id	sid	tid	token	lemma	pos
ma-55	1	1	2022	2022	NUM
ma-55	1	2	ada	ada	PROPN
ma-55	1	3	academica	academica	PROPN
ma-55	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-55	1	5	.	.	PUNCT
ma-55	2	1	j.	j.	PROPN
ma-55	2	2	math	math	PROPN
ma-55	2	3	.	.	PUNCT
ma-55	3	1	anal	anal	ADJ
ma-55	3	2	.	.	PUNCT
ma-55	3	3	2	2	NUM
ma-55	3	4	(	(	PUNCT
ma-55	3	5	2022	2022	NUM
ma-55	3	6	)	)	PUNCT
ma-55	3	7	5doi	5doi	NOUN
ma-55	3	8	:	:	PUNCT
ma-55	3	9	10.28924	10.28924	NUM
ma-55	3	10	/	/	SYM
ma-55	3	11	ada	ada	NOUN
ma-55	3	12	/	/	SYM
ma-55	3	13	ma.2.5	ma.2.5	PROPN
ma-55	3	14	unilateral	unilateral	ADJ
ma-55	3	15	problem	problem	NOUN
ma-55	3	16	for	for	ADP
ma-55	3	17	a	a	DET
ma-55	3	18	viscoelastic	viscoelastic	ADJ
ma-55	3	19	beam	beam	NOUN
ma-55	3	20	equation	equation	NOUN
ma-55	3	21	type	type	NOUN
ma-55	3	22	p	p	NOUN
ma-55	3	23	-	-	PUNCT
ma-55	3	24	laplacian	laplacian	NOUN
ma-55	3	25	with	with	ADP
ma-55	3	26	strong	strong	ADJ
ma-55	3	27	damping	damp	VERB
ma-55	3	28	and	and	CCONJ
ma-55	3	29	logarithmic	logarithmic	ADJ
ma-55	3	30	source	source	NOUN
ma-55	3	31	ducival	ducival	PROPN
ma-55	3	32	c.	c.	PROPN
ma-55	3	33	pereira1	pereira1	PROPN
ma-55	3	34	,	,	PUNCT
ma-55	3	35	geraldo	geraldo	PROPN
ma-55	3	36	m.	m.	PROPN
ma-55	3	37	de	de	X
ma-55	3	38	araújo2	araújo2	PROPN
ma-55	3	39	,	,	PUNCT
ma-55	3	40	carlos	carlos	PROPN
ma-55	3	41	a.	a.	PROPN
ma-55	3	42	raposo3,∗	raposo3,∗	VERB
ma-55	3	43	1department	1department	NUM
ma-55	3	44	of	of	ADP
ma-55	3	45	mathematics	mathematic	NOUN
ma-55	3	46	,	,	PUNCT
ma-55	3	47	state	state	NOUN
ma-55	3	48	university	university	PROPN
ma-55	3	49	of	of	ADP
ma-55	3	50	pará	pará	PROPN
ma-55	3	51	,	,	PUNCT
ma-55	3	52	belém	belém	NOUN
ma-55	3	53	,	,	PUNCT
ma-55	3	54	pa	pa	PROPN
ma-55	3	55	,	,	PUNCT
ma-55	3	56	66113	66113	NUM
ma-55	3	57	-	-	SYM
ma-55	3	58	200	200	NUM
ma-55	3	59	,	,	PUNCT
ma-55	3	60	brazil	brazil	PROPN
ma-55	3	61	ducival@uepa.br	ducival@uepa.br	VERB
ma-55	4	1	2department	2department	NUM
ma-55	4	2	of	of	ADP
ma-55	4	3	mathematics	mathematic	NOUN
ma-55	4	4	,	,	PUNCT
ma-55	4	5	federal	federal	ADJ
ma-55	4	6	university	university	PROPN
ma-55	4	7	of	of	ADP
ma-55	4	8	pará	pará	PROPN
ma-55	4	9	,	,	PUNCT
ma-55	4	10	belém	belém	NOUN
ma-55	4	11	,	,	PUNCT
ma-55	4	12	pa	pa	PROPN
ma-55	4	13	,	,	PUNCT
ma-55	4	14	66075	66075	NUM
ma-55	4	15	-	-	SYM
ma-55	4	16	110	110	NUM
ma-55	4	17	,	,	PUNCT
ma-55	4	18	brazil	brazil	PROPN
ma-55	4	19	gera@ufpa.br	gera@ufpa.br	PROPN
ma-55	5	1	3department	3department	NUM
ma-55	5	2	of	of	ADP
ma-55	5	3	mathematics	mathematic	NOUN
ma-55	5	4	,	,	PUNCT
ma-55	5	5	federal	federal	ADJ
ma-55	5	6	university	university	PROPN
ma-55	5	7	of	of	ADP
ma-55	5	8	são	são	PROPN
ma-55	5	9	joão	joão	PROPN
ma-55	5	10	del	del	PROPN
ma-55	5	11	-	-	PROPN
ma-55	5	12	rei	rei	PROPN
ma-55	5	13	,	,	PUNCT
ma-55	5	14	são	são	PROPN
ma-55	5	15	joão	joão	PROPN
ma-55	5	16	del	del	PROPN
ma-55	5	17	-	-	PROPN
ma-55	5	18	rei	rei	PROPN
ma-55	5	19	,	,	PUNCT
ma-55	5	20	36307	36307	NUM
ma-55	5	21	-	-	SYM
ma-55	5	22	352	352	NUM
ma-55	5	23	,	,	PUNCT
ma-55	5	24	brazil	brazil	PROPN
ma-55	5	25	∗correspondence	∗correspondence	NOUN
ma-55	5	26	:	:	PUNCT
ma-55	5	27	raposo@ufsj.edu.br	raposo@ufsj.edu.br	NOUN
ma-55	5	28	abstract	abstract	NOUN
ma-55	5	29	.	.	PUNCT
ma-55	6	1	in	in	ADP
ma-55	6	2	this	this	DET
ma-55	6	3	manuscript	manuscript	NOUN
ma-55	6	4	,	,	PUNCT
ma-55	6	5	we	we	PRON
ma-55	6	6	investigate	investigate	VERB
ma-55	6	7	the	the	DET
ma-55	6	8	unilateral	unilateral	ADJ
ma-55	6	9	problem	problem	NOUN
ma-55	6	10	for	for	ADP
ma-55	6	11	a	a	DET
ma-55	6	12	viscoelastic	viscoelastic	ADJ
ma-55	6	13	beam	beam	NOUN
ma-55	6	14	equationof	equationof	PROPN
ma-55	6	15	p	p	PROPN
ma-55	6	16	-	-	PUNCT
ma-55	6	17	laplacian	laplacian	ADJ
ma-55	6	18	type	type	NOUN
ma-55	6	19	.	.	PUNCT
ma-55	7	1	the	the	DET
ma-55	7	2	competition	competition	NOUN
ma-55	7	3	of	of	ADP
ma-55	7	4	the	the	DET
ma-55	7	5	strong	strong	ADJ
ma-55	7	6	damping	damping	NOUN
ma-55	7	7	versus	versus	ADP
ma-55	7	8	the	the	DET
ma-55	7	9	logarithmic	logarithmic	ADJ
ma-55	7	10	source	source	NOUN
ma-55	7	11	term	term	NOUN
ma-55	7	12	isconsidered	isconsidere	VERB
ma-55	7	13	.	.	PUNCT
ma-55	8	1	we	we	PRON
ma-55	8	2	use	use	VERB
ma-55	8	3	the	the	DET
ma-55	8	4	potential	potential	ADJ
ma-55	8	5	well	well	NOUN
ma-55	8	6	theory	theory	NOUN
ma-55	8	7	.	.	PUNCT
ma-55	9	1	taking	take	VERB
ma-55	9	2	into	into	ADP
ma-55	9	3	account	account	NOUN
ma-55	9	4	the	the	DET
ma-55	9	5	initial	initial	ADJ
ma-55	9	6	data	datum	NOUN
ma-55	9	7	is	be	AUX
ma-55	9	8	in	in	ADP
ma-55	9	9	the	the	DET
ma-55	9	10	stabilityset	stabilityset	NOUN
ma-55	9	11	created	create	VERB
ma-55	9	12	by	by	ADP
ma-55	9	13	the	the	DET
ma-55	9	14	nehari	nehari	PROPN
ma-55	9	15	surface	surface	NOUN
ma-55	9	16	,	,	PUNCT
ma-55	9	17	we	we	PRON
ma-55	9	18	prove	prove	VERB
ma-55	9	19	the	the	DET
ma-55	9	20	existence	existence	NOUN
ma-55	9	21	and	and	CCONJ
ma-55	9	22	uniqueness	uniqueness	NOUN
ma-55	9	23	of	of	ADP
ma-55	9	24	global	global	ADJ
ma-55	9	25	solutions	solution	NOUN
ma-55	9	26	by	by	ADP
ma-55	9	27	usingthe	usingthe	DET
ma-55	9	28	penalization	penalization	NOUN
ma-55	9	29	method	method	NOUN
ma-55	9	30	and	and	CCONJ
ma-55	9	31	faedo	faedo	NOUN
ma-55	9	32	-	-	PUNCT
ma-55	9	33	galerkin	galerkin	NOUN
ma-55	9	34	’s	’s	PART
ma-55	9	35	approximation	approximation	NOUN
ma-55	9	36	.	.	PUNCT
ma-55	10	1	1	1	X
ma-55	10	2	.	.	X
ma-55	10	3	introduction	introduction	NOUN
ma-55	10	4	we	we	PRON
ma-55	10	5	denote	denote	VERB
ma-55	10	6	the	the	DET
ma-55	10	7	p	p	PROPN
ma-55	10	8	-	-	PUNCT
ma-55	10	9	laplacian	laplacian	ADJ
ma-55	10	10	operator	operator	NOUN
ma-55	10	11	by	by	ADP
ma-55	10	12	∆pu	∆pu	NOUN
ma-55	10	13	=	=	SYM
ma-55	10	14	div	div	X
ma-55	10	15	(	(	PUNCT
ma-55	10	16	|∇u|p−2∇u	|∇u|p−2∇u	NUM
ma-55	10	17	)	)	PUNCT
ma-55	10	18	,	,	PUNCT
ma-55	10	19	which	which	PRON
ma-55	10	20	can	can	AUX
ma-55	10	21	be	be	AUX
ma-55	10	22	extended	extend	VERB
ma-55	10	23	to	to	ADP
ma-55	10	24	amonotone	amonotone	NOUN
ma-55	10	25	,	,	PUNCT
ma-55	10	26	bounded	bound	VERB
ma-55	10	27	,	,	PUNCT
ma-55	10	28	hemicontinuos	hemicontinuo	NOUN
ma-55	10	29	and	and	CCONJ
ma-55	10	30	coercive	coercive	ADJ
ma-55	10	31	operator	operator	NOUN
ma-55	10	32	between	between	ADP
ma-55	10	33	the	the	DET
ma-55	10	34	spaces	space	NOUN
ma-55	10	35	w	w	PROPN
ma-55	10	36	1,p	1,p	PROPN
ma-55	10	37	0	0	SYM
ma-55	10	38	(	(	PUNCT
ma-55	10	39	ω	ω	NOUN
ma-55	10	40	)	)	PUNCT
ma-55	10	41	and	and	CCONJ
ma-55	10	42	its	its	PRON
ma-55	10	43	dualby	dualby	NOUN
ma-55	10	44	−∆p	−∆p	NOUN
ma-55	10	45	:	:	PUNCT
ma-55	10	46	w	w	PROPN
ma-55	10	47	1,p	1,p	PROPN
ma-55	10	48	0	0	NUM
ma-55	10	49	(	(	PUNCT
ma-55	10	50	ω)→	ω)→	NOUN
ma-55	10	51	w−1,q(ω	w−1,q(ω	PROPN
ma-55	10	52	)	)	PUNCT
ma-55	10	53	,	,	PUNCT
ma-55	10	54	〈	〈	PROPN
ma-55	10	55	−∆pu	−∆pu	ADP
ma-55	10	56	,	,	PUNCT
ma-55	10	57	v〉p	v〉p	PROPN
ma-55	10	58	=	=	SYM
ma-55	10	59	∫	∫	PROPN
ma-55	10	60	ω	ω	PROPN
ma-55	10	61	|∇u|p−2∇u	|∇u|p−2∇u	PROPN
ma-55	10	62	·	·	PUNCT
ma-55	10	63	∇v	∇v	ADJ
ma-55	10	64	dx	dx	PROPN
ma-55	10	65	.	.	PUNCT
ma-55	11	1	in	in	ADP
ma-55	11	2	[	[	X
ma-55	11	3	3	3	X
ma-55	11	4	]	]	PUNCT
ma-55	11	5	the	the	DET
ma-55	11	6	authors	author	NOUN
ma-55	11	7	establish	establish	VERB
ma-55	11	8	existence	existence	NOUN
ma-55	11	9	of	of	ADP
ma-55	11	10	global	global	ADJ
ma-55	11	11	solution	solution	NOUN
ma-55	11	12	to	to	ADP
ma-55	11	13	the	the	DET
ma-55	11	14	problem	problem	NOUN
ma-55	11	15	utt	utt	NOUN
ma-55	12	1	+	+	CCONJ
ma-55	12	2	∆2u	∆2u	NOUN
ma-55	12	3	−	−	PROPN
ma-55	12	4	∆pu	∆pu	NOUN
ma-55	12	5	+	+	CCONJ
ma-55	12	6	∫	∫	PROPN
ma-55	12	7	t	t	PROPN
ma-55	12	8	0	0	NUM
ma-55	12	9	g(t	g(t	PROPN
ma-55	12	10	−	−	PROPN
ma-55	12	11	s)∆u(s)ds	s)∆u(s)ds	ADJ
ma-55	12	12	−	−	NOUN
ma-55	12	13	∆ut	∆ut	AUX
ma-55	12	14	+	+	X
ma-55	12	15	f	f	X
ma-55	12	16	(	(	PUNCT
ma-55	12	17	u	u	NOUN
ma-55	12	18	)	)	PUNCT
ma-55	12	19	=	=	SYM
ma-55	12	20	0	0	NUM
ma-55	12	21	in	in	ADP
ma-55	12	22	ω×	ω×	NOUN
ma-55	12	23	r+	r+	X
ma-55	12	24	,	,	PUNCT
ma-55	12	25	(	(	PUNCT
ma-55	12	26	1.1	1.1	NUM
ma-55	12	27	)	)	PUNCT
ma-55	12	28	u	u	NOUN
ma-55	12	29	=	=	PUNCT
ma-55	12	30	∆u	∆u	PROPN
ma-55	12	31	=	=	SYM
ma-55	12	32	0	0	NUM
ma-55	12	33	on	on	ADP
ma-55	12	34	γ×	γ×	PROPN
ma-55	12	35	r+	r+	PUNCT
ma-55	12	36	,	,	PUNCT
ma-55	12	37	(	(	PUNCT
ma-55	12	38	1.2	1.2	NUM
ma-55	12	39	)	)	PUNCT
ma-55	12	40	u(x	u(x	NOUN
ma-55	12	41	,	,	PUNCT
ma-55	12	42	0	0	NUM
ma-55	12	43	)	)	PUNCT
ma-55	12	44	=	=	SYM
ma-55	12	45	u0	u0	ADJ
ma-55	12	46	,	,	PUNCT
ma-55	12	47	ut(x	ut(x	NOUN
ma-55	12	48	,	,	PUNCT
ma-55	12	49	0	0	NUM
ma-55	12	50	)	)	PUNCT
ma-55	12	51	=	=	NOUN
ma-55	12	52	u1	u1	PROPN
ma-55	12	53	in	in	ADP
ma-55	12	54	ω	ω	PROPN
ma-55	12	55	,	,	PUNCT
ma-55	12	56	(	(	PUNCT
ma-55	12	57	1.3	1.3	NUM
ma-55	12	58	)	)	PUNCT
ma-55	12	59	where	where	SCONJ
ma-55	12	60	ω	ω	PROPN
ma-55	12	61	is	be	AUX
ma-55	12	62	a	a	DET
ma-55	12	63	bounded	bounded	ADJ
ma-55	12	64	domain	domain	NOUN
ma-55	12	65	of	of	ADP
ma-55	12	66	rn	rn	PROPN
ma-55	12	67	with	with	ADP
ma-55	12	68	smooth	smooth	ADJ
ma-55	12	69	boundary	boundary	ADJ
ma-55	12	70	γ	γ	X
ma-55	12	71	=	=	SYM
ma-55	12	72	∂ω.equations	∂ω.equation	NOUN
ma-55	12	73	of	of	ADP
ma-55	12	74	the	the	DET
ma-55	12	75	type	type	NOUN
ma-55	12	76	(	(	PUNCT
ma-55	12	77	1.1	1.1	NUM
ma-55	12	78	)	)	PUNCT
ma-55	12	79	are	be	AUX
ma-55	12	80	related	relate	VERB
ma-55	12	81	to	to	ADP
ma-55	12	82	models	model	NOUN
ma-55	12	83	of	of	ADP
ma-55	12	84	elastoplastic	elastoplastic	ADJ
ma-55	12	85	microstructure	microstructure	ADJ
ma-55	12	86	flows	flow	NOUN
ma-55	12	87	.	.	PUNCT
ma-55	13	1	asconsidered	asconsidere	VERB
ma-55	13	2	by	by	ADP
ma-55	13	3	an	an	DET
ma-55	13	4	and	and	CCONJ
ma-55	13	5	peirce	peirce	NOUN
ma-55	14	1	[	[	X
ma-55	14	2	1	1	NUM
ma-55	14	3	,	,	PUNCT
ma-55	14	4	2	2	NUM
