic	parabolic	ADJ
ejde-502	511	8	equations	equation	NOUN
ejde-502	511	9	and	and	CCONJ
ejde-502	511	10	blow	blow	NOUN
ejde-502	511	11	-	-	PUNCT
ejde-502	511	12	up	up	NOUN
ejde-502	511	13	in	in	ADP
ejde-502	511	14	rn	rn	PROPN
ejde-502	511	15	,	,	PUNCT
ejde-502	511	16	proc	proc	PROPN
ejde-502	511	17	.	.	PUNCT
ejde-502	512	1	roy	roy	PROPN
ejde-502	512	2	.	.	PROPN
ejde-502	512	3	soc	soc	PROPN
ejde-502	512	4	.	.	PUNCT
ejde-502	513	1	edinburgh	edinburgh	PROPN
ejde-502	513	2	section	section	PROPN
ejde-502	513	3	a	a	PRON
ejde-502	513	4	,	,	PUNCT
ejde-502	513	5	128	128	NUM
ejde-502	513	6	(	(	PUNCT
ejde-502	513	7	1998	1998	NUM
ejde-502	513	8	)	)	PUNCT
ejde-502	513	9	,	,	PUNCT
ejde-502	513	10	no	no	INTJ
ejde-502	513	11	.	.	NOUN
ejde-502	513	12	1	1	NUM
ejde-502	513	13	,	,	PUNCT
ejde-502	513	14	123	123	NUM
ejde-502	513	15	-	-	SYM
ejde-502	513	16	136	136	NUM
ejde-502	513	17	.	.	PUNCT
ejde-502	514	1	[	[	X
ejde-502	514	2	32	32	NUM
ejde-502	514	3	]	]	PUNCT
ejde-502	514	4	p.	p.	NOUN
ejde-502	514	5	quittner	quittner	PROPN
ejde-502	514	6	,	,	PUNCT
ejde-502	514	7	ph	ph	PROPN
ejde-502	514	8	.	.	PROPN
ejde-502	514	9	souplet	souplet	NOUN
ejde-502	514	10	;	;	PUNCT
ejde-502	514	11	superlinear	superlinear	ADJ
ejde-502	514	12	parabolic	parabolic	PROPN
ejde-502	514	13	problems	problem	NOUN
ejde-502	514	14	.	.	PUNCT
ejde-502	515	1	blow	blow	NOUN
ejde-502	515	2	-	-	PUNCT
ejde-502	515	3	up	up	NOUN
ejde-502	515	4	,	,	PUNCT
ejde-502	515	5	global	global	ADJ
ejde-502	515	6	existence	existence	NOUN
ejde-502	515	7	and	and	CCONJ
ejde-502	515	8	steady	steady	ADJ
ejde-502	515	9	states	state	NOUN
ejde-502	515	10	,	,	PUNCT
ejde-502	515	11	birkhauser	birkhaus	ADJ
ejde-502	515	12	advanced	advanced	ADJ
ejde-502	515	13	texts	text	NOUN
ejde-502	515	14	,	,	PUNCT
ejde-502	515	15	birkhauser	birkhaus	ADJ
ejde-502	515	16	verlag	verlag	NOUN
ejde-502	515	17	,	,	PUNCT
ejde-502	515	18	basel	basel	PROPN
ejde-502	515	19	,	,	PUNCT
ejde-502	515	20	2007	2007	NUM
ejde-502	515	21	.	.	PUNCT
ejde-502	516	1	[	[	X
ejde-502	516	2	33	33	NUM
ejde-502	516	3	]	]	PUNCT
ejde-502	516	4	p.	p.	PROPN
ejde-502	516	5	e.	e.	PROPN
ejde-502	516	6	sacks	sacks	PROPN
ejde-502	516	7	;	;	PUNCT
ejde-502	516	8	the	the	DET
ejde-502	516	9	initial	initial	ADJ
ejde-502	516	10	and	and	CCONJ
ejde-502	516	11	boundary	boundary	ADJ
ejde-502	516	12	problem	problem	NOUN
ejde-502	516	13	for	for	ADP
ejde-502	516	14	a	a	DET
ejde-502	516	15	class	class	NOUN
ejde-502	516	16	of	of	ADP
ejde-502	516	17	degenerate	degenerate	ADJ
ejde-502	516	18	parabolic	parabolic	NOUN
ejde-502	516	19	equations	equation	NOUN
ejde-502	516	20	,	,	PUNCT
ejde-502	516	21	comm	comm	NOUN
ejde-502	516	22	.	.	PUNCT
ejde-502	517	1	partial	partial	ADJ
ejde-502	517	2	differential	differential	NOUN
ejde-502	517	3	equations	equation	NOUN
ejde-502	517	4	,	,	PUNCT
ejde-502	517	5	8	8	NUM
ejde-502	517	6	(	(	PUNCT
ejde-502	517	7	1983	1983	NUM