ma-55	14	5	]	]	PUNCT
ma-55	14	6	,	,	PUNCT
ma-55	14	7	they	they	PRON
ma-55	14	8	are	be	AUX
ma-55	14	9	essentially	essentially	ADV
ma-55	14	10	of	of	ADP
ma-55	14	11	the	the	DET
ma-55	14	12	form	form	NOUN
ma-55	14	13	utt	utt	INTJ
ma-55	15	1	+	+	CCONJ
ma-55	15	2	uxxxx	uxxxx	ADJ
ma-55	15	3	−	−	PROPN
ma-55	15	4	a(u2	a(u2	NOUN
ma-55	15	5	x	x	PUNCT
ma-55	15	6	)	)	PUNCT
ma-55	15	7	x	x	SYM
ma-55	15	8	=	=	SYM
ma-55	15	9	0	0	X
ma-55	15	10	.	.	PUNCT
ma-55	15	11	received	receive	VERB
ma-55	15	12	:	:	PUNCT
ma-55	15	13	13	13	NUM
ma-55	15	14	nov	nov	PROPN
ma-55	15	15	2021	2021	NUM
ma-55	15	16	.	.	PUNCT
ma-55	16	1	key	key	ADJ
ma-55	16	2	words	word	NOUN
ma-55	16	3	and	and	CCONJ
ma-55	16	4	phrases	phrase	NOUN
ma-55	16	5	.	.	PUNCT
ma-55	17	1	unilateral	unilateral	ADJ
ma-55	17	2	problem	problem	NOUN
ma-55	17	3	;	;	PUNCT
ma-55	17	4	viscoelastic	viscoelastic	ADJ
ma-55	17	5	beam	beam	NOUN
ma-55	17	6	equation	equation	NOUN
ma-55	17	7	type	type	NOUN
ma-55	17	8	p	p	NOUN
ma-55	17	9	-	-	PUNCT
ma-55	17	10	laplacian	laplacian	ADJ
ma-55	17	11	;	;	PUNCT
ma-55	17	12	logarithmic	logarithmic	ADJ
ma-55	17	13	source.1	source.1	PROPN
ma-55	17	14	https://adac.ee	https://adac.ee	PROPN
ma-55	17	15	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	17	16	eur	eur	PROPN
ma-55	17	17	.	.	PUNCT
ma-55	18	1	j.	j.	PROPN
ma-55	18	2	math	math	PROPN
ma-55	18	3	.	.	PUNCT
ma-55	19	1	anal	anal	PROPN
ma-55	19	2	.	.	PUNCT
ma-55	20	1	10.28924	10.28924	NUM
ma-55	20	2	/	/	SYM
ma-55	20	3	ada	ada	PROPN
ma-55	20	4	/	/	SYM
ma-55	20	5	ma.2.5	ma.2.5	PROPN
ma-55	20	6	2a	2a	NUM
ma-55	20	7	more	more	ADV
ma-55	20	8	general	general	ADJ
ma-55	20	9	equation	equation	NOUN
ma-55	20	10	,	,	PUNCT
ma-55	20	11	utt	utt	PROPN
ma-55	20	12	+	+	CCONJ
ma-55	20	13	∆2u	∆2u	NOUN
ma-55	20	14	−	−	PROPN
ma-55	20	15	div(σ(|∇u|2)∇u)−	div(σ(|∇u|2)∇u)−	NOUN
ma-55	20	16	∆ut	∆ut	NOUN
ma-55	20	17	+	+	X
ma-55	20	18	h1(ut	h1(ut	PROPN
ma-55	20	19	)	)	PUNCT
ma-55	21	1	+	+	NUM
ma-55	21	2	h2(u	h2(u	X
ma-55	21	3	)	)	PUNCT
ma-55	21	4	=	=	SYM
ma-55	21	5	h3(x	h3(x	PROPN
ma-55	21	6	)	)	PUNCT
ma-55	21	7	,	,	PUNCT
ma-55	21	8	was	be	AUX
ma-55	21	9	considered	consider	VERB
ma-55	21	10	by	by	ADP
ma-55	21	11	yang	yang	PROPN
ma-55	21	12	et	et	PROPN
ma-55	21	13	al	al	PROPN
ma-55	22	1	[	[	X
ma-55	22	2	22–24	22–24	NUM
ma-55	22	3	]	]	PUNCT
ma-55	22	4	.	.	PUNCT
ma-55	23	1	they	they	PRON
ma-55	23	2	studied	study	VERB
ma-55	23	3	de	de	ADP
ma-55	23	4	existence	existence	NOUN
ma-55	23	5	of	of	ADP
ma-55	23	6	attractors	attractor	NOUN
ma-55	23	7	and	and	CCONJ
ma-55	23	8	their	their	PRON
ma-55	23	9	hausdorffdimensions	hausdorffdimension	NOUN
ma-55	23	10	.	.	PUNCT
ma-55	24	1	another	another	DET
ma-55	24	2	related	relate	VERB
ma-55	24	3	equation	equation	NOUN
ma-55	24	4	is	be	AUX
ma-55	24	5	utt	utt	ADJ
ma-55	24	6	+	+	CCONJ
ma-55	24	7	∆2u	∆2u	NOUN
ma-55	24	8	−	−	PROPN
ma-55	24	9	div(f0(∇u	div(f0(∇u	PROPN
ma-55	24	10	)	)	PUNCT
ma-55	24	11	)	)	PUNCT
ma-55	25	1	+	+	CCONJ
ma-55	25	2	kut	kut	PROPN
ma-55	25	3	=	=	SYM
ma-55	25	4	∆(f1(u))−	∆(f1(u))−	PROPN
ma-55	25	5	f2(u	f2(u	PROPN
ma-55	25	6	)	)	PUNCT
ma-55	25	7	,	,	PUNCT
ma-55	25	8	which	which	PRON
ma-55	25	9	was	be	AUX
ma-55	25	10	considered	consider	VERB
ma-55	25	11	by	by	ADP
ma-55	25	12	chueshov	chueshov	NOUN
ma-55	25	13	and	and	CCONJ
ma-55	25	14	lasiecka	lasiecka	PROPN
ma-55	26	1	[	[	X
ma-55	26	2	12]the	12]the	NUM
ma-55	26	3	problem	problem	NOUN
ma-55	26	4	(	(	PUNCT
ma-55	26	5	1.1	1.1	NUM
ma-55	26	6	)	)	PUNCT
ma-55	26	7	,	,	PUNCT
ma-55	26	8	with	with	ADP
ma-55	26	9	its	its	PRON
ma-55	26	10	memory	memory	NOUN
ma-55	26	11	term	term	NOUN
ma-55	26	12	∫	∫	PROPN
ma-55	26	13	t	t	PROPN
ma-55	26	14	0	0	NUM
ma-55	26	15	g(t−	g(t−	PROPN
ma-55	26	16	s)∆u(s)ds	s)∆u(s)ds	PROPN
ma-55	26	17	,	,	PUNCT
ma-55	26	18	can	can	AUX
ma-55	26	19	be	be	AUX
ma-55	26	20	regarded	regard	VERB
ma-55	26	21	as	as	ADP
ma-55	26	22	a	a	DET
ma-55	26	23	fourth	fourth	ADJ
ma-55	26	24	-	-	PUNCT
ma-55	26	25	orderviscoelastic	orderviscoelastic	ADJ
ma-55	26	26	plate	plate	NOUN
ma-55	26	27	equation	equation	NOUN
ma-55	26	28	with	with	ADP
ma-55	26	29	a	a	DET
ma-55	26	30	lower	low	ADJ
ma-55	26	31	order	order	NOUN
ma-55	26	32	perturbation	perturbation	NOUN
ma-55	26	33	of	of	ADP
ma-55	26	34	the	the	DET
ma-55	26	35	p	p	ADJ
ma-55	26	36	-	-	PUNCT
ma-55	26	37	laplacian	laplacian	ADJ
ma-55	26	38	type	type	NOUN
ma-55	26	39	.	.	PUNCT
ma-55	27	1	this	this	DET
ma-55	27	2	kind	kind	ADJ
ma-55	27	3	ofproblem	ofproblem	NOUN
ma-55	27	4	can	can	AUX
ma-55	27	5	be	be	AUX
ma-55	27	6	also	also	ADV
ma-55	27	7	regarded	regard	VERB
ma-55	27	8	as	as	ADP
ma-55	27	9	an	an	DET
ma-55	27	10	elastoplastic	elastoplastic	ADJ
ma-55	27	11	flow	flow	NOUN
ma-55	27	12	equation	equation	NOUN
ma-55	27	13	with	with	ADP
ma-55	27	14	some	some	DET
ma-55	27	15	kind	kind	NOUN
ma-55	27	16	of	of	ADP
ma-55	27	17	memory	memory	NOUN
ma-55	27	18	effect.we	effect.we	PRON
ma-55	27	19	observe	observe	VERB
ma-55	27	20	that	that	SCONJ
ma-55	27	21	for	for	ADP
ma-55	27	22	viscoelastic	viscoelastic	ADJ
ma-55	27	23	plate	plate	NOUN
ma-55	27	24	equation	equation	NOUN
ma-55	27	25	,	,	PUNCT
ma-55	27	26	it	it	PRON
ma-55	27	27	is	be	AUX
ma-55	27	28	usual	usual	ADJ
ma-55	27	29	consider	consider	VERB
ma-55	27	30	a	a	DET
ma-55	27	31	memory	memory	NOUN
ma-55	27	32	of	of	ADP
ma-55	27	33	the	the	DET
ma-55	27	34	form∫	form∫	PROPN
ma-55	27	35	t	t	NOUN
ma-55	27	36	0	0	NUM
ma-55	27	37	g(t	g(t	PROPN
ma-55	27	38	−	−	PROPN
ma-55	27	39	s)∆2u(s)ds	s)∆2u(s)ds	PROPN
ma-55	27	40	,	,	PUNCT
ma-55	27	41	see	see	VERB
ma-55	27	42	for	for	ADP
ma-55	27	43	instance	instance	NOUN
ma-55	27	44	[	[	X
ma-55	27	45	10	10	NUM
ma-55	27	46	]	]	PUNCT
ma-55	27	47	.	.	PUNCT
ma-55	28	1	however	however	ADV
ma-55	28	2	,	,	PUNCT
ma-55	28	3	because	because	SCONJ
ma-55	28	4	the	the	DET
ma-55	28	5	main	main	ADJ
ma-55	28	6	dissipation	dissipation	NOUN
ma-55	28	7	of	of	ADP
ma-55	28	8	the	the	DET
ma-55	28	9	system	system	NOUN
ma-55	28	10	(	(	PUNCT
ma-55	28	11	1.1	1.1	NUM
ma-55	28	12	)	)	PUNCT
ma-55	28	13	is	be	AUX
ma-55	28	14	given	give	VERB
ma-55	28	15	by	by	ADP
ma-55	28	16	strongdamping	strongdampe	VERB
ma-55	28	17	−∆ut	−∆ut	NOUN
ma-55	28	18	,	,	PUNCT
ma-55	28	19	here	here	ADV
ma-55	28	20	we	we	PRON
ma-55	28	21	consider	consider	VERB
ma-55	28	22	a	a	DET
ma-55	28	23	weaker	weak	ADJ
ma-55	28	24	memory	memory	NOUN
ma-55	28	25	,	,	PUNCT
ma-55	28	26	acting	act	VERB
ma-55	28	27	only	only	ADV
ma-55	28	28	on	on	ADP
ma-55	28	29	∆u	∆u	PROPN
ma-55	28	30	.	.	PUNCT
ma-55	29	1	there	there	PRON
ma-55	29	2	is	be	VERB
ma-55	29	3	a	a	DET
ma-55	29	4	large	large	ADJ
ma-55	29	5	literatureabout	literatureabout	NOUN
ma-55	29	6	stability	stability	NOUN
ma-55	29	7	in	in	ADP
ma-55	29	8	viscoelasticity	viscoelasticity	NOUN
ma-55	29	9	.	.	PUNCT
ma-55	30	1	we	we	PRON
ma-55	30	2	refer	refer	VERB
ma-55	30	3	the	the	DET
ma-55	30	4	reader	reader	NOUN
ma-55	30	5	to	to	ADP
ma-55	30	6	[	[	PUNCT
ma-55	30	7	11,13].a	11,13].a	NUM
ma-55	30	8	nonlinear	nonlinear	ADJ
ma-55	30	9	perturbation	perturbation	NOUN
ma-55	30	10	of	of	ADP
ma-55	30	11	problem	problem	NOUN
ma-55	30	12	(	(	PUNCT
ma-55	30	13	1.1	1.1	NUM
ma-55	30	14	)	)	PUNCT
ma-55	30	15	is	be	AUX
ma-55	30	16	given	give	VERB
ma-55	30	17	by	by	ADP
ma-55	30	18	utt	utt	ADJ
ma-55	30	19	+	+	CCONJ
ma-55	30	20	∆2u	∆2u	NOUN
ma-55	30	21	−	−	PROPN
ma-55	30	22	∆pu	∆pu	NOUN
ma-55	30	23	+	+	CCONJ
ma-55	30	24	∫	∫	PROPN
ma-55	30	25	t	t	PROPN
ma-55	30	26	0	0	NUM
ma-55	30	27	g(t	g(t	PROPN
ma-55	30	28	−	−	PROPN
ma-55	30	29	s)∆u(s)ds	s)∆u(s)ds	ADJ
ma-55	30	30	−	−	NOUN
ma-55	30	31	∆ut	∆ut	AUX
ma-55	31	1	+	+	X
ma-55	31	2	f	f	X
ma-55	31	3	(	(	PUNCT
ma-55	31	4	u	u	NOUN
ma-55	31	5	)	)	PUNCT
ma-55	31	6	≥	≥	NOUN
ma-55	31	7	0	0	NUM
ma-55	31	8	.	.	PUNCT
ma-55	32	1	(	(	PUNCT
ma-55	32	2	1.4	1.4	NUM
ma-55	32	3	)	)	PUNCT
ma-55	32	4	variational	variational	ADJ
ma-55	32	5	inequality	inequality	NOUN
ma-55	32	6	theory	theory	NOUN
ma-55	32	7	was	be	AUX
ma-55	32	8	introduced	introduce	VERB
ma-55	32	9	by	by	ADP
ma-55	32	10	hartman	hartman	PROPN
ma-55	32	11	and	and	CCONJ
ma-55	32	12	stampacchia	stampacchia	PROPN
ma-55	32	13	(	(	PUNCT
ma-55	32	14	1966	1966	NUM
ma-55	32	15	)	)	PUNCT
ma-55	33	1	[	[	X
ma-55	33	2	14	14	NUM
ma-55	33	3	]	]	PUNCT
ma-55	33	4	as	as	ADP
ma-55	33	5	a	a	DET
ma-55	33	6	toolfor	toolfor	NOUN
ma-55	33	7	the	the	DET
ma-55	33	8	study	study	NOUN
ma-55	33	9	of	of	ADP
ma-55	33	10	partial	partial	ADJ
ma-55	33	11	differential	differential	ADJ
ma-55	33	12	equations	equation	NOUN
ma-55	33	13	with	with	ADP
ma-55	33	14	applications	application	NOUN
ma-55	33	15	principally	principally	ADV
ma-55	33	16	in	in	ADP
ma-55	33	17	mechanics.in	mechanics.in	PRON
ma-55	33	18	[	[	X
ma-55	33	19	7	7	X
ma-55	33	20	]	]	PUNCT
ma-55	33	21	the	the	DET
ma-55	33	22	authors	author	NOUN
ma-55	33	23	investigated	investigate	VERB
ma-55	33	24	the	the	DET
ma-55	33	25	unilateral	unilateral	ADJ
ma-55	33	26	problem	problem	NOUN
ma-55	33	27	associated	associate	VERB
ma-55	33	28	with	with	ADP
ma-55	33	29	this	this	DET
ma-55	33	30	perturbation	perturbation	NOUN
ma-55	33	31	,	,	PUNCT
ma-55	33	32	thatis	thatis	NOUN
ma-55	33	33	,	,	PUNCT
ma-55	33	34	a	a	DET
ma-55	33	35	variational	variational	ADJ
ma-55	33	36	inequality	inequality	NOUN
ma-55	33	37	given	give	VERB
ma-55	33	38	for	for	ADP
ma-55	33	39	(	(	PUNCT
ma-55	33	40	1.4	1.4	NUM
ma-55	33	41	)	)	PUNCT
ma-55	33	42	(	(	PUNCT
ma-55	33	43	see	see	VERB
ma-55	33	44	[	[	X
ma-55	33	45	16	16	NUM
ma-55	33	46	]	]	PUNCT
ma-55	33	47	)	)	PUNCT
ma-55	33	48	.	.	PUNCT
ma-55	34	1	making	make	VERB
ma-55	34	2	use	use	NOUN
ma-55	34	3	of	of	ADP
ma-55	34	4	the	the	DET
ma-55	34	5	penalization	penalization	NOUN
ma-55	34	6	method	method	NOUN
ma-55	34	7	andgalerkin	andgalerkin	X
ma-55	34	8	’s	’s	PART
ma-55	34	9	approximations	approximation	NOUN
ma-55	34	10	,	,	PUNCT
ma-55	34	11	they	they	PRON
ma-55	34	12	established	establish	VERB
ma-55	34	13	existence	existence	NOUN
ma-55	34	14	and	and	CCONJ
ma-55	34	15	the	the	DET
ma-55	34	16	uniqueness	uniqueness	NOUN
ma-55	34	17	of	of	ADP
ma-55	34	18	strong	strong	ADJ
ma-55	34	19	solutions.the	solutions.the	DET
ma-55	34	20	unilateral	unilateral	ADJ
ma-55	34	21	problem	problem	NOUN
ma-55	34	22	is	be	AUX
ma-55	34	23	very	very	ADV
ma-55	34	24	interesting	interesting	ADJ
ma-55	34	25	because	because	SCONJ
ma-55	34	26	,	,	PUNCT
ma-55	34	27	in	in	ADP
ma-55	34	28	general	general	ADJ
ma-55	34	29	,	,	PUNCT
ma-55	34	30	dynamic	dynamic	ADJ
ma-55	34	31	contact	contact	NOUN
ma-55	34	32	problems	problem	NOUN
ma-55	34	33	arecharacterized	arecharacterize	VERB
ma-55	34	34	by	by	ADP
ma-55	34	35	nonlinear	nonlinear	ADJ
ma-55	34	36	hyperbolic	hyperbolic	ADJ
ma-55	34	37	variational	variational	ADJ
ma-55	34	38	inequalities	inequality	NOUN
ma-55	34	39	.	.	PUNCT
ma-55	35	1	variational	variational	ADJ
ma-55	35	2	inequality	inequality	NOUN
ma-55	35	3	theory	theory	NOUN
ma-55	35	4	wasintroduced	wasintroduce	VERB
ma-55	35	5	by	by	ADP
ma-55	35	6	hartman	hartman	PROPN
ma-55	35	7	and	and	CCONJ
ma-55	35	8	stampacchia	stampacchia	PROPN
ma-55	35	9	(	(	PUNCT
ma-55	35	10	1966	1966	NUM
ma-55	35	11	)	)	PUNCT
ma-55	36	1	[	[	X
ma-55	36	2	14	14	NUM
ma-55	36	3	]	]	PUNCT
ma-55	36	4	as	as	ADP
ma-55	36	5	a	a	DET
ma-55	36	6	tool	tool	NOUN
ma-55	36	7	for	for	ADP
ma-55	36	8	the	the	DET
ma-55	36	9	study	study	NOUN
ma-55	36	10	of	of	ADP
ma-55	36	11	partial	partial	ADJ
ma-55	36	12	differentialequations	differentialequation	NOUN
ma-55	36	13	with	with	ADP
ma-55	36	14	applications	application	NOUN
ma-55	36	15	principally	principally	ADV
ma-55	36	16	in	in	ADP
ma-55	36	17	mechanics	mechanic	NOUN
ma-55	36	18	.	.	PUNCT
ma-55	37	1	bensoussan	bensoussan	ADJ
ma-55	37	2	and	and	CCONJ
ma-55	37	3	lions	lion	NOUN
ma-55	37	4	(	(	PUNCT
ma-55	37	5	1982	1982	NUM
ma-55	37	6	)	)	PUNCT
ma-55	38	1	[	[	X
ma-55	38	2	9	9	NUM
ma-55	38	3	]	]	PUNCT
ma-55	38	4	used	use	VERB
ma-55	38	5	vari	vari	ADJ
ma-55	38	6	-	-	ADJ
ma-55	38	7	ational	ational	ADJ
ma-55	38	8	inequalities	inequality	NOUN
ma-55	38	9	initially	initially	ADV
ma-55	38	10	in	in	ADP
ma-55	38	11	the	the	DET
ma-55	38	12	study	study	NOUN
ma-55	38	13	of	of	ADP
ma-55	38	14	stochastic	stochastic	ADJ
ma-55	38	15	control	control	NOUN
ma-55	38	16	.	.	PUNCT
ma-55	39	1	in	in	ADP
ma-55	39	2	[	[	X
ma-55	39	3	5	5	NUM
ma-55	39	4	]	]	PUNCT
ma-55	39	5	was	be	AUX
ma-55	39	6	obtained	obtain	VERB
ma-55	39	7	a	a	DET
ma-55	39	8	variationalinequality	variationalinequality	NOUN
ma-55	39	9	for	for	ADP
ma-55	39	10	the	the	DET
ma-55	39	11	navier	navier	NOUN
ma-55	39	12	-	-	PUNCT
ma-55	39	13	stokes	stoke	NOUN
ma-55	39	14	operator	operator	NOUN
ma-55	39	15	with	with	ADP
ma-55	39	16	variable	variable	ADJ
ma-55	39	17	viscosity	viscosity	NOUN
ma-55	39	18	.	.	PUNCT
ma-55	40	1	in	in	ADP
ma-55	40	2	[	[	X
ma-55	40	3	6	6	NUM
ma-55	40	4	]	]	PUNCT
ma-55	40	5	was	be	AUX
ma-55	40	6	studied	study	VERB
ma-55	40	7	the	the	DET
ma-55	40	8	contactproblem	contactproblem	NOUN
ma-55	40	9	on	on	ADP
ma-55	40	10	the	the	DET
ma-55	40	11	oldroyd	oldroyd	ADJ
ma-55	40	12	model	model	NOUN
ma-55	40	13	of	of	ADP
ma-55	40	14	viscoelastic	viscoelastic	NOUN
ma-55	40	15	fluids	fluid	NOUN
ma-55	40	16	.	.	PUNCT
ma-55	41	1	by	by	ADP
ma-55	41	2	using	use	VERB
ma-55	41	3	results	result	NOUN
ma-55	41	4	from	from	ADP
ma-55	41	5	the	the	DET
ma-55	41	6	theory	theory	NOUN
ma-55	41	7	of	of	ADP
ma-55	41	8	monotoneoperators	monotoneoperator	NOUN
ma-55	41	9	,	,	PUNCT
ma-55	41	10	was	be	AUX
ma-55	41	11	established	establish	VERB
ma-55	41	12	the	the	DET
ma-55	41	13	existence	existence	NOUN
ma-55	41	14	of	of	ADP
ma-55	41	15	weak	weak	ADJ
ma-55	41	16	solutions	solution	NOUN
ma-55	41	17	.	.	PUNCT
ma-55	42	1	in	in	ADP
ma-55	42	2	[	[	X
ma-55	42	3	8	8	NUM
ma-55	42	4	]	]	PUNCT
ma-55	42	5	was	be	AUX
ma-55	42	6	studied	study	VERB
ma-55	42	7	the	the	DET
ma-55	42	8	problem	problem	NOUN
ma-55	42	9	forparabolic	forparabolic	ADJ
ma-55	42	10	variational	variational	ADJ
ma-55	42	11	inequalities	inequality	NOUN
ma-55	42	12	with	with	ADP
ma-55	42	13	volterra	volterra	PROPN
ma-55	42	14	type	type	PROPN
ma-55	42	15	operators	operator	NOUN
ma-55	42	16	.	.	PUNCT
ma-55	43	1	the	the	DET
ma-55	43	2	authors	author	NOUN
ma-55	43	3	proved	prove	VERB
ma-55	43	4	the	the	DET
ma-55	43	5	existenceand	existenceand	NOUN
ma-55	43	6	the	the	DET
ma-55	43	7	uniqueness	uniqueness	NOUN
ma-55	43	8	of	of	ADP
ma-55	43	9	the	the	DET
ma-55	43	10	solution	solution	NOUN
ma-55	43	11	.	.	PUNCT
ma-55	44	1	for	for	ADP
ma-55	44	2	contact	contact	NOUN
ma-55	44	3	problems	problem	NOUN
ma-55	44	4	on	on	ADP
ma-55	44	5	elasticity	elasticity	NOUN
ma-55	44	6	and	and	CCONJ
ma-55	44	7	finite	finite	ADJ
ma-55	44	8	element	element	NOUN
ma-55	44	9	method	method	NOUN
ma-55	44	10	,	,	PUNCT
ma-55	44	11	see	see	VERB
ma-55	44	12	kikuchi	kikuchi	PROPN
ma-55	44	13	-	-	PUNCT
ma-55	44	14	oden	oden	PROPN
ma-55	45	1	[	[	X
ma-55	45	2	15	15	NUM
ma-55	45	3	]	]	PUNCT
ma-55	45	4	and	and	CCONJ
ma-55	45	5	reference	reference	NOUN
ma-55	45	6	therein	therein	ADV
ma-55	45	7	.	.	PUNCT
ma-55	46	1	in	in	ADP
ma-55	46	2	[	[	X
ma-55	46	3	18	18	NUM
ma-55	46	4	]	]	PUNCT
ma-55	46	5	was	be	AUX
ma-55	46	6	studied	study	VERB
ma-55	46	7	the	the	DET
ma-55	46	8	unilateral	unilateral	ADJ
ma-55	46	9	problem	problem	NOUN
ma-55	46	10	for	for	ADP
ma-55	46	11	the	the	DET
ma-55	46	12	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	46	13	eur	eur	PROPN
ma-55	46	14	.	.	PUNCT
ma-55	47	1	j.	j.	PROPN
ma-55	47	2	math	math	PROPN
ma-55	47	3	.	.	PUNCT
ma-55	48	1	anal	anal	PROPN
ma-55	48	2	.	.	PUNCT
ma-55	49	1	10.28924	10.28924	NUM
ma-55	49	2	/	/	SYM
ma-55	49	3	ada	ada	PROPN
ma-55	49	4	/	/	SYM
ma-55	49	5	ma.2.5	ma.2.5	PROPN
ma-55	49	6	3klein	3klein	NUM
ma-55	49	7	-	-	PUNCT
ma-55	49	8	gordon	gordon	NOUN
ma-55	49	9	operator	operator	NOUN
ma-55	49	10	with	with	ADP
ma-55	49	11	the	the	DET
ma-55	49	12	nonlinearity	nonlinearity	NOUN
ma-55	49	13	of	of	ADP
ma-55	49	14	kirchhoff	kirchhoff	NOUN
ma-55	49	15	-	-	PUNCT
ma-55	49	16	carrier	carrier	NOUN
ma-55	49	17	type	type	NOUN
ma-55	49	18	.	.	PUNCT
ma-55	50	1	by	by	ADP
ma-55	50	2	using	use	VERB
ma-55	50	3	an	an	DET
ma-55	50	4	appropriate	appropriate	ADJ
ma-55	50	5	pe	pe	ADJ
ma-55	50	6	-	-	PUNCT
ma-55	50	7	nalization	nalization	NOUN
ma-55	50	8	was	be	AUX
ma-55	50	9	shown	show	VERB
ma-55	50	10	the	the	DET
ma-55	50	11	existence	existence	NOUN
ma-55	50	12	and	and	CCONJ
ma-55	50	13	uniqueness	uniqueness	NOUN
ma-55	50	14	of	of	ADP
ma-55	50	15	solutions	solution	NOUN
ma-55	50	16	for	for	ADP
ma-55	50	17	the	the	DET
ma-55	50	18	perturbed	perturb	VERB
ma-55	50	19	equation	equation	NOUN
ma-55	50	20	.	.	PUNCT
ma-55	51	1	in	in	ADP
ma-55	51	2	[	[	X
ma-55	51	3	19]was	19]was	NUM
ma-55	51	4	considered	consider	VERB
ma-55	51	5	the	the	DET
ma-55	51	6	unilateral	unilateral	ADJ
ma-55	51	7	problem	problem	NOUN
ma-55	51	8	for	for	ADP
ma-55	51	9	a	a	DET
ma-55	51	10	nonlinear	nonlinear	ADJ
ma-55	51	11	wave	wave	NOUN
ma-55	51	12	equation	equation	NOUN
ma-55	51	13	with	with	ADP
ma-55	51	14	p	p	NOUN
ma-55	51	15	-	-	PUNCT
ma-55	51	16	laplacian	laplacian	ADJ
ma-55	51	17	operatorand	operatorand	NOUN
ma-55	51	18	source	source	NOUN
ma-55	51	19	term	term	NOUN
ma-55	51	20	.	.	PUNCT
ma-55	52	1	by	by	ADP
ma-55	52	2	using	use	VERB
ma-55	52	3	an	an	DET
ma-55	52	4	appropriate	appropriate	ADJ
ma-55	52	5	penalization	penalization	NOUN
ma-55	52	6	,	,	PUNCT
ma-55	52	7	authors	author	NOUN
ma-55	52	8	obtained	obtain	VERB
ma-55	52	9	a	a	DET
ma-55	52	10	variational	variational	NOUN
ma-55	52	11	inequalityfor	inequalityfor	SCONJ
ma-55	52	12	the	the	DET
ma-55	52	13	equation	equation	NOUN
ma-55	52	14	perturbed	perturb	VERB
ma-55	52	15	and	and	CCONJ
ma-55	52	16	then	then	ADV
ma-55	52	17	the	the	DET
ma-55	52	18	existence	existence	NOUN
ma-55	52	19	of	of	ADP
ma-55	52	20	solutions	solution	NOUN
ma-55	52	21	was	be	AUX
ma-55	52	22	proved.in	proved.in	PRON
ma-55	52	23	this	this	DET
ma-55	52	24	work	work	NOUN
ma-55	52	25	,	,	PUNCT
ma-55	52	26	we	we	PRON
ma-55	52	27	propose	propose	VERB
ma-55	52	28	to	to	PART
ma-55	52	29	investigate	investigate	VERB
ma-55	52	30	the	the	DET
ma-55	52	31	existence	existence	NOUN
ma-55	52	32	and	and	CCONJ
ma-55	52	33	uniqueness	uniqueness	NOUN
ma-55	52	34	of	of	ADP
ma-55	52	35	solutions	solution	NOUN
ma-55	52	36	for	for	ADP
ma-55	52	37	the	the	DET
ma-55	52	38	vari	vari	ADJ
ma-55	52	39	-	-	PUNCT
ma-55	52	40	ational	ational	ADJ
ma-55	52	41	inequality	inequality	NOUN
ma-55	52	42	associated	associate	VERB
ma-55	52	43	with	with	ADP
ma-55	52	44	the	the	DET
ma-55	52	45	problem	problem	NOUN
ma-55	52	46	(	(	PUNCT
ma-55	52	47	1.4	1.4	NUM
ma-55	52	48	)	)	PUNCT
ma-55	52	49	with	with	ADP
ma-55	52	50	the	the	DET
ma-55	52	51	source	source	NOUN
ma-55	52	52	term	term	NOUN
ma-55	52	53	f	f	PROPN
ma-55	52	54	(	(	PUNCT
ma-55	52	55	u	u	NOUN
ma-55	52	56	)	)	PUNCT
ma-55	52	57	=	=	SYM
ma-55	52	58	−|u|r−2u	−|u|r−2u	X
ma-55	52	59	ln	ln	ADV
ma-55	52	60	|u|.more	|u|.more	X
ma-55	53	1	precisely	precisely	ADV
ma-55	53	2	,	,	PUNCT
ma-55	53	3	we	we	PRON
ma-55	53	4	investigate	investigate	VERB
ma-55	53	5	the	the	DET
ma-55	53	6	existence	existence	NOUN
ma-55	53	7	and	and	CCONJ
ma-55	53	8	uniqueness	uniqueness	NOUN
ma-55	53	9	of	of	ADP
ma-55	53	10	solutions	solution	NOUN
ma-55	53	11	for	for	ADP
ma-55	53	12	the	the	DET
ma-55	53	13	unilateral	unilateral	ADJ
ma-55	53	14	problem	problem	NOUN
ma-55	53	15	utt	utt	NOUN
ma-55	54	1	+	+	CCONJ
ma-55	54	2	∆2u	∆2u	NOUN
ma-55	54	3	−	−	PROPN
ma-55	54	4	∆pu	∆pu	NOUN
ma-55	54	5	+	+	CCONJ
ma-55	54	6	∫	∫	PROPN
ma-55	54	7	t	t	PROPN
ma-55	54	8	0	0	NUM
ma-55	54	9	g(t	g(t	PROPN
ma-55	54	10	−	−	PROPN
ma-55	54	11	s)∆u(s)ds	s)∆u(s)ds	PROPN
ma-55	54	12	−	−	PROPN