ejde-502	517	8	)	)	PUNCT
ejde-502	517	9	,	,	PUNCT
ejde-502	517	10	no	no	INTJ
ejde-502	517	11	.	.	NOUN
ejde-502	517	12	7	7	NUM
ejde-502	517	13	,	,	PUNCT
ejde-502	517	14	693	693	NUM
ejde-502	517	15	-	-	SYM
ejde-502	517	16	733	733	NUM
ejde-502	517	17	.	.	PUNCT
ejde-502	518	1	[	[	X
ejde-502	518	2	34	34	NUM
ejde-502	518	3	]	]	PUNCT
ejde-502	518	4	a.	a.	NOUN
ejde-502	518	5	a.	a.	PROPN
ejde-502	518	6	samarskii	samarskii	PROPN
ejde-502	518	7	,	,	PUNCT
ejde-502	518	8	v.	v.	PROPN
ejde-502	518	9	a.	a.	PROPN
ejde-502	518	10	galaktionov	galaktionov	PROPN
ejde-502	518	11	,	,	PUNCT
ejde-502	518	12	s.	s.	PROPN
ejde-502	518	13	p.	p.	PROPN
ejde-502	518	14	kurdyumov	kurdyumov	PROPN
ejde-502	518	15	,	,	PUNCT
ejde-502	518	16	a.	a.	PROPN
ejde-502	518	17	p.	p.	PROPN
ejde-502	518	18	mikhailov	mikhailov	PROPN
ejde-502	518	19	;	;	PUNCT
ejde-502	518	20	blow	blow	NOUN
ejde-502	518	21	-	-	PUNCT
ejde-502	518	22	up	up	NOUN
ejde-502	518	23	in	in	ADP
ejde-502	518	24	quasilinear	quasilinear	PROPN
ejde-502	518	25	parabolic	parabolic	NOUN
ejde-502	518	26	problems	problem	NOUN
ejde-502	518	27	,	,	PUNCT
ejde-502	518	28	de	de	X
ejde-502	518	29	gruyter	gruyter	NOUN
ejde-502	518	30	expositions	exposition	NOUN
ejde-502	518	31	in	in	ADP
ejde-502	518	32	mathematics	mathematic	NOUN
ejde-502	518	33	,	,	PUNCT
ejde-502	518	34	19	19	NUM
ejde-502	518	35	,	,	PUNCT
ejde-502	518	36	w.	w.	PROPN
ejde-502	518	37	de	de	PROPN
ejde-502	518	38	gruyter	gruyter	PROPN
ejde-502	518	39	,	,	PUNCT
ejde-502	518	40	berlin	berlin	PROPN
ejde-502	518	41	,	,	PUNCT
ejde-502	518	42	1995	1995	NUM
ejde-502	518	43	.	.	PUNCT
ejde-502	519	1	[	[	X
ejde-502	519	2	35	35	NUM
ejde-502	519	3	]	]	X
ejde-502	519	4	r.	r.	PROPN
ejde-502	519	5	suzuki	suzuki	PROPN
ejde-502	519	6	;	;	PUNCT
ejde-502	519	7	existence	existence	NOUN
ejde-502	519	8	and	and	CCONJ
ejde-502	519	9	nonexistence	nonexistence	NOUN
ejde-502	519	10	of	of	ADP
ejde-502	519	11	global	global	ADJ
ejde-502	519	12	solutions	solution	NOUN
ejde-502	519	13	of	of	ADP
ejde-502	519	14	quasilinear	quasilinear	PROPN
ejde-502	519	15	parabolic	parabolic	PROPN
ejde-502	519	16	equations	equation	NOUN
ejde-502	519	17	,	,	PUNCT
ejde-502	519	18	j.	j.	PROPN
ejde-502	519	19	math	math	PROPN
ejde-502	519	20	.	.	PUNCT
ejde-502	520	1	soc	soc	PROPN
ejde-502	520	2	.	.	PUNCT
ejde-502	521	1	japan	japan	PROPN
ejde-502	521	2	,	,	PUNCT
ejde-502	521	3	54	54	NUM
ejde-502	521	4	(	(	PUNCT
ejde-502	521	5	2002	2002	NUM
ejde-502	521	6	)	)	PUNCT
ejde-502	521	7	,	,	PUNCT
ejde-502	521	8	no	no	INTJ
ejde-502	521	9	.	.	NOUN
ejde-502	521	10	4	4	NUM
ejde-502	521	11	,	,	PUNCT
ejde-502	521	12	747	747	NUM
ejde-502	521	13	-	-	SYM
ejde-502	521	14	792	792	NUM
ejde-502	521	15	.	.	PUNCT
ejde-502	522	1	ejde-2023/72	ejde-2023/72	NOUN
ejde-502	522	2	reaction	reaction	NOUN
ejde-502	522	3	-	-	PUNCT
ejde-502	522	4	diffusion	diffusion	NOUN
ejde-502	522	5	with	with	ADP