ma-55	54	13	∆ut	∆ut	NOUN
ma-55	54	14	≥	≥	X
ma-55	54	15	|u|r−2u	|u|r−2u	PROPN
ma-55	54	16	ln	ln	ADJ
ma-55	54	17	|u|	|u|	PROPN
ma-55	54	18	in	in	ADP
ma-55	54	19	ω×r+	ω×r+	NUM
ma-55	54	20	,	,	PUNCT
ma-55	54	21	(	(	PUNCT
ma-55	54	22	1.5	1.5	NUM
ma-55	54	23	)	)	PUNCT
ma-55	54	24	u(x	u(x	NOUN
ma-55	54	25	,	,	PUNCT
ma-55	54	26	0	0	NUM
ma-55	54	27	)	)	PUNCT
ma-55	54	28	=	=	SYM
ma-55	54	29	u0(x	u0(x	NOUN
ma-55	54	30	)	)	PUNCT
ma-55	54	31	,	,	PUNCT
ma-55	54	32	ut(x	ut(x	NOUN
ma-55	54	33	,	,	PUNCT
ma-55	54	34	0	0	NUM
ma-55	54	35	)	)	PUNCT
ma-55	54	36	=	=	SYM
ma-55	55	1	u1(x	u1(x	NOUN
ma-55	55	2	)	)	PUNCT
ma-55	55	3	in	in	ADP
ma-55	55	4	ω	ω	NUM
ma-55	55	5	,	,	PUNCT
ma-55	55	6	(	(	PUNCT
ma-55	55	7	1.6	1.6	NUM
ma-55	55	8	)	)	PUNCT
ma-55	55	9	u(x	u(x	NOUN
ma-55	55	10	,	,	PUNCT
ma-55	55	11	t	t	NOUN
ma-55	55	12	)	)	PUNCT
ma-55	55	13	=	=	VERB
ma-55	56	1	∆u(x	∆u(x	ADV
ma-55	56	2	,	,	PUNCT
ma-55	56	3	t	t	PROPN
ma-55	56	4	)	)	PUNCT
ma-55	56	5	=	=	SYM
ma-55	56	6	0	0	NUM
ma-55	56	7	on	on	ADP
ma-55	56	8	γ×	γ×	PROPN
ma-55	56	9	r+	r+	X
ma-55	56	10	.	.	PUNCT
ma-55	57	1	(	(	PUNCT
ma-55	57	2	1.7	1.7	NUM
ma-55	57	3	)	)	PUNCT
ma-55	57	4	this	this	DET
ma-55	57	5	work	work	NOUN
ma-55	57	6	is	be	AUX
ma-55	57	7	organized	organize	VERB
ma-55	57	8	as	as	SCONJ
ma-55	57	9	follows	follow	VERB
ma-55	57	10	:	:	PUNCT
ma-55	57	11	in	in	ADP
ma-55	57	12	section	section	NOUN
ma-55	57	13	2	2	NUM
ma-55	57	14	we	we	PRON
ma-55	57	15	introduce	introduce	VERB
ma-55	57	16	the	the	DET
ma-55	57	17	notation	notation	NOUN
ma-55	57	18	and	and	CCONJ
ma-55	57	19	some	some	DET
ma-55	57	20	well	well	ADJ
ma-55	57	21	-	-	PUNCT
ma-55	57	22	knownresults	knownresult	NOUN
ma-55	57	23	.	.	PUNCT
ma-55	58	1	in	in	ADP
ma-55	58	2	section	section	NOUN
ma-55	58	3	3	3	NUM
ma-55	58	4	we	we	PRON
ma-55	58	5	introduce	introduce	VERB
ma-55	58	6	the	the	DET
ma-55	58	7	potential	potential	ADJ
ma-55	58	8	theory	theory	NOUN
ma-55	58	9	suitable	suitable	ADJ
ma-55	58	10	for	for	ADP
ma-55	58	11	our	our	PRON
ma-55	58	12	problem	problem	NOUN
ma-55	58	13	.	.	PUNCT
ma-55	59	1	in	in	ADP
ma-55	59	2	section	section	NOUN
ma-55	59	3	4	4	NUM
ma-55	59	4	definestrong	definestrong	NOUN
ma-55	59	5	solution	solution	NOUN
ma-55	59	6	to	to	ADP
ma-55	59	7	the	the	DET
ma-55	59	8	boundary	boundary	ADJ
ma-55	59	9	value	value	NOUN
ma-55	59	10	problem	problem	NOUN
ma-55	59	11	(	(	PUNCT
ma-55	59	12	1.5)-(1.7	1.5)-(1.7	NUM
ma-55	59	13	)	)	PUNCT
ma-55	59	14	and	and	CCONJ
ma-55	59	15	present	present	VERB
ma-55	59	16	the	the	DET
ma-55	59	17	theorem	theorem	NOUN
ma-55	59	18	of	of	ADP
ma-55	59	19	existence	existence	NOUN
ma-55	59	20	ofstrong	ofstrong	NOUN
ma-55	59	21	solution	solution	NOUN
ma-55	59	22	.	.	PUNCT
ma-55	60	1	in	in	ADP
ma-55	60	2	section	section	NOUN
ma-55	60	3	5	5	NUM
ma-55	60	4	we	we	PRON
ma-55	60	5	apply	apply	VERB
ma-55	60	6	the	the	DET
ma-55	60	7	penalization	penalization	NOUN
ma-55	60	8	method	method	NOUN
ma-55	60	9	.	.	PUNCT
ma-55	61	1	the	the	DET
ma-55	61	2	existence	existence	NOUN
ma-55	61	3	of	of	ADP
ma-55	61	4	global	global	ADJ
ma-55	61	5	solutionsis	solutionsis	NOUN
ma-55	61	6	given	give	VERB
ma-55	61	7	by	by	ADP
ma-55	61	8	using	use	VERB
ma-55	61	9	faedo	faedo	ADJ
ma-55	61	10	-	-	PUNCT
ma-55	61	11	galerkin	galerkin	NOUN
ma-55	61	12	approximation	approximation	NOUN
ma-55	61	13	.	.	PUNCT
ma-55	62	1	finally	finally	ADV
ma-55	62	2	,	,	PUNCT
ma-55	62	3	in	in	ADP
ma-55	62	4	section	section	NOUN
ma-55	62	5	6	6	NUM
ma-55	62	6	we	we	PRON
ma-55	62	7	prove	prove	VERB
ma-55	62	8	the	the	DET
ma-55	62	9	result	result	NOUN
ma-55	62	10	ofuniqueness	ofuniqueness	ADJ
ma-55	62	11	.	.	PUNCT
ma-55	63	1	2	2	X
ma-55	63	2	.	.	X
ma-55	63	3	preliminaries	preliminary	NOUN
ma-55	63	4	let	let	VERB
ma-55	63	5	ω	ω	NOUN
ma-55	63	6	be	be	AUX
ma-55	63	7	a	a	DET
ma-55	63	8	bounded	bounded	ADJ
ma-55	63	9	domain	domain	NOUN
ma-55	63	10	in	in	ADP
ma-55	63	11	rn	rn	PROPN
ma-55	63	12	with	with	ADP
ma-55	63	13	the	the	DET
ma-55	63	14	boundary	boundary	ADJ
ma-55	63	15	γ	γ	NOUN
ma-55	63	16	of	of	ADP
ma-55	63	17	class	class	PROPN
ma-55	63	18	c2	c2	PROPN
ma-55	63	19	.	.	PUNCT
ma-55	64	1	for	for	ADP
ma-55	64	2	t	t	PROPN
ma-55	64	3	>	>	X
ma-55	64	4	0	0	PROPN
ma-55	64	5	,	,	PUNCT
ma-55	64	6	we	we	PRON
ma-55	64	7	denote	denote	VERB
ma-55	64	8	by	by	ADP
ma-55	64	9	qthe	qthe	DET
ma-55	64	10	cylinder	cylinder	NOUN
ma-55	64	11	ω×(0	ω×(0	PROPN
ma-55	64	12	,	,	PUNCT
ma-55	64	13	t	t	PROPN
ma-55	64	14	)	)	PUNCT
ma-55	64	15	,	,	PUNCT
ma-55	64	16	with	with	ADP
ma-55	64	17	lateral	lateral	ADJ
ma-55	64	18	boundary	boundary	ADJ
ma-55	64	19	σ	σ	PROPN
ma-55	64	20	=	=	SYM
ma-55	64	21	γ×(0	γ×(0	PROPN
ma-55	64	22	,	,	PUNCT
ma-55	64	23	t	t	NOUN
ma-55	64	24	)	)	PUNCT
ma-55	64	25	.	.	PUNCT
ma-55	65	1	by	by	ADP
ma-55	65	2	〈	〈	PROPN
ma-55	65	3	·	·	PROPN
ma-55	65	4	,	,	PUNCT
ma-55	65	5	·	·	PUNCT
ma-55	65	6	〉	〉	NOUN
ma-55	65	7	we	we	PRON
ma-55	65	8	will	will	AUX
ma-55	65	9	represent	represent	VERB
ma-55	65	10	the	the	DET
ma-55	65	11	dualitypairing	dualitypairing	NOUN
ma-55	65	12	between	between	ADP
ma-55	65	13	a	a	DET
ma-55	65	14	banach	banach	NOUN
ma-55	65	15	space	space	NOUN
ma-55	65	16	x	x	PUNCT
ma-55	65	17	and	and	CCONJ
ma-55	65	18	x	x	SYM
ma-55	65	19	′	′	NOUN
ma-55	65	20	,	,	PUNCT
ma-55	65	21	x	x	X
ma-55	65	22	′	′	NUM
ma-55	65	23	being	be	AUX
ma-55	65	24	the	the	DET
ma-55	65	25	topological	topological	ADJ
ma-55	65	26	dual	dual	ADJ
ma-55	65	27	of	of	ADP
ma-55	65	28	the	the	DET
ma-55	65	29	space	space	NOUN
ma-55	65	30	x	x	X
ma-55	65	31	,	,	PUNCT
ma-55	65	32	and	and	CCONJ
ma-55	65	33	by	by	ADP
ma-55	65	34	c	c	NOUN
ma-55	65	35	we	we	PRON
ma-55	65	36	denote	denote	VERB
ma-55	65	37	various	various	ADJ
ma-55	65	38	positive	positive	ADJ
ma-55	65	39	constants	constant	NOUN
ma-55	65	40	.	.	PUNCT
ma-55	66	1	the	the	DET
ma-55	66	2	inner	inner	ADJ
ma-55	66	3	product	product	NOUN
ma-55	66	4	in	in	ADP
ma-55	66	5	h1	h1	NOUN
ma-55	66	6	0(ω	0(ω	NUM
ma-55	66	7	)	)	PUNCT
ma-55	66	8	and	and	CCONJ
ma-55	66	9	l2(ω	l2(ω	NOUN
ma-55	66	10	)	)	PUNCT
ma-55	66	11	,	,	PUNCT
ma-55	66	12	respectively	respectively	ADV
ma-55	66	13	,	,	PUNCT
ma-55	66	14	willbe	willbe	NOUN
ma-55	66	15	denoted	denote	VERB
ma-55	66	16	by	by	ADP
ma-55	66	17	(	(	PUNCT
ma-55	66	18	∇·,∇	∇·,∇	PROPN
ma-55	66	19	·	·	PUNCT
ma-55	66	20	)	)	PUNCT
ma-55	66	21	,	,	PUNCT
ma-55	66	22	(	(	PUNCT
ma-55	66	23	·	·	PUNCT
ma-55	66	24	,	,	PUNCT
ma-55	66	25	·	·	PUNCT
ma-55	66	26	)	)	PUNCT
ma-55	66	27	.	.	PUNCT
ma-55	67	1	the	the	DET
ma-55	67	2	norm	norm	NOUN
ma-55	67	3	in	in	ADP
ma-55	67	4	lp(ω	lp(ω	PROPN
ma-55	67	5	)	)	PUNCT
ma-55	67	6	will	will	AUX
ma-55	67	7	be	be	AUX
ma-55	67	8	denoted	denote	VERB
ma-55	67	9	by	by	ADP
ma-55	67	10	|	|	ADV
ma-55	67	11	·	·	PUNCT
ma-55	67	12	|p	|p	VERB
ma-55	67	13	.the	.the	PRON
ma-55	67	14	inequality	inequality	NOUN
ma-55	67	15	(	(	PUNCT
ma-55	67	16	1.5	1.5	NUM
ma-55	67	17	)	)	PUNCT
ma-55	67	18	must	must	AUX
ma-55	67	19	be	be	AUX
ma-55	67	20	satisfied	satisfied	ADJ
ma-55	67	21	in	in	ADP
ma-55	67	22	the	the	DET
ma-55	67	23	following	follow	VERB
ma-55	67	24	sense	sense	NOUN
ma-55	67	25	.	.	PUNCT
ma-55	68	1	let	let	VERB
ma-55	68	2	k	k	NOUN
ma-55	68	3	=	=	PRON
ma-55	68	4	{	{	PUNCT
ma-55	68	5	v	v	NUM
ma-55	68	6	∈	∈	PROPN
ma-55	68	7	h1	h1	NOUN
ma-55	68	8	0(ω	0(ω	NUM
ma-55	68	9	)	)	PUNCT
ma-55	68	10	;	;	PUNCT
ma-55	68	11	v	v	NUM
ma-55	68	12	≥	≥	NOUN
ma-55	68	13	0	0	NUM
ma-55	69	1	a.e	a.e	PROPN
ma-55	69	2	.	.	PROPN
ma-55	70	1	in	in	ADP
ma-55	70	2	ω	ω	PROPN
ma-55	70	3	}	}	PUNCT
ma-55	70	4	be	be	AUX
ma-55	70	5	a	a	DET
ma-55	70	6	closed	closed	ADJ
ma-55	70	7	and	and	CCONJ
ma-55	70	8	convex	convex	NOUN
ma-55	70	9	subset	subset	NOUN
ma-55	70	10	of	of	ADP
ma-55	70	11	h1	h1	PROPN
ma-55	70	12	0(ω	0(ω	NUM
ma-55	70	13	)	)	PUNCT
ma-55	70	14	,	,	PUNCT
ma-55	70	15	the	the	DET
ma-55	70	16	unilateral	unilateral	ADJ
ma-55	70	17	problem	problem	NOUN
ma-55	70	18	consists	consist	VERB
ma-55	70	19	to	to	PART
ma-55	70	20	find	find	VERB
ma-55	70	21	a	a	DET
ma-55	70	22	solution	solution	NOUN
ma-55	70	23	u(x	u(x	NOUN
ma-55	70	24	,	,	PUNCT
ma-55	70	25	t)satisfying∫	t)satisfying∫	PROPN
ma-55	70	26	q	q	X
ma-55	70	27	(	(	PUNCT
ma-55	70	28	utt	utt	ADJ
ma-55	70	29	+	+	CCONJ
ma-55	70	30	∆2u	∆2u	NOUN
ma-55	70	31	−	−	PROPN
ma-55	70	32	∆pu	∆pu	NOUN
ma-55	70	33	+	+	CCONJ
ma-55	70	34	∫	∫	PROPN
ma-55	70	35	t	t	PROPN
ma-55	70	36	0	0	NUM
ma-55	70	37	g(t	g(t	PROPN
ma-55	70	38	−	−	PROPN
ma-55	70	39	s)∆u(s)ds	s)∆u(s)ds	ADJ
ma-55	70	40	−	−	PROPN
ma-55	70	41	∆ut	∆ut	NOUN
ma-55	70	42	−	−	PROPN
ma-55	70	43	|u|r−2u	|u|r−2u	PROPN
ma-55	70	44	ln	ln	ADJ
ma-55	70	45	|u|)(v	|u|)(v	PROPN
ma-55	70	46	−	−	PROPN
ma-55	70	47	ut	ut	PROPN
ma-55	70	48	)	)	PUNCT
ma-55	70	49	≥	≥	NOUN
ma-55	70	50	0	0	NUM
ma-55	70	51	,	,	PUNCT
ma-55	70	52	(	(	PUNCT
ma-55	70	53	2.1	2.1	NUM
ma-55	70	54	)	)	PUNCT
ma-55	70	55	for	for	ADP
ma-55	70	56	all	all	DET
ma-55	70	57	v	v	ADP
ma-55	70	58	∈	∈	PROPN
ma-55	70	59	k	k	NOUN
ma-55	70	60	with	with	ADP
ma-55	70	61	ut(x	ut(x	NOUN
ma-55	70	62	,	,	PUNCT
ma-55	70	63	t	t	X
ma-55	70	64	)	)	PUNCT
ma-55	70	65	∈	∈	PROPN
ma-55	71	1	k	k	PROPN
ma-55	71	2	a.e	a.e	PROPN
ma-55	71	3	.	.	PROPN
ma-55	72	1	on	on	ADP
ma-55	72	2	[	[	X
ma-55	72	3	0	0	NUM
ma-55	72	4	,	,	PUNCT
ma-55	72	5	t	t	NOUN
ma-55	72	6	]	]	PUNCT
ma-55	72	7	and	and	CCONJ
ma-55	72	8	the	the	DET
ma-55	72	9	initial	initial	ADJ
ma-55	72	10	and	and	CCONJ
ma-55	72	11	boundary	boundary	ADJ
ma-55	72	12	data	datum	NOUN
ma-55	72	13	u	u	NOUN
ma-55	72	14	=	=	PUNCT
ma-55	72	15	∆u	∆u	PROPN
ma-55	72	16	=	=	SYM
ma-55	72	17	0	0	NUM
ma-55	72	18	in	in	ADP
ma-55	72	19	γ×	γ×	PROPN
ma-55	72	20	(	(	PUNCT
ma-55	72	21	0	0	NUM
ma-55	72	22	,	,	PUNCT
ma-55	72	23	t	t	NOUN
ma-55	72	24	)	)	PUNCT
ma-55	72	25	,	,	PUNCT
ma-55	72	26	(	(	PUNCT
ma-55	72	27	2.2	2.2	NUM
ma-55	72	28	)	)	PUNCT
ma-55	72	29	u(x	u(x	NOUN
ma-55	72	30	,	,	PUNCT
ma-55	72	31	0	0	NUM
ma-55	72	32	)	)	PUNCT
ma-55	72	33	=	=	SYM
ma-55	72	34	u0	u0	ADJ
ma-55	72	35	,	,	PUNCT
ma-55	72	36	ut(x	ut(x	NOUN
ma-55	72	37	,	,	PUNCT
ma-55	72	38	0	0	NUM
ma-55	72	39	)	)	PUNCT
ma-55	72	40	=	=	NOUN
ma-55	72	41	u1	u1	NOUN
ma-55	72	42	in	in	ADP
ma-55	72	43	ω	ω	PROPN
ma-55	72	44	.	.	PUNCT
ma-55	73	1	(	(	PUNCT
ma-55	73	2	2.3	2.3	NUM
ma-55	73	3	)	)	PUNCT
ma-55	73	4	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	73	5	eur	eur	PROPN
ma-55	73	6	.	.	PUNCT
ma-55	74	1	j.	j.	PROPN
ma-55	74	2	math	math	PROPN
ma-55	74	3	.	.	PUNCT
ma-55	75	1	anal	anal	PROPN
ma-55	75	2	.	.	PUNCT
ma-55	76	1	10.28924	10.28924	NUM
ma-55	76	2	/	/	SYM
ma-55	76	3	ada	ada	PROPN
ma-55	76	4	/	/	SYM
ma-55	76	5	ma.2.5	ma.2.5	PROPN
ma-55	76	6	4to	4to	ADJ
ma-55	76	7	study	study	NOUN
ma-55	76	8	the	the	DET
ma-55	76	9	existence	existence	NOUN
ma-55	76	10	and	and	CCONJ
ma-55	76	11	uniqueness	uniqueness	NOUN
ma-55	76	12	of	of	ADP
ma-55	76	13	the	the	DET
ma-55	76	14	problem	problem	NOUN
ma-55	76	15	(	(	PUNCT
ma-55	76	16	1.5)-(1.7	1.5)-(1.7	NUM
ma-55	76	17	)	)	PUNCT
ma-55	76	18	,	,	PUNCT
ma-55	76	19	let	let	VERB
ma-55	76	20	us	we	PRON
ma-55	76	21	consider	consider	VERB
ma-55	76	22	the	the	DET
ma-55	76	23	followinghypotheses	followinghypothese	NOUN
ma-55	76	24	:	:	PUNCT
ma-55	76	25	h1	h1	X
ma-55	76	26	.	.	PUNCT
ma-55	76	27	suppose	suppose	VERB
ma-55	76	28	that	that	SCONJ
ma-55	76	29			PROPN
ma-55	76	30	2	2	NUM
ma-55	76	31	≤	≤	NOUN
ma-55	76	32	p	p	NOUN
ma-55	76	33	,	,	PUNCT
ma-55	76	34	if	if	SCONJ
ma-55	76	35	n	n	CCONJ
ma-55	76	36	=	=	SYM
ma-55	76	37	1	1	NUM
ma-55	76	38	,	,	PUNCT
ma-55	76	39	2	2	NUM
ma-55	76	40	2	2	NUM
ma-55	76	41	≤	≤	NOUN
ma-55	76	42	p	p	NOUN
ma-55	76	43	≤	≤	ADJ
ma-55	76	44	2n	2n	NUM
ma-55	76	45	−	−	ADP
ma-55	76	46	2	2	NUM
ma-55	76	47	n	n	NOUN
ma-55	76	48	−	−	PROPN
ma-55	76	49	2	2	NUM
ma-55	76	50	,	,	PUNCT
ma-55	76	51	if	if	SCONJ
ma-55	76	52	n	n	PRON
ma-55	76	53	≥	≥	NOUN
ma-55	76	54	3.h2	3.h2	NUM
ma-55	76	55	.	.	PUNCT
ma-55	77	1	with	with	ADP
ma-55	77	2	respect	respect	NOUN
ma-55	77	3	to	to	ADP
ma-55	77	4	the	the	DET
ma-55	77	5	power	power	NOUN
ma-55	77	6	r	r	NOUN
ma-55	77	7	,	,	PUNCT
ma-55	77	8	let	let	VERB
ma-55	77	9	us	we	PRON
ma-55	77	10	suppose	suppose	VERB
ma-55	77	11	that	that	PROPN
ma-55	77	12	2	2	NUM
ma-55	77	13	<	<	X
ma-55	77	14	r	r	NOUN
ma-55	77	15	<	<	X
ma-55	77	16	+	+	NOUN
ma-55	77	17	∞	∞	PROPN
ma-55	77	18	,	,	PUNCT
ma-55	77	19	if	if	SCONJ
ma-55	77	20	n	n	NOUN
ma-55	77	21	=	=	SYM
ma-55	77	22	1	1	NUM
ma-55	77	23	,	,	PUNCT
ma-55	77	24	2	2	NUM
ma-55	77	25	2	2	NUM
ma-55	77	26	<	<	X
ma-55	77	27	r	r	X
ma-55	77	28	<	<	X
ma-55	77	29	2n	2n	NUM
ma-55	77	30	n	n	CCONJ
ma-55	77	31	−	−	PROPN
ma-55	77	32	2	2	NUM
ma-55	77	33	,	,	PUNCT
ma-55	77	34	if	if	SCONJ
ma-55	77	35	n	n	PRON
ma-55	77	36	≥	≥	NOUN
ma-55	77	37	3	3	NUM
ma-55	77	38	.	.	NUM
ma-55	77	39	h3	h3	NOUN
ma-55	77	40	.	.	PUNCT
ma-55	78	1	with	with	ADP
ma-55	78	2	respect	respect	NOUN
ma-55	78	3	to	to	ADP
ma-55	78	4	the	the	DET
ma-55	78	5	function	function	NOUN
ma-55	78	6	g	g	NOUN
ma-55	78	7	:	:	PUNCT
ma-55	78	8	[	[	X
ma-55	78	9	0,+∞)→	0,+∞)→	NUM
ma-55	78	10	r	r	NOUN
ma-55	78	11	,	,	PUNCT
ma-55	78	12	we	we	PRON
ma-55	78	13	will	will	AUX
ma-55	78	14	assume	assume	VERB
ma-55	78	15	that	that	SCONJ
ma-55	78	16	g	g	PROPN
ma-55	78	17	∈	∈	PROPN
ma-55	78	18	c1[0	c1[0	PROPN
ma-55	78	19	,	,	PUNCT
ma-55	78	20	t	t	X
ma-55	78	21	]	]	PUNCT
ma-55	78	22	and	and	CCONJ
ma-55	78	23	g(0	g(0	NOUN
ma-55	78	24	)	)	PUNCT
ma-55	78	25	>	>	X
ma-55	78	26	0	0	NUM
ma-55	78	27	,	,	PUNCT
ma-55	78	28	i	i	PRON
ma-55	78	29	=	=	PROPN
ma-55	78	30	1−	1−	NUM
ma-55	78	31	µ	µ	X
ma-55	78	32	∫	∫	PROPN
ma-55	78	33	∞	∞	PROPN
ma-55	78	34	0	0	NUM
ma-55	78	35	g(s)ds	g(s)ds	NOUN
ma-55	78	36	>	>	X
ma-55	78	37	0	0	PROPN
ma-55	78	38	,	,	PUNCT
ma-55	78	39	where	where	SCONJ
ma-55	78	40	µ	µ	X
ma-55	78	41	>	>	X
ma-55	78	42	0	0	NUM
ma-55	78	43	is	be	AUX
ma-55	78	44	the	the	PRON
ma-55	78	45	embedding	embed	VERB
ma-55	78	46	constant	constant	ADJ
ma-55	78	47	for	for	ADP
ma-55	78	48	|∇u|	|∇u|	ADJ
ma-55	78	49	≤	≤	NUM
ma-55	78	50	õ	õ	PUNCT
ma-55	79	1	|∆u|	|∆u|	PROPN
ma-55	79	2	,	,	PUNCT
ma-55	79	3	for	for	ADP
ma-55	79	4	all	all	PRON
ma-55	79	5	u	u	PROPN
ma-55	79	6	∈	∈	PROPN
ma-55	79	7	h1	h1	NOUN
ma-55	79	8	0(ω	0(ω	ADV
ma-55	79	9	)	)	PUNCT
ma-55	79	10	∩h2(ω	∩h2(ω	ADJ
ma-55	79	11	)	)	PUNCT
ma-55	79	12	.	.	PUNCT
ma-55	80	1	h4	h4	PROPN
ma-55	80	2	.	.	PUNCT
ma-55	81	1	there	there	PRON
ma-55	81	2	exists	exist	VERB
ma-55	81	3	a	a	DET
ma-55	81	4	constant	constant	ADJ
ma-55	81	5	k1	k1	NOUN
ma-55	81	6	>	>	X
ma-55	81	7	0	0	NUM
ma-55	81	8	such	such	ADJ
ma-55	81	9	that	that	SCONJ
ma-55	81	10	g′(t	g′(t	NOUN
ma-55	81	11	)	)	PUNCT
ma-55	81	12	≤	≤	NOUN
ma-55	82	1	−k1g(t	−k1g(t	NUM
ma-55	82	2	)	)	PUNCT
ma-55	82	3	,	,	PUNCT
ma-55	82	4	∀t	∀t	PROPN
ma-55	82	5	≥	≥	NOUN
ma-55	82	6	0	0	NUM
ma-55	82	7	.	.	PUNCT
ma-55	83	1	by	by	ADP
ma-55	83	2	h1	h1	PROPN
ma-55	83	3	we	we	PRON
ma-55	83	4	have	have	VERB
ma-55	83	5	h1	h1	NOUN
ma-55	83	6	0(ω	0(ω	ADV
ma-55	83	7	)	)	PUNCT
ma-55	84	1	∩h2(ω	∩h2(ω	ADJ
ma-55	84	2	)	)	PUNCT
ma-55	84	3	↪	↪	PROPN
ma-55	84	4	→	→	SYM
ma-55	84	5	w	w	NOUN
ma-55	84	6	1,2(p−1	1,2(p−1	NUM
ma-55	84	7	)	)	PUNCT
ma-55	84	8	0	0	NUM
ma-55	85	1	(	(	PUNCT
ma-55	85	2	ω	ω	NOUN
ma-55	85	3	)	)	PUNCT
ma-55	85	4	↪	↪	PROPN
ma-55	85	5	→	→	SYM
ma-55	85	6	h1	h1	NOUN
ma-55	85	7	0(ω	0(ω	ADV
ma-55	85	8	)	)	PUNCT
ma-55	86	1	↪	↪	PROPN
ma-55	86	2	→	→	SYM
ma-55	86	3	l2(ω	l2(ω	NOUN
ma-55	86	4	)	)	PUNCT
ma-55	86	5	.	.	PUNCT
ma-55	87	1	the	the	DET
ma-55	87	2	lemmas	lemmas	PROPN
ma-55	87	3	below	below	ADV
ma-55	87	4	will	will	AUX
ma-55	87	5	be	be	AUX
ma-55	87	6	a	a	DET
ma-55	87	7	important	important	ADJ
ma-55	87	8	role	role	NOUN
ma-55	87	9	in	in	ADP
ma-55	87	10	this	this	DET
ma-55	87	11	manuscript	manuscript	NOUN
ma-55	87	12	.	.	PUNCT
ma-55	88	1	lemma	lemma	PROPN
ma-55	88	2	2.1	2.1	NUM
ma-55	88	3	.	.	PUNCT
ma-55	89	1	(	(	PUNCT
ma-55	89	2	sobolev	sobolev	PROPN
ma-55	89	3	poincaré	poincaré	PROPN
ma-55	89	4	inequality	inequality	PROPN
ma-55	89	5	)	)	PUNCT
ma-55	89	6	let	let	VERB
ma-55	89	7	p	p	PRON
ma-55	89	8	be	be	AUX
ma-55	89	9	a	a	DET
ma-55	89	10	number	number	NOUN
ma-55	89	11	with	with	ADP
ma-55	89	12	2	2	NUM
ma-55	89	13	<	<	X
ma-55	89	14	p	p	X
ma-55	89	15	<	<	X
ma-55	89	16	∞	∞	PROPN
ma-55	89	17	if	if	SCONJ
ma-55	89	18	n	n	NOUN
ma-55	89	19	=	=	SYM
ma-55	89	20	1	1	NUM
ma-55	89	21	,	,	PUNCT
ma-55	89	22	2	2	NUM
ma-55	89	23	or	or	CCONJ
ma-55	89	24	2	2	NUM
ma-55	89	25	≤	≤	NOUN
ma-55	89	26	p	p	NOUN
ma-55	89	27	≤	≤	NUM
ma-55	89	28	2n	2n	NUM
ma-55	89	29	n	n	CCONJ
ma-55	89	30	−	−	PROPN
ma-55	89	31	2	2	NUM
ma-55	89	32	if	if	SCONJ
ma-55	89	33	n	n	PRON
ma-55	89	34	≥	≥	NOUN
ma-55	89	35	3	3	NUM
ma-55	89	36	,	,	PUNCT
ma-55	89	37	then	then	ADV
ma-55	89	38	there	there	PRON
ma-55	89	39	exists	exist	VERB
ma-55	89	40	a	a	DET
ma-55	89	41	constant	constant	ADJ
ma-55	89	42	c	c	NOUN
ma-55	89	43	>	>	X
ma-55	89	44	0	0	NUM
ma-55	89	45	such	such	ADJ
ma-55	89	46	that	that	PRON
ma-55	89	47	|u|p	|u|p	PROPN
ma-55	89	48	≤	≤	ADJ
ma-55	89	49	c|∇u|,∀u	c|∇u|,∀u	ADP
ma-55	89	50	∈	∈	PROPN
ma-55	89	51	h1	h1	NOUN
ma-55	89	52	0(ω	0(ω	ADV
ma-55	89	53	)	)	PUNCT
ma-55	89	54	lemma	lemma	PROPN
ma-55	89	55	2.2	2.2	NUM
ma-55	89	56	.	.	PUNCT
ma-55	90	1	(	(	PUNCT
ma-55	90	2	technical	technical	ADJ
ma-55	90	3	lemma	lemma	PROPN
ma-55	90	4	)	)	PUNCT
ma-55	90	5	for	for	ADP
ma-55	90	6	v	v	PROPN
ma-55	90	7	∈	∈	PROPN
ma-55	90	8	c1(0	c1(0	PROPN
ma-55	90	9	,	,	PUNCT
ma-55	90	10	t	t	PROPN
ma-55	90	11	;	;	PUNCT
ma-55	90	12	h1	h1	NOUN
ma-55	90	13	0(ω	0(ω	ADJ
ma-55	90	14	)	)	PUNCT
ma-55	90	15	)	)	PUNCT
ma-55	90	16	,	,	PUNCT
ma-55	90	17	we	we	PRON
ma-55	90	18	have∫	have∫	VERB
ma-55	90	19	ω	ω	NUM
ma-55	90	20	∫	∫	PROPN
ma-55	90	21	t	t	PROPN
ma-55	90	22	0	0	NUM
ma-55	90	23	g(t	g(t	PROPN
ma-55	90	24	−	−	PROPN
ma-55	90	25	s)∇v	s)∇v	ADJ
ma-55	90	26	·	·	PUNCT
ma-55	90	27	∇vtdsdx	∇vtdsdx	X
ma-55	91	1	=	=	SYM
ma-55	91	2	1	1	NUM
ma-55	91	3	2	2	NUM
ma-55	91	4	(	(	PUNCT
ma-55	91	5	g′	g′	X
ma-55	91	6	�	�	PROPN
ma-55	91	7	∇v)(t)−	∇v)(t)−	PROPN
ma-55	91	8	1	1	NUM
ma-55	91	9	2	2	NUM
ma-55	91	10	g(t)|∇v(t)|2	g(t)|∇v(t)|2	PROPN
ma-55	91	11	−	−	NOUN
ma-55	91	12	1	1	NUM
ma-55	91	13	2	2	NUM
ma-55	91	14	d	d	NOUN
ma-55	91	15	dt	dt	X
ma-55	91	16	[	[	PUNCT
ma-55	91	17	(	(	PUNCT
ma-55	91	18	g	g	PROPN
ma-55	91	19	�	�	PROPN
ma-55	91	20	∇v)(t)−	∇v)(t)−	PROPN
ma-55	91	21	(	(	PUNCT
ma-55	91	22	∫	∫	PROPN
ma-55	91	23	t	t	PROPN
ma-55	91	24	0	0	NUM
ma-55	91	25	g(s)ds	g(s)ds	PROPN
ma-55	91	26	)	)	PUNCT
ma-55	91	27	|∇v(t)|2	|∇v(t)|2	PROPN
ma-55	91	28	]	]	PUNCT
ma-55	91	29	,	,	PUNCT
ma-55	91	30	where	where	SCONJ
ma-55	91	31	(	(	PUNCT
ma-55	91	32	g	g	PROPN
ma-55	91	33	�	�	PROPN
ma-55	91	34	∇u)(t	∇u)(t	PROPN
ma-55	91	35	)	)	PUNCT
ma-55	91	36	=	=	SYM
ma-55	92	1	∫	∫	PROPN
ma-55	92	2	t	t	PROPN
ma-55	92	3	0	0	NUM
ma-55	92	4	g(t	g(t	PROPN
ma-55	92	5	−	−	PROPN
ma-55	92	6	s)|∇u(s)−∇u(t)|2ds	s)|∇u(s)−∇u(t)|2ds	NOUN
ma-55	92	7	.	.	PUNCT
ma-55	93	1	proof	proof	NOUN
ma-55	93	2	.	.	PUNCT
ma-55	94	1	differentiating	differentiate	VERB
ma-55	94	2	the	the	DET
ma-55	94	3	term	term	NOUN
ma-55	94	4	(	(	PUNCT
ma-55	94	5	g	g	PROPN
ma-55	94	6	�	�	PROPN
ma-55	94	7	∇u)(t	∇u)(t	PROPN
ma-55	94	8	)	)	PUNCT
ma-55	94	9	we	we	PRON
ma-55	94	10	arrive	arrive	VERB
ma-55	94	11	to	to	ADP
ma-55	94	12	the	the	DET
ma-55	94	13	above	above	ADJ
ma-55	94	14	inequality	inequality	NOUN
ma-55	94	15	.	.	PUNCT
ma-55	95	1	�	�	PROPN
ma-55	95	2	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	95	3	eur	eur	PROPN
ma-55	95	4	.	.	PUNCT
ma-55	96	1	j.	j.	PROPN
ma-55	96	2	math	math	PROPN
ma-55	96	3	.	.	PUNCT
ma-55	97	1	anal	anal	PROPN
ma-55	97	2	.	.	PUNCT
ma-55	98	1	10.28924	10.28924	NUM
ma-55	98	2	/	/	SYM
ma-55	98	3	ada	ada	PROPN
ma-55	98	4	/	/	SYM
ma-55	98	5	ma.2.5	ma.2.5	PROPN
ma-55	98	6	53	53	NUM
ma-55	98	7	.	.	PUNCT