ejde-502	522	6	weighted	weight	VERB
ejde-502	522	7	strong	strong	ADJ
ejde-502	522	8	reaction	reaction	NOUN
ejde-502	522	9	21	21	NUM
ejde-502	523	1	[	[	SYM
ejde-502	523	2	36	36	NUM
ejde-502	523	3	]	]	X
ejde-502	523	4	j.	j.	PROPN
ejde-502	523	5	l.	l.	PROPN
ejde-502	524	1	vázquez	vázquez	PROPN
ejde-502	524	2	;	;	PUNCT
ejde-502	524	3	the	the	DET
ejde-502	524	4	porous	porous	ADJ
ejde-502	524	5	medium	medium	ADJ
ejde-502	524	6	equation	equation	NOUN
ejde-502	524	7	.	.	PUNCT
ejde-502	525	1	mathematical	mathematical	ADJ
ejde-502	525	2	theory	theory	NOUN
ejde-502	525	3	,	,	PUNCT
ejde-502	525	4	oxford	oxford	PROPN
ejde-502	525	5	monographs	monograph	NOUN
ejde-502	525	6	in	in	ADP
ejde-502	525	7	mathematics	mathematic	NOUN
ejde-502	525	8	,	,	PUNCT
ejde-502	525	9	oxford	oxford	PROPN
ejde-502	525	10	university	university	PROPN
ejde-502	525	11	press	press	NOUN
ejde-502	525	12	,	,	PUNCT
ejde-502	525	13	2007	2007	NUM
ejde-502	525	14	.	.	PUNCT
ejde-502	526	1	razvan	razvan	PROPN
ejde-502	526	2	gabriel	gabriel	PROPN
ejde-502	526	3	iagar	iagar	PROPN
ejde-502	526	4	departamento	departamento	PROPN
ejde-502	526	5	de	de	PROPN
ejde-502	526	6	matemática	matemática	PROPN
ejde-502	526	7	aplicada	aplicada	PROPN
ejde-502	526	8	,	,	PUNCT
ejde-502	526	9	ciencia	ciencia	PROPN
ejde-502	526	10	e	e	PROPN
ejde-502	526	11	ingenieria	ingenieria	PROPN
ejde-502	526	12	de	de	X
ejde-502	526	13	materiales	materiale	VERB
ejde-502	526	14	y	y	PROPN
ejde-502	526	15	tecnologia	tecnologia	NOUN
ejde-502	526	16	electrónica	electrónica	PROPN
ejde-502	526	17	,	,	PUNCT
ejde-502	526	18	universidad	universidad	PROPN
ejde-502	526	19	rey	rey	PROPN
ejde-502	526	20	juan	juan	PROPN
ejde-502	526	21	carlos	carlos	PROPN
ejde-502	526	22	,	,	PUNCT
ejde-502	526	23	móstoles	móstoles	PROPN
ejde-502	526	24	,	,	PUNCT
ejde-502	526	25	28933	28933	NUM
ejde-502	526	26	,	,	PUNCT
ejde-502	526	27	madrid	madrid	PROPN
ejde-502	526	28	,	,	PUNCT
ejde-502	526	29	spain	spain	PROPN
ejde-502	526	30	email	email	NOUN
ejde-502	526	31	address	address	NOUN
ejde-502	526	32	:	:	PUNCT
ejde-502	526	33	razvan.iagar@urjc.es	razvan.iagar@urjc.es	VERB
ejde-502	526	34	ana	ana	PROPN
ejde-502	526	35	i.	i.	PROPN
ejde-502	526	36	muñoz	muñoz	PROPN
ejde-502	526	37	departamento	departamento	PROPN
ejde-502	526	38	de	de	PROPN
ejde-502	526	39	matemática	matemática	PROPN
ejde-502	526	40	aplicada	aplicada	PROPN
ejde-502	526	41	,	,	PUNCT
ejde-502	526	42	ciencia	ciencia	PROPN
ejde-502	526	43	e	e	PROPN
ejde-502	526	44	ingenieria	ingenieria	PROPN
ejde-502	526	45	de	de	X
ejde-502	526	46	materiales	materiale	VERB
ejde-502	526	47	y	y	PROPN
ejde-502	526	48	tecnologia	tecnologia	NOUN
ejde-502	526	49	electrónica	electrónica	PROPN
ejde-502	526	50	,	,	PUNCT
ejde-502	526	51	universidad	universidad	PROPN
ejde-502	526	52	rey	rey	PROPN
ejde-502	526	53	juan	juan	PROPN
ejde-502	526	54	carlos	carlos	PROPN
ejde-502	526	55	,	,	PUNCT