ma-55	99	1	potential	potential	ADJ
ma-55	99	2	well	well	ADV
ma-55	99	3	in	in	ADP
ma-55	99	4	this	this	DET
ma-55	99	5	section	section	NOUN
ma-55	99	6	,	,	PUNCT
ma-55	99	7	we	we	PRON
ma-55	99	8	use	use	VERB
ma-55	99	9	the	the	DET
ma-55	99	10	potential	potential	ADJ
ma-55	99	11	well	well	NOUN
ma-55	99	12	theory	theory	NOUN
ma-55	99	13	,	,	PUNCT
ma-55	99	14	a	a	DET
ma-55	99	15	power	power	NOUN
ma-55	99	16	full	full	ADJ
ma-55	99	17	tool	tool	NOUN
ma-55	99	18	in	in	ADP
ma-55	99	19	the	the	DET
ma-55	99	20	study	study	NOUN
ma-55	99	21	of	of	ADP
ma-55	99	22	the	the	DET
ma-55	99	23	globalexistence	globalexistence	NOUN
ma-55	99	24	of	of	ADP
ma-55	99	25	solution	solution	NOUN
ma-55	99	26	in	in	ADP
ma-55	99	27	partial	partial	ADJ
ma-55	99	28	differential	differential	NOUN
ma-55	99	29	equation	equation	NOUN
ma-55	99	30	.	.	PUNCT
ma-55	100	1	see	see	VERB
ma-55	100	2	payne	payne	PROPN
ma-55	100	3	-	-	PUNCT
ma-55	100	4	sattinger	sattinger	NOUN
ma-55	100	5	[	[	X
ma-55	100	6	17	17	NUM
ma-55	100	7	]	]	PUNCT
ma-55	100	8	.	.	PUNCT
ma-55	101	1	it	it	PRON
ma-55	101	2	is	be	AUX
ma-55	101	3	well	well	ADV
ma-55	101	4	-	-	PUNCT
ma-55	101	5	knownthat	knownthat	ADJ
ma-55	101	6	the	the	DET
ma-55	101	7	energy	energy	NOUN
ma-55	101	8	of	of	ADP
ma-55	101	9	a	a	DET
ma-55	101	10	pde	pde	NOUN
ma-55	101	11	system	system	NOUN
ma-55	101	12	,	,	PUNCT
ma-55	101	13	in	in	ADP
ma-55	101	14	some	some	DET
ma-55	101	15	sense	sense	NOUN
ma-55	101	16	,	,	PUNCT
ma-55	101	17	splits	split	VERB
ma-55	101	18	into	into	ADP
ma-55	101	19	kinetic	kinetic	ADJ
ma-55	101	20	and	and	CCONJ
ma-55	101	21	potential	potential	ADJ
ma-55	101	22	energy	energy	NOUN
ma-55	101	23	.	.	PUNCT
ma-55	102	1	the	the	DET
ma-55	102	2	sourceterm	sourceterm	NOUN
ma-55	102	3	induces	induce	VERB
ma-55	102	4	potential	potential	ADJ
ma-55	102	5	energy	energy	NOUN
ma-55	102	6	in	in	ADP
ma-55	102	7	the	the	DET
ma-55	102	8	system	system	NOUN
ma-55	102	9	that	that	PRON
ma-55	102	10	acts	act	VERB
ma-55	102	11	in	in	ADP
ma-55	102	12	opposition	opposition	NOUN
ma-55	102	13	to	to	ADP
ma-55	102	14	the	the	DET
ma-55	102	15	effect	effect	NOUN
ma-55	102	16	of	of	ADP
ma-55	102	17	the	the	DET
ma-55	102	18	stabilizingmechanism	stabilizingmechanism	NOUN
ma-55	102	19	.	.	PUNCT
ma-55	103	1	in	in	ADP
ma-55	103	2	this	this	DET
ma-55	103	3	sense	sense	NOUN
ma-55	103	4	,	,	PUNCT
ma-55	103	5	it	it	PRON
ma-55	103	6	is	be	AUX
ma-55	103	7	possible	possible	ADJ
ma-55	103	8	that	that	SCONJ
ma-55	103	9	the	the	DET
ma-55	103	10	energy	energy	NOUN
ma-55	103	11	from	from	ADP
ma-55	103	12	the	the	DET
ma-55	103	13	source	source	NOUN
ma-55	103	14	term	term	NOUN
ma-55	103	15	destabilizes	destabilize	VERB
ma-55	103	16	all	all	DET
ma-55	103	17	thesystem	thesystem	NOUN
ma-55	103	18	and	and	CCONJ
ma-55	103	19	produces	produce	VERB
ma-55	103	20	a	a	DET
ma-55	103	21	blow	blow	NOUN
ma-55	103	22	-	-	PUNCT
ma-55	103	23	up	up	NOUN
ma-55	103	24	in	in	ADP
ma-55	103	25	a	a	DET
ma-55	103	26	finite	finite	ADJ
ma-55	103	27	time	time	NOUN
ma-55	103	28	.	.	PUNCT
ma-55	104	1	to	to	PART
ma-55	104	2	provide	provide	VERB
ma-55	104	3	a	a	DET
ma-55	104	4	global	global	ADJ
ma-55	104	5	solution	solution	NOUN
ma-55	104	6	,	,	PUNCT
ma-55	104	7	we	we	PRON
ma-55	104	8	are	be	AUX
ma-55	104	9	able	able	ADJ
ma-55	104	10	toconstruct	toconstruct	VERB
ma-55	104	11	a	a	DET
ma-55	104	12	stability	stability	NOUN
ma-55	104	13	set	set	VERB
ma-55	104	14	corresponding	correspond	VERB
ma-55	104	15	to	to	ADP
ma-55	104	16	the	the	DET
ma-55	104	17	source	source	NOUN
ma-55	104	18	term	term	NOUN
ma-55	104	19	created	create	VERB
ma-55	104	20	from	from	ADP
ma-55	104	21	the	the	DET
ma-55	104	22	nehari	nehari	NOUN
ma-55	104	23	manifold	manifold	NOUN
ma-55	104	24	,	,	PUNCT
ma-55	104	25	see	see	VERB
ma-55	104	26	y.ye	y.ye	PROPN
ma-55	104	27	[	[	X
ma-55	104	28	20	20	NUM
ma-55	104	29	]	]	PUNCT
ma-55	104	30	.	.	PUNCT
ma-55	105	1	in	in	ADP
ma-55	105	2	the	the	DET
ma-55	105	3	stability	stability	NOUN
ma-55	105	4	set	set	NOUN
ma-55	105	5	,	,	PUNCT
ma-55	105	6	there	there	PRON
ma-55	105	7	exists	exist	VERB
ma-55	105	8	a	a	DET
ma-55	105	9	valley	valley	NOUN
ma-55	105	10	or	or	CCONJ
ma-55	105	11	a	a	DET
ma-55	105	12	well	well	NOUN
ma-55	105	13	of	of	ADP
ma-55	105	14	the	the	DET
ma-55	105	15	depth	depth	NOUN
ma-55	105	16	d	d	NOUN
ma-55	105	17	created	create	VERB
ma-55	105	18	in	in	ADP
ma-55	105	19	the	the	DET
ma-55	105	20	potentialenergy	potentialenergy	NOUN
ma-55	105	21	.	.	PUNCT
ma-55	106	1	if	if	SCONJ
ma-55	106	2	d	d	NOUN
ma-55	106	3	is	be	AUX
ma-55	106	4	strictly	strictly	ADV
ma-55	106	5	positive	positive	ADJ
ma-55	106	6	,	,	PUNCT
ma-55	106	7	then	then	ADV
ma-55	106	8	we	we	PRON
ma-55	106	9	find	find	VERB
ma-55	106	10	that	that	SCONJ
ma-55	106	11	,	,	PUNCT
ma-55	106	12	for	for	ADP
ma-55	106	13	solutions	solution	NOUN
ma-55	106	14	with	with	ADP
ma-55	106	15	the	the	DET
ma-55	106	16	initial	initial	ADJ
ma-55	106	17	data	datum	NOUN
ma-55	106	18	in	in	ADP
ma-55	106	19	the	the	DET
ma-55	106	20	goodpart	goodpart	NOUN
ma-55	106	21	of	of	ADP
ma-55	106	22	the	the	DET
ma-55	106	23	potential	potential	ADJ
ma-55	106	24	well	well	ADV
ma-55	106	25	,	,	PUNCT
ma-55	106	26	the	the	DET
ma-55	106	27	potential	potential	ADJ
ma-55	106	28	energy	energy	NOUN
ma-55	106	29	of	of	ADP
ma-55	106	30	the	the	DET
ma-55	106	31	solution	solution	NOUN
ma-55	106	32	can	can	AUX
ma-55	106	33	never	never	ADV
ma-55	106	34	escape	escape	VERB
ma-55	106	35	the	the	DET
ma-55	106	36	potential	potential	ADJ
ma-55	106	37	well.in	well.in	PROPN
ma-55	106	38	general	general	NOUN
ma-55	106	39	,	,	PUNCT
ma-55	106	40	the	the	DET
ma-55	106	41	energy	energy	NOUN
ma-55	106	42	from	from	ADP
ma-55	106	43	the	the	DET
ma-55	106	44	source	source	NOUN
ma-55	106	45	term	term	NOUN
ma-55	106	46	causes	cause	VERB
ma-55	106	47	the	the	DET
ma-55	106	48	blow	blow	NOUN
ma-55	106	49	-	-	PUNCT
ma-55	106	50	up	up	NOUN
ma-55	106	51	in	in	ADP
ma-55	106	52	a	a	DET
ma-55	106	53	finite	finite	ADJ
ma-55	106	54	time	time	NOUN
ma-55	106	55	.	.	PUNCT
ma-55	107	1	however	however	ADV
ma-55	107	2	,	,	PUNCT
ma-55	107	3	thegood	thegood	ADJ
ma-55	107	4	part	part	NOUN
ma-55	107	5	of	of	ADP
ma-55	107	6	the	the	DET
ma-55	107	7	potential	potential	ADJ
ma-55	107	8	well	well	NOUN
ma-55	107	9	is	be	AUX
ma-55	107	10	an	an	DET
ma-55	107	11	invariant	invariant	ADJ
ma-55	107	12	set	set	NOUN
ma-55	107	13	where	where	SCONJ
ma-55	107	14	it	it	PRON
ma-55	107	15	remains	remain	VERB
ma-55	107	16	bounded	bound	VERB
ma-55	107	17	.	.	PUNCT
ma-55	108	1	as	as	ADP
ma-55	108	2	a	a	DET
ma-55	108	3	result	result	NOUN
ma-55	108	4	,	,	PUNCT
ma-55	108	5	the	the	DET
ma-55	108	6	totalenergy	totalenergy	NOUN
ma-55	108	7	of	of	ADP
ma-55	108	8	the	the	DET
ma-55	108	9	solution	solution	NOUN
ma-55	108	10	remains	remain	VERB
ma-55	108	11	finite	finite	ADJ
ma-55	108	12	for	for	ADP
ma-55	108	13	any	any	DET
ma-55	108	14	time	time	NOUN
ma-55	108	15	interval	interval	NOUN
ma-55	108	16	[	[	X
ma-55	108	17	0	0	NUM
ma-55	108	18	,	,	PUNCT
ma-55	108	19	t	t	X
ma-55	108	20	]	]	PUNCT
ma-55	108	21	,	,	PUNCT
ma-55	108	22	providing	provide	VERB
ma-55	108	23	the	the	DET
ma-55	108	24	global	global	ADJ
ma-55	108	25	existence	existence	NOUN
ma-55	108	26	ofthe	ofthe	NOUN
ma-55	108	27	solution.for	solution.for	ADP
ma-55	108	28	the	the	DET
ma-55	108	29	model	model	NOUN
ma-55	108	30	considered	consider	VERB
ma-55	108	31	here	here	ADV
ma-55	108	32	,	,	PUNCT
ma-55	108	33	the	the	DET
ma-55	108	34	total	total	ADJ
ma-55	108	35	energy	energy	NOUN
ma-55	108	36	is	be	AUX
ma-55	108	37	given	give	VERB
ma-55	108	38	by	by	ADP
ma-55	108	39	e(t	e(t	NOUN
ma-55	108	40	)	)	PUNCT
ma-55	109	1	=	=	SYM
ma-55	109	2	1	1	NUM
ma-55	109	3	2	2	NUM
ma-55	109	4	[	[	PUNCT
ma-55	109	5	|ut(t)|2	|ut(t)|2	NOUN
ma-55	109	6	+	+	CCONJ
ma-55	109	7	|∆u(t)|2	|∆u(t)|2	NOUN
ma-55	109	8	+	+	CCONJ
ma-55	109	9	2	2	NUM
ma-55	109	10	p	p	NOUN
ma-55	109	11	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	110	1	+	+	CCONJ
ma-55	110	2	(	(	PUNCT
ma-55	110	3	g	g	PROPN
ma-55	110	4	�	�	PROPN
ma-55	110	5	∇u)(t	∇u)(t	PROPN
ma-55	110	6	)	)	PUNCT
ma-55	110	7	−	−	PROPN
ma-55	111	1	(	(	PUNCT
ma-55	111	2	∫	∫	PROPN
ma-55	111	3	t	t	PROPN
ma-55	111	4	0	0	NUM
ma-55	111	5	g(s)ds	g(s)ds	PROPN
ma-55	111	6	)	)	PUNCT
ma-55	111	7	|∇u(t)|2	|∇u(t)|2	ADV
ma-55	111	8	+	+	CCONJ
ma-55	111	9	2	2	NUM
ma-55	111	10	r2	r2	NOUN
ma-55	111	11	|u(t)|rr	|u(t)|rr	ADJ
ma-55	111	12	−	−	NOUN
ma-55	111	13	2	2	NUM
ma-55	111	14	r	r	NOUN
ma-55	111	15	∫	∫	PROPN
ma-55	111	16	ω	ω	NUM
ma-55	111	17	|u(t)|r	|u(t)|r	PROPN
ma-55	111	18	ln	ln	ADJ
ma-55	111	19	|u(t)|dx	|u(t)|dx	NOUN
ma-55	111	20	]	]	PUNCT
ma-55	111	21	(	(	PUNCT
ma-55	111	22	3.1	3.1	NUM
ma-55	111	23	)	)	PUNCT
ma-55	111	24	and	and	CCONJ
ma-55	111	25	satisfies	satisfie	NOUN
ma-55	111	26	d	d	X
ma-55	111	27	dt	dt	X
ma-55	111	28	e(t	e(t	PROPN
ma-55	111	29	)	)	PUNCT
ma-55	111	30	≤	≤	NOUN
ma-55	112	1	−|∇ut(t)|2	−|∇ut(t)|2	PROPN
ma-55	112	2	.	.	PUNCT
ma-55	113	1	(	(	PUNCT
ma-55	113	2	3.2	3.2	NUM
ma-55	113	3	)	)	PUNCT
ma-55	113	4	from	from	ADP
ma-55	113	5	(	(	PUNCT
ma-55	113	6	h3	h3	NOUN
ma-55	113	7	)	)	PUNCT
ma-55	113	8	we	we	PRON
ma-55	113	9	get	get	VERB
ma-55	113	10	i(t	i(t	NOUN
ma-55	113	11	)	)	PUNCT
ma-55	114	1	=	=	SYM
ma-55	114	2	1	1	NUM
ma-55	114	3	2	2	NUM
ma-55	114	4	[	[	PUNCT
ma-55	114	5	|ut(t)|2	|ut(t)|2	NOUN
ma-55	114	6	+	+	CCONJ
ma-55	114	7	(	(	PUNCT
ma-55	114	8	1−	1−	NUM
ma-55	114	9	µ	µ	NUM
ma-55	114	10	∫	∫	PROPN
ma-55	114	11	t	t	PROPN
ma-55	114	12	0	0	NUM
ma-55	114	13	g(s)ds	g(s)ds	PROPN
ma-55	114	14	)	)	PUNCT
ma-55	114	15	|∆u(t)|2	|∆u(t)|2	NOUN
ma-55	114	16	+	+	PROPN
ma-55	114	17	(	(	PUNCT
ma-55	114	18	g	g	PROPN
ma-55	114	19	�	�	PROPN
ma-55	114	20	∇u(t)+	∇u(t)+	PROPN
ma-55	114	21	2	2	NUM
ma-55	114	22	p	p	NOUN
ma-55	114	23	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	114	24	+	+	CCONJ