ejde-502	526	56	móstoles	móstoles	PROPN
ejde-502	526	57	,	,	PUNCT
ejde-502	526	58	28933	28933	NUM
ejde-502	526	59	,	,	PUNCT
ejde-502	526	60	madrid	madrid	PROPN
ejde-502	526	61	,	,	PUNCT
ejde-502	526	62	spain	spain	PROPN
ejde-502	526	63	email	email	NOUN
ejde-502	526	64	address	address	NOUN
ejde-502	526	65	:	:	PUNCT
ejde-502	527	1	anaisabel.munoz@urjc.es	anaisabel.munoz@urjc.es	NOUN
ejde-502	527	2	ariel	ariel	PROPN
ejde-502	527	3	sánchez	sánchez	PROPN
ejde-502	527	4	departamento	departamento	PROPN
ejde-502	527	5	de	de	PROPN
ejde-502	527	6	matemática	matemática	PROPN
ejde-502	527	7	aplicada	aplicada	PROPN
ejde-502	527	8	,	,	PUNCT
ejde-502	527	9	ciencia	ciencia	PROPN
ejde-502	527	10	e	e	PROPN
ejde-502	527	11	ingenieria	ingenieria	PROPN
ejde-502	527	12	de	de	X
ejde-502	527	13	materiales	materiale	VERB
ejde-502	527	14	y	y	PROPN
ejde-502	527	15	tecnologia	tecnologia	NOUN
ejde-502	527	16	electrónica	electrónica	PROPN
ejde-502	527	17	,	,	PUNCT
ejde-502	527	18	universidad	universidad	PROPN
ejde-502	527	19	rey	rey	PROPN
ejde-502	527	20	juan	juan	PROPN
ejde-502	527	21	carlos	carlos	PROPN
ejde-502	527	22	,	,	PUNCT
ejde-502	527	23	móstoles	móstoles	PROPN
ejde-502	527	24	,	,	PUNCT
ejde-502	527	25	28933	28933	NUM
ejde-502	527	26	,	,	PUNCT
ejde-502	527	27	madrid	madrid	PROPN
ejde-502	527	28	,	,	PUNCT
ejde-502	527	29	spain	spain	PROPN
ejde-502	527	30	email	email	NOUN
ejde-502	527	31	address	address	NOUN
ejde-502	527	32	:	:	PUNCT
ejde-502	527	33	ariel.sanchez@urjc.es	ariel.sanchez@urjc.es	NOUN
ejde-502	527	34	1	1	NUM
ejde-502	527	35	.	.	PUNCT
ejde-502	527	36	introduction	introduction	NOUN
ejde-502	527	37	2	2	NUM
ejde-502	527	38	.	.	PUNCT
ejde-502	527	39	existence	existence	NOUN
ejde-502	527	40	and	and	CCONJ
ejde-502	527	41	finite	finite	VERB
ejde-502	527	42	speed	speed	NOUN
ejde-502	527	43	of	of	ADP
ejde-502	527	44	propagation	propagation	NOUN
ejde-502	527	45	when	when	SCONJ
ejde-502	527	46	m+p2	m+p2	PROPN
ejde-502	527	47	3	3	NUM
ejde-502	527	48	.	.	PUNCT
ejde-502	527	49	non	non	ADJ
ejde-502	527	50	-	-	NOUN
ejde-502	527	51	uniqueness	uniqueness	ADJ
ejde-502	527	52	for	for	ADP
ejde-502	527	53	m+p2	m+p2	NOUN
ejde-502	527	54	4	4	NUM
ejde-502	527	55	.	.	PUNCT
ejde-502	528	1	aronson	aronson	PROPN
ejde-502	528	2	-	-	PUNCT
ejde-502	528	3	bénilan	bénilan	PROPN
ejde-502	528	4	estimates	estimate	VERB
ejde-502	528	5	when	when	SCONJ
ejde-502	528	6	m+p<2	m+p<2	PROPN
ejde-502	528	7	5	5	NUM
ejde-502	528	8	.	.	PUNCT
ejde-502	528	9	infinite	infinite	ADJ
ejde-502	528	10	speed	speed	NOUN
ejde-502	528	11	of	of	ADP
ejde-502	528	12	propagation	propagation	NOUN
ejde-502	528	13	when	when	SCONJ
ejde-502	528	14	m+p<2	m+p<2	X
ejde-502	528	15	extensions	extension	NOUN
ejde-502	528	16	and	and	CCONJ
ejde-502	528	17	open	open	ADJ
ejde-502	528	18	problems	problem	NOUN
ejde-502	528	19	acknowledgments	acknowledgment	NOUN
ejde-502	528	20	references	reference	NOUN