ma-55	114	25	2	2	NUM
ma-55	114	26	r2	r2	NOUN
ma-55	114	27	|u(t)|rr	|u(t)|rr	ADJ
ma-55	114	28	−	−	NOUN
ma-55	114	29	2	2	NUM
ma-55	115	1	r	r	NOUN
ma-55	115	2	∫	∫	PROPN
ma-55	115	3	ω	ω	NUM
ma-55	115	4	|u(t)|r	|u(t)|r	PROPN
ma-55	115	5	ln	ln	ADJ
ma-55	115	6	|u(t)|dx	|u(t)|dx	NOUN
ma-55	115	7	]	]	PUNCT
ma-55	115	8	.	.	PUNCT
ma-55	116	1	(	(	PUNCT
ma-55	116	2	3.3	3.3	NUM
ma-55	116	3	)	)	PUNCT
ma-55	116	4	then	then	ADV
ma-55	116	5	,	,	PUNCT
ma-55	116	6	we	we	PRON
ma-55	116	7	introduce	introduce	VERB
ma-55	116	8	the	the	DET
ma-55	116	9	functional	functional	ADJ
ma-55	116	10	j	j	NOUN
ma-55	116	11	:	:	PUNCT
ma-55	116	12	h1	h1	VERB
ma-55	116	13	0(ω	0(ω	ADV
ma-55	116	14	)	)	PUNCT
ma-55	117	1	∩h2(ω)→	∩h2(ω)→	NOUN
ma-55	117	2	r	r	NOUN
ma-55	117	3	defined	define	VERB
ma-55	117	4	by	by	ADP
ma-55	117	5	j(u	j(u	PROPN
ma-55	117	6	)	)	PUNCT
ma-55	117	7	=	=	NOUN
ma-55	117	8	1	1	NUM
ma-55	117	9	2	2	NUM
ma-55	117	10	[	[	X
ma-55	117	11	(	(	PUNCT
ma-55	117	12	1−	1−	NUM
ma-55	117	13	µ	µ	NUM
ma-55	117	14	∫	∫	PROPN
ma-55	117	15	t	t	PROPN
ma-55	117	16	0	0	NUM
ma-55	117	17	g(s)ds	g(s)ds	PROPN
ma-55	117	18	)	)	PUNCT
ma-55	117	19	|∆u|2	|∆u|2	PUNCT
ma-55	118	1	+	+	SYM
ma-55	118	2	2	2	NUM
ma-55	118	3	p	p	NOUN
ma-55	118	4	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	118	5	+	+	CCONJ
ma-55	118	6	(	(	PUNCT
ma-55	118	7	g	g	PROPN
ma-55	118	8	�	�	PROPN
ma-55	118	9	∇u)(t	∇u)(t	PROPN
ma-55	118	10	)	)	PUNCT
ma-55	119	1	+	+	CCONJ
ma-55	119	2	2	2	NUM
ma-55	119	3	r2	r2	NOUN
ma-55	119	4	|u(t)|rr	|u(t)|rr	ADJ
ma-55	119	5	−	−	NOUN
ma-55	119	6	2	2	NUM
ma-55	119	7	r	r	NOUN
ma-55	119	8	∫	∫	PROPN
ma-55	119	9	ω	ω	NUM
ma-55	119	10	|u(t)|r	|u(t)|r	PROPN
ma-55	119	11	ln	ln	ADJ
ma-55	119	12	|u(t)|dx	|u(t)|dx	NOUN
ma-55	119	13	]	]	PUNCT
ma-55	119	14	.	.	PUNCT
ma-55	120	1	(	(	PUNCT
ma-55	120	2	3.4	3.4	NUM
ma-55	120	3	)	)	PUNCT
ma-55	120	4	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	120	5	eur	eur	PROPN
ma-55	120	6	.	.	PUNCT
ma-55	121	1	j.	j.	PROPN
ma-55	121	2	math	math	PROPN
ma-55	121	3	.	.	PUNCT
ma-55	122	1	anal	anal	PROPN
ma-55	122	2	.	.	PUNCT
ma-55	123	1	10.28924	10.28924	NUM
ma-55	123	2	/	/	SYM
ma-55	123	3	ada	ada	PROPN
ma-55	123	4	/	/	SYM
ma-55	123	5	ma.2.5	ma.2.5	PROPN
ma-55	123	6	6for	6for	ADP
ma-55	123	7	u	u	NOUN
ma-55	123	8	∈	∈	PROPN
ma-55	123	9	h1	h1	NOUN
ma-55	123	10	0(ω	0(ω	ADV
ma-55	123	11	)	)	PUNCT
ma-55	124	1	∩h2(ω	∩h2(ω	ADJ
ma-55	124	2	)	)	PUNCT
ma-55	124	3	,	,	PUNCT
ma-55	124	4	we	we	PRON
ma-55	124	5	have	have	VERB
ma-55	124	6	j(λu	j(λu	NOUN
ma-55	124	7	)	)	PUNCT
ma-55	124	8	=	=	SYM
ma-55	124	9	λ2	λ2	NOUN
ma-55	124	10	2	2	NUM
ma-55	124	11	(	(	PUNCT
ma-55	124	12	1−	1−	NUM
ma-55	124	13	µ	µ	NUM
ma-55	124	14	∫	∫	PROPN
ma-55	124	15	t	t	PROPN
ma-55	124	16	0	0	NUM
ma-55	124	17	g(s)ds	g(s)ds	PROPN
ma-55	124	18	)	)	PUNCT
ma-55	124	19	|∆u|2	|∆u|2	PUNCT
ma-55	125	1	+	+	CCONJ
ma-55	125	2	λp	λp	X
ma-55	125	3	p	p	X
ma-55	125	4	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	125	5	+	+	CCONJ
ma-55	125	6	λ	λ	PROPN
ma-55	125	7	2	2	NUM
ma-55	125	8	(	(	PUNCT
ma-55	125	9	g	g	PROPN
ma-55	125	10	�	�	PROPN
ma-55	125	11	∇u)(t	∇u)(t	PROPN
ma-55	125	12	)	)	PUNCT
ma-55	125	13	+	+	NUM
ma-55	125	14	λr	λr	ADP
ma-55	125	15	r2	r2	PROPN
ma-55	125	16	|u(t)|rr	|u(t)|rr	ADJ
ma-55	125	17	−	−	NOUN
ma-55	125	18	λr	λr	ADP
ma-55	125	19	r	r	NOUN
ma-55	125	20	∫	∫	PROPN
ma-55	125	21	ω	ω	NUM
ma-55	125	22	|u(t)|r	|u(t)|r	PROPN
ma-55	125	23	ln	ln	PROPN
ma-55	125	24	|u(t)|dx	|u(t)|dx	NOUN
ma-55	125	25	.	.	PUNCT
ma-55	126	1	(	(	PUNCT
ma-55	126	2	3.5	3.5	NUM
ma-55	126	3	)	)	PUNCT
ma-55	126	4	associated	associate	VERB
ma-55	126	5	with	with	ADP
ma-55	126	6	j	j	PROPN
ma-55	126	7	,	,	PUNCT
ma-55	126	8	we	we	PRON
ma-55	126	9	have	have	VERB
ma-55	126	10	the	the	DET
ma-55	126	11	well	well	ADV
ma-55	126	12	-	-	PUNCT
ma-55	126	13	known	know	VERB
ma-55	126	14	nehari	nehari	NOUN
ma-55	126	15	manifold	manifold	NOUN
ma-55	126	16	given	give	VERB
ma-55	126	17	by	by	ADP
ma-55	126	18	n	n	CCONJ
ma-55	126	19	def	def	NOUN
ma-55	126	20	=	=	SYM
ma-55	126	21	{	{	PUNCT
ma-55	126	22	u	u	NOUN
ma-55	126	23	∈	∈	PROPN
ma-55	126	24	h1	h1	NOUN
ma-55	126	25	0(ω	0(ω	ADV
ma-55	126	26	)	)	PUNCT
ma-55	126	27	∩h2(ω	∩h2(ω	ADJ
ma-55	126	28	)	)	PUNCT
ma-55	126	29	\	\	NOUN
ma-55	126	30	{	{	PUNCT
ma-55	126	31	0	0	NUM
ma-55	126	32	}	}	PUNCT
ma-55	126	33	;	;	PUNCT
ma-55	126	34	[	[	PUNCT
ma-55	126	35	d	d	X
ma-55	126	36	dλ	dλ	NOUN
ma-55	126	37	j(λu	j(λu	PROPN
ma-55	126	38	)	)	PUNCT
ma-55	126	39	]	]	PUNCT
ma-55	127	1	λ=1	λ=1	PUNCT
ma-55	127	2	=	=	NOUN
ma-55	127	3	0	0	NUM
ma-55	127	4	}	}	PUNCT
ma-55	127	5	(	(	PUNCT
ma-55	127	6	3.6	3.6	NUM
ma-55	127	7	)	)	PUNCT
ma-55	127	8	or	or	CCONJ
ma-55	127	9	equivalently	equivalently	ADV
ma-55	127	10	,	,	PUNCT
ma-55	127	11	n	n	PROPN
ma-55	127	12	=	=	PRON
ma-55	127	13	{	{	PUNCT
ma-55	127	14	u	u	NOUN
ma-55	127	15	∈	∈	PROPN
ma-55	127	16	h1	h1	NOUN
ma-55	127	17	0(ω	0(ω	ADV
ma-55	127	18	)	)	PUNCT
ma-55	127	19	∩h2(ω	∩h2(ω	ADJ
ma-55	127	20	)	)	PUNCT
ma-55	127	21	)	)	PUNCT
ma-55	127	22	\	\	NOUN
ma-55	128	1	{	{	PUNCT
ma-55	128	2	0	0	NUM
ma-55	128	3	}	}	PUNCT
ma-55	128	4	;	;	PUNCT
ma-55	128	5	(	(	PUNCT
ma-55	128	6	1−	1−	NUM
ma-55	128	7	µ	µ	NUM
ma-55	128	8	∫	∫	PROPN
ma-55	128	9	t	t	PROPN
ma-55	128	10	0	0	NUM
ma-55	128	11	g(s)ds	g(s)ds	PROPN
ma-55	128	12	)	)	PUNCT
ma-55	128	13	|∆u|2	|∆u|2	PUNCT
ma-55	129	1	+	+	NUM
ma-55	129	2	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	129	3	+	+	CCONJ
ma-55	129	4	1	1	NUM
ma-55	129	5	2	2	NUM
ma-55	129	6	(	(	PUNCT
ma-55	129	7	g	g	PROPN
ma-55	129	8	�	�	PROPN
ma-55	129	9	∇u)(t	∇u)(t	PROPN
ma-55	129	10	)	)	PUNCT
ma-55	129	11	=	=	SYM
ma-55	129	12	∫	∫	PROPN
ma-55	129	13	ω	ω	NUM
ma-55	129	14	|u(t)|r	|u(t)|r	PROPN
ma-55	129	15	ln	ln	ADJ
ma-55	129	16	|u(t)|dx	|u(t)|dx	PROPN
ma-55	129	17	}	}	PUNCT
ma-55	129	18	.	.	PUNCT
ma-55	130	1	(	(	PUNCT
ma-55	130	2	3.7	3.7	NUM
ma-55	130	3	)	)	PUNCT
ma-55	130	4	we	we	PRON
ma-55	130	5	define	define	VERB
ma-55	130	6	as	as	ADP
ma-55	130	7	in	in	ADP
ma-55	130	8	the	the	DET
ma-55	130	9	mountain	mountain	NOUN
ma-55	130	10	pass	pass	NOUN
ma-55	130	11	theorem	theorem	NOUN
ma-55	130	12	due	due	ADP
ma-55	130	13	to	to	ADP
ma-55	130	14	ambrosetti	ambrosetti	NOUN
ma-55	130	15	and	and	CCONJ
ma-55	130	16	rabinowitz	rabinowitz	VERB
ma-55	130	17	[	[	X
ma-55	130	18	4	4	NUM
ma-55	130	19	]	]	X
ma-55	130	20	d	d	X
ma-55	130	21	def	def	PROPN
ma-55	130	22	=	=	SYM
ma-55	130	23	inf	inf	PROPN
ma-55	130	24	u∈(h1	u∈(h1	NOUN
ma-55	130	25	0(ω)∩h2(ω)\{0	0(ω)∩h2(ω)\{0	PROPN
ma-55	130	26	}	}	PUNCT
ma-55	130	27	sup	sup	NOUN
ma-55	130	28	λ≥0	λ≥0	PROPN
ma-55	130	29	j(λu	j(λu	PROPN
ma-55	130	30	)	)	PUNCT
ma-55	130	31	.	.	PUNCT
ma-55	131	1	similar	similar	ADJ
ma-55	131	2	to	to	ADP
ma-55	131	3	the	the	DET
ma-55	131	4	result	result	NOUN
ma-55	131	5	in	in	ADP
ma-55	131	6	[	[	X
ma-55	131	7	21	21	NUM
ma-55	131	8	]	]	X
ma-55	131	9	one	one	PRON
ma-55	131	10	has	have	AUX
ma-55	131	11	0	0	NUM
ma-55	131	12	<	<	X
ma-55	131	13	d	d	X
ma-55	131	14	=	=	SYM
ma-55	131	15	inf	inf	PROPN
ma-55	131	16	u∈n	u∈n	NOUN
ma-55	131	17	j(u).now	j(u).now	NOUN
ma-55	131	18	,	,	PUNCT
ma-55	131	19	we	we	PRON
ma-55	131	20	introduce	introduce	VERB
ma-55	131	21	w	w	NOUN
ma-55	131	22	=	=	SYM
ma-55	131	23	{	{	PUNCT
ma-55	131	24	u	u	NOUN
ma-55	131	25	∈	∈	PROPN
ma-55	131	26	h1	h1	NOUN
ma-55	131	27	0(ω	0(ω	ADV
ma-55	131	28	)	)	PUNCT
ma-55	131	29	∩h2(ω	∩h2(ω	ADJ
ma-55	131	30	)	)	PUNCT
ma-55	131	31	;	;	PUNCT
ma-55	131	32	j(u	j(u	PROPN
ma-55	131	33	)	)	PUNCT
ma-55	131	34	<	<	X
ma-55	132	1	d	d	X
ma-55	132	2	}	}	PUNCT
ma-55	132	3	∪	∪	X
ma-55	132	4	{	{	PUNCT
ma-55	132	5	0	0	NUM
ma-55	132	6	}	}	PUNCT
ma-55	132	7	and	and	CCONJ
ma-55	132	8	partition	partition	VERB
ma-55	132	9	it	it	PRON
ma-55	132	10	into	into	ADP
ma-55	132	11	two	two	NUM
ma-55	132	12	sets	set	NOUN
ma-55	132	13	w	w	NOUN
ma-55	132	14	=	=	SYM
ma-55	132	15	w1	w1	NOUN
ma-55	132	16	∪w2	∪w2	NOUN
ma-55	132	17	as	as	SCONJ
ma-55	132	18	follows	follow	VERB
ma-55	132	19	w1	w1	NOUN
ma-55	132	20	=	=	SYM
ma-55	132	21	{	{	PUNCT
ma-55	132	22	u	u	NOUN
ma-55	132	23	∈	∈	PROPN
ma-55	132	24	w	w	NOUN
ma-55	132	25	;	;	PUNCT
ma-55	132	26	(	(	PUNCT
ma-55	132	27	1−	1−	NUM
ma-55	132	28	µ	µ	NUM
ma-55	132	29	∫	∫	PROPN
ma-55	132	30	t	t	PROPN
ma-55	132	31	0	0	NUM
ma-55	132	32	g(s)ds	g(s)ds	PROPN
ma-55	132	33	)	)	PUNCT
ma-55	132	34	|∆u|2	|∆u|2	PUNCT
ma-55	133	1	+	+	SYM
ma-55	133	2	2	2	NUM
ma-55	133	3	p	p	NOUN
ma-55	133	4	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	133	5	+	+	CCONJ
ma-55	133	6	(	(	PUNCT
ma-55	133	7	g	g	PROPN
ma-55	133	8	�	�	PROPN
ma-55	133	9	∇u)(t	∇u)(t	PROPN
ma-55	133	10	)	)	PUNCT
ma-55	134	1	+	+	CCONJ
ma-55	134	2	2	2	NUM
ma-55	134	3	r2	r2	NOUN
ma-55	134	4	|u(t)|rr	|u(t)|rr	X
ma-55	134	5	>	>	SYM
ma-55	134	6	2	2	NUM
ma-55	134	7	r	r	NOUN
ma-55	134	8	∫	∫	PROPN
ma-55	134	9	ω	ω	NUM
ma-55	134	10	|u(t)|r	|u(t)|r	PROPN
ma-55	134	11	ln	ln	ADJ
ma-55	134	12	|u(t)|dx	|u(t)|dx	NOUN
ma-55	134	13	}	}	PUNCT
ma-55	134	14	∪	∪	VERB
ma-55	134	15	{	{	PUNCT
ma-55	134	16	0	0	NUM
ma-55	134	17	}	}	PUNCT
ma-55	134	18	(	(	PUNCT
ma-55	134	19	3.8	3.8	NUM
ma-55	134	20	)	)	PUNCT
ma-55	134	21	and	and	CCONJ
ma-55	134	22	w2	w2	NOUN
ma-55	134	23	=	=	SYM
ma-55	134	24	{	{	PUNCT
ma-55	134	25	u	u	NOUN
ma-55	134	26	∈	∈	PROPN
ma-55	134	27	w	w	NOUN
ma-55	134	28	;	;	PUNCT
ma-55	134	29	(	(	PUNCT
ma-55	134	30	1−	1−	NUM
ma-55	134	31	µ	µ	NUM
ma-55	134	32	∫	∫	PROPN
ma-55	134	33	t	t	PROPN
ma-55	134	34	0	0	NUM
ma-55	134	35	g(s)ds	g(s)ds	PROPN
ma-55	134	36	)	)	PUNCT
ma-55	134	37	|∆u|2	|∆u|2	PUNCT
ma-55	135	1	+	+	SYM
ma-55	135	2	2	2	NUM
ma-55	135	3	p	p	NOUN
ma-55	135	4	|∇u(t)|pp	|∇u(t)|pp	NOUN
ma-55	135	5	+	+	CCONJ
ma-55	135	6	(	(	PUNCT
ma-55	135	7	g	g	PROPN
ma-55	135	8	�	�	PROPN
ma-55	135	9	∇u)(t	∇u)(t	PROPN
ma-55	135	10	)	)	PUNCT
ma-55	136	1	+	+	CCONJ
ma-55	136	2	2	2	NUM
ma-55	136	3	r2	r2	NOUN
ma-55	136	4	|u(t)|rr	|u(t)|rr	X
ma-55	136	5	<	<	X
ma-55	136	6	2	2	NUM
ma-55	136	7	r	r	NOUN
ma-55	136	8	∫	∫	PROPN
ma-55	136	9	ω	ω	NUM
ma-55	136	10	|u(t)|r	|u(t)|r	PROPN
ma-55	136	11	ln	ln	ADJ
ma-55	136	12	|u(t)|dx	|u(t)|dx	PROPN
ma-55	136	13	}	}	PUNCT
ma-55	136	14	.	.	PUNCT
ma-55	137	1	(	(	PUNCT
ma-55	137	2	3.9	3.9	NUM
ma-55	137	3	)	)	PUNCT
ma-55	137	4	so	so	ADV
ma-55	137	5	,	,	PUNCT
ma-55	137	6	we	we	PRON
ma-55	137	7	define	define	VERB
ma-55	137	8	by	by	ADP
ma-55	137	9	w1	w1	NOUN
ma-55	137	10	the	the	DET
ma-55	137	11	set	set	NOUN
ma-55	137	12	of	of	ADP
ma-55	137	13	stability	stability	NOUN
ma-55	137	14	for	for	ADP
ma-55	137	15	the	the	DET
ma-55	137	16	problem	problem	NOUN
ma-55	137	17	(	(	PUNCT
ma-55	137	18	1.5)-(1.7	1.5)-(1.7	NUM
ma-55	137	19	)	)	PUNCT
ma-55	137	20	,	,	PUNCT
ma-55	137	21	and	and	CCONJ
ma-55	137	22	before	before	ADP
ma-55	137	23	starting	start	VERB
ma-55	137	24	the	the	DET
ma-55	137	25	sectionof	sectionof	PROPN
ma-55	137	26	existence	existence	NOUN
ma-55	137	27	and	and	CCONJ
ma-55	137	28	uniqueness	uniqueness	NOUN
ma-55	137	29	of	of	ADP
ma-55	137	30	solution	solution	NOUN
ma-55	137	31	,	,	PUNCT
ma-55	137	32	we	we	PRON
ma-55	137	33	will	will	AUX
ma-55	137	34	prove	prove	VERB
ma-55	137	35	thatw1	thatw1	PROPN
ma-55	137	36	is	be	AUX
ma-55	137	37	invariant	invariant	ADJ
ma-55	137	38	set	set	NOUN
ma-55	137	39	for	for	ADP
ma-55	137	40	sub	sub	ADJ
ma-55	137	41	-	-	ADJ
ma-55	137	42	critical	critical	ADJ
ma-55	137	43	initialenergy	initialenergy	NOUN
ma-55	137	44	.	.	PUNCT
ma-55	138	1	proposition	proposition	NOUN
ma-55	138	2	1	1	NUM
ma-55	138	3	.	.	PUNCT
ma-55	139	1	let	let	VERB
ma-55	139	2	u0	u0	PROPN
ma-55	139	3	∈	∈	PROPN
ma-55	139	4	w1	w1	NOUN
ma-55	139	5	and	and	CCONJ
ma-55	139	6	u1	u1	PROPN
ma-55	139	7	∈	∈	PROPN
ma-55	139	8	h1	h1	NOUN
ma-55	139	9	0(ω	0(ω	NUM
ma-55	139	10	)	)	PUNCT
ma-55	139	11	.	.	PUNCT
ma-55	140	1	if	if	SCONJ
ma-55	140	2	e(0	e(0	NOUN
ma-55	140	3	)	)	PUNCT
ma-55	140	4	<	<	X
ma-55	141	1	d	d	X
ma-55	141	2	then	then	ADV
ma-55	141	3	u(t	u(t	NOUN
ma-55	141	4	)	)	PUNCT
ma-55	141	5	∈	∈	PROPN
ma-55	141	6	w1	w1	NOUN
ma-55	141	7	.	.	PUNCT
ma-55	142	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	142	2	eur	eur	PROPN
ma-55	142	3	.	.	PUNCT
ma-55	143	1	j.	j.	PROPN
ma-55	143	2	math	math	PROPN
ma-55	143	3	.	.	PUNCT
ma-55	144	1	anal	anal	PROPN
ma-55	144	2	.	.	PUNCT
ma-55	145	1	10.28924	10.28924	NUM
ma-55	145	2	/	/	SYM
ma-55	145	3	ada	ada	PROPN
ma-55	145	4	/	/	SYM
ma-55	145	5	ma.2.5	ma.2.5	PROPN
ma-55	145	6	7	7	NUM
ma-55	145	7	proof	proof	NOUN
ma-55	145	8	.	.	PUNCT
ma-55	146	1	let	let	VERB
ma-55	146	2	t	t	PROPN
ma-55	146	3	>	>	X
ma-55	146	4	0	0	PUNCT
ma-55	146	5	be	be	AUX
ma-55	146	6	the	the	DET
ma-55	146	7	maximum	maximum	ADJ
ma-55	146	8	existence	existence	NOUN
ma-55	146	9	time	time	NOUN
ma-55	146	10	.	.	PUNCT
ma-55	147	1	from	from	ADP
ma-55	147	2	(	(	PUNCT
ma-55	147	3	3.2	3.2	NUM
ma-55	147	4	)	)	PUNCT
ma-55	147	5	we	we	PRON
ma-55	147	6	get	get	VERB
ma-55	147	7	e(t	e(t	NOUN
ma-55	147	8	)	)	PUNCT
ma-55	147	9	≤	≤	NOUN
ma-55	148	1	e(0	e(0	NOUN
ma-55	148	2	)	)	PUNCT
ma-55	148	3	<	<	X
ma-55	149	1	d	d	X
ma-55	149	2	,	,	PUNCT
ma-55	149	3	for	for	ADP
ma-55	149	4	all	all	DET
ma-55	149	5	t	t	NOUN
ma-55	149	6	∈	∈	PROPN
ma-55	150	1	[	[	X
ma-55	150	2	0	0	NUM
ma-55	150	3	,	,	PUNCT
ma-55	150	4	t	t	NOUN
ma-55	150	5	)	)	PUNCT
ma-55	150	6	.	.	PUNCT
ma-55	151	1	and	and	CCONJ
ma-55	151	2	then	then	ADV
ma-55	151	3	,	,	PUNCT
ma-55	151	4	1	1	NUM
ma-55	151	5	2	2	NUM
ma-55	151	6	∫	∫	PROPN
ma-55	151	7	ω	ω	PROPN
ma-55	151	8	|ut(t)|2	|ut(t)|2	PROPN
ma-55	151	9	dx	dx	PROPN
ma-55	152	1	+	+	X
ma-55	152	2	j(u(t	j(u(t	PROPN
ma-55	152	3	)	)	PUNCT
ma-55	152	4	)	)	PUNCT
ma-55	153	1	<	<	X
ma-55	154	1	d	d	X
ma-55	154	2	,	,	PUNCT
ma-55	154	3	for	for	ADP
ma-55	154	4	all	all	DET
ma-55	154	5	t	t	NOUN
ma-55	154	6	∈	∈	PROPN
ma-55	154	7	[	[	X
ma-55	154	8	0	0	NUM
ma-55	154	9	,	,	PUNCT
ma-55	154	10	t	t	NOUN
ma-55	154	11	)	)	PUNCT
ma-55	154	12	.	.	PUNCT
ma-55	155	1	(	(	PUNCT
ma-55	155	2	3.10	3.10	NUM
ma-55	155	3	)	)	PUNCT
ma-55	155	4	arguing	argue	VERB
ma-55	155	5	by	by	ADP
ma-55	155	6	contradiction	contradiction	NOUN
ma-55	155	7	,	,	PUNCT
ma-55	155	8	we	we	PRON
ma-55	155	9	suppose	suppose	VERB
ma-55	155	10	that	that	SCONJ
ma-55	155	11	there	there	PRON
ma-55	155	12	exists	exist	VERB
ma-55	155	13	a	a	DET
ma-55	155	14	first	first	ADJ
ma-55	155	15	t0	t0	PROPN
ma-55	155	16	∈	∈	PROPN
ma-55	155	17	(	(	PUNCT
ma-55	155	18	0	0	NUM
ma-55	155	19	,	,	PUNCT
ma-55	155	20	t	t	NOUN
ma-55	155	21	)	)	PUNCT
ma-55	155	22	such	such	ADJ
ma-55	155	23	that	that	DET
ma-55	155	24	i(u(t0	i(u(t0	NOUN
ma-55	155	25	)	)	PUNCT
ma-55	155	26	)	)	PUNCT
ma-55	156	1	=	=	SYM
ma-55	156	2	0and	0and	PROPN
ma-55	156	3	i(u(t	i(u(t	VERB
ma-55	156	4	)	)	PUNCT
ma-55	156	5	)	)	PUNCT
ma-55	156	6	>	>	X
ma-55	156	7	0	0	PUNCT
ma-55	157	1	for	for	ADP
ma-55	157	2	all	all	DET
ma-55	157	3	0	0	NUM
ma-55	157	4	≤	≤	NOUN
ma-55	157	5	t	t	PROPN
ma-55	157	6	<	<	X
ma-55	157	7	t0	t0	PROPN
ma-55	157	8	,	,	PUNCT
ma-55	157	9	that	that	ADV
ma-55	157	10	is	is	ADV
ma-55	157	11	,	,	PUNCT
ma-55	157	12	(	(	PUNCT
ma-55	157	13	1−	1−	NUM
ma-55	157	14	µ	µ	NUM
ma-55	157	15	∫	∫	PROPN
ma-55	157	16	t	t	PROPN
ma-55	157	17	0	0	NUM
ma-55	157	18	g(s)ds	g(s)ds	PROPN
ma-55	157	19	)	)	PUNCT
ma-55	157	20	|∆u(t0)|2	|∆u(t0)|2	NOUN
ma-55	157	21	+	+	CCONJ
ma-55	157	22	2	2	NUM
ma-55	157	23	p	p	NOUN
ma-55	157	24	|∇u(t0)|pp	|∇u(t0)|pp	PUNCT
ma-55	157	25	+	+	CCONJ
ma-55	157	26	(	(	PUNCT
ma-55	157	27	g	g	PROPN
ma-55	157	28	�	�	PROPN
ma-55	157	29	∇u)(t0	∇u)(t0	PROPN
ma-55	157	30	)	)	PUNCT
ma-55	157	31	+	+	CCONJ
ma-55	157	32	2	2	NUM
ma-55	157	33	r2	r2	NOUN
ma-55	157	34	|u(t0)|rr	|u(t0)|rr	NOUN
ma-55	157	35	=	=	SYM
ma-55	157	36	2	2	NUM
ma-55	157	37	r	r	NOUN
ma-55	157	38	∫	∫	PROPN
ma-55	157	39	ω	ω	X
ma-55	157	40	|u(t0)|r	|u(t0)|r	PROPN
ma-55	157	41	ln	ln	ADJ
ma-55	157	42	|u(t0)|dx	|u(t0)|dx	NOUN
ma-55	157	43	from	from	ADP
ma-55	157	44	the	the	DET
ma-55	157	45	definition	definition	NOUN
ma-55	157	46	of	of	ADP
ma-55	157	47	n	n	PROPN
ma-55	157	48	,	,	PUNCT
ma-55	157	49	we	we	PRON
ma-55	157	50	have	have	VERB
ma-55	157	51	that	that	DET
ma-55	157	52	u(t0	u(t0	NOUN
ma-55	157	53	)	)	PUNCT
ma-55	157	54	∈	∈	PROPN
ma-55	157	55	n	n	NOUN
ma-55	157	56	,	,	PUNCT
ma-55	157	57	which	which	PRON
ma-55	157	58	leads	lead	VERB
ma-55	157	59	to	to	ADP
ma-55	157	60	j(u(t0	j(u(t0	PROPN
ma-55	157	61	)	)	PUNCT
ma-55	157	62	)	)	PUNCT
ma-55	157	63	≥	≥	PROPN
ma-55	157	64	inf	inf	NOUN
ma-55	157	65	u(t)∈n	u(t)∈n	PROPN
ma-55	157	66	j(u(t	j(u(t	PROPN
ma-55	157	67	)	)	PUNCT
ma-55	157	68	)	)	PUNCT
ma-55	158	1	=	=	PUNCT
ma-55	158	2	d.	d.	NOUN
ma-55	158	3	we	we	PRON
ma-55	158	4	deduce	deduce	VERB
ma-55	158	5	1	1	NUM
ma-55	158	6	2	2	NUM
ma-55	158	7	∫	∫	PROPN
ma-55	158	8	ω	ω	PROPN
ma-55	158	9	|ut(t0)|2	|ut(t0)|2	PROPN
ma-55	158	10	dx	dx	PROPN
ma-55	159	1	+	+	CCONJ
ma-55	159	2	j(u(t0	j(u(t0	PROPN
ma-55	159	3	)	)	PUNCT
ma-55	159	4	)	)	PUNCT
ma-55	160	1	≥	≥	NOUN
ma-55	161	1	d	d	NOUN
ma-55	161	2	,	,	PUNCT
ma-55	161	3	which	which	PRON
ma-55	161	4	contradicts	contradict	VERB
ma-55	161	5	with	with	ADP
ma-55	161	6	(	(	PUNCT
ma-55	161	7	3.10	3.10	NUM
ma-55	161	8	)	)	PUNCT
ma-55	161	9	.	.	PUNCT
ma-55	162	1	then	then	ADV
ma-55	162	2	u(t	u(t	NOUN
ma-55	162	3	)	)	PUNCT
ma-55	162	4	∈	∈	PROPN
ma-55	162	5	w1	w1	NOUN
ma-55	162	6	for	for	ADP
ma-55	162	7	all	all	DET
ma-55	162	8	t	t	NOUN
ma-55	162	9	∈	∈	PROPN
ma-55	163	1	[	[	X
ma-55	163	2	0	0	NUM
ma-55	163	3	,	,	PUNCT
ma-55	163	4	t	t	NOUN
ma-55	163	5	)	)	PUNCT
ma-55	163	6	.	.	PUNCT
ma-55	164	1	�	�	PROPN
ma-55	164	2	4	4	NUM
ma-55	164	3	.	.	PUNCT
ma-55	164	4	existence	existence	NOUN
ma-55	164	5	of	of	ADP
ma-55	164	6	strong	strong	ADJ
ma-55	164	7	solutions	solution	NOUN
ma-55	164	8	next	next	ADV
ma-55	164	9	,	,	PUNCT
ma-55	164	10	we	we	PRON
ma-55	164	11	shall	shall	AUX
ma-55	164	12	state	state	VERB
ma-55	164	13	the	the	DET
ma-55	164	14	main	main	ADJ
ma-55	164	15	results	result	NOUN
ma-55	164	16	of	of	ADP
ma-55	164	17	this	this	DET
ma-55	164	18	paper	paper	NOUN
ma-55	164	19	.	.	PUNCT
ma-55	165	1	theorem	theorem	VERB
ma-55	165	2	4.1	4.1	NUM
ma-55	165	3	.	.	PUNCT
ma-55	166	1	consider	consider	VERB
ma-55	166	2	the	the	DET
ma-55	166	3	space	space	NOUN
ma-55	166	4	h3	h3	NOUN
ma-55	166	5	γ(ω	γ(ω	PROPN
ma-55	166	6	)	)	PUNCT
ma-55	167	1	=	=	PRON
ma-55	167	2	{	{	PUNCT
ma-55	167	3	u	u	X
ma-55	167	4	∈	∈	PROPN
ma-55	167	5	h3(ω)|u	h3(ω)|u	X
ma-55	167	6	=	=	PUNCT
ma-55	168	1	∆u	∆u	PROPN
ma-55	168	2	=	=	SYM
ma-55	168	3	0	0	NUM
ma-55	168	4	on	on	ADP
ma-55	168	5	γ	γ	NOUN
ma-55	168	6	}	}	PUNCT
ma-55	168	7	.	.	PUNCT
ma-55	169	1	if	if	SCONJ
ma-55	169	2	u0	u0	PROPN
ma-55	169	3	∈	∈	PROPN
ma-55	169	4	w1	w1	NOUN
ma-55	169	5	∩h3	∩h3	PROPN
ma-55	169	6	γ(ω	γ(ω	PROPN
ma-55	169	7	)	)	PUNCT
ma-55	169	8	,	,	PUNCT
ma-55	169	9	j(u0	j(u0	NOUN
ma-55	169	10	)	)	PUNCT
ma-55	169	11	<	<	X
ma-55	170	1	d	d	X
ma-55	170	2	,	,	PUNCT
ma-55	170	3	u1	u1	PROPN
ma-55	170	4	∈	∈	PROPN
ma-55	170	5	h1	h1	NOUN
ma-55	170	6	0(ω	0(ω	NUM
ma-55	170	7	)	)	PUNCT
ma-55	170	8	and	and	CCONJ
ma-55	170	9	the	the	DET
ma-55	170	10	hypothesis	hypothesis	NOUN
ma-55	170	11	(	(	PUNCT
ma-55	170	12	h1)-(h4	h1)-(h4	NOUN
ma-55	170	13	)	)	PUNCT
ma-55	170	14	holds	hold	VERB
ma-55	170	15	,	,	PUNCT
ma-55	170	16	then	then	ADV
ma-55	170	17	there	there	PRON
ma-55	170	18	exists	exist	VERB
ma-55	170	19	a	a	DET
ma-55	170	20	function	function	NOUN
ma-55	170	21	u	u	NOUN
ma-55	170	22	:	:	PUNCT
ma-55	170	23	ω×	ω×	PUNCT
ma-55	170	24	(	(	PUNCT
ma-55	170	25	0	0	NUM
ma-55	170	26	,	,	PUNCT
ma-55	170	27	t	t	NOUN
ma-55	170	28	)	)	PUNCT
ma-55	170	29	→	→	PUNCT
ma-55	170	30	r	r	NOUN
ma-55	170	31	such	such	ADJ
ma-55	170	32	that	that	SCONJ
ma-55	170	33	u	u	PROPN
ma-55	170	34	∈	∈	PROPN
ma-55	170	35	l∞(0	l∞(0	PRON
ma-55	170	36	,	,	PUNCT
ma-55	170	37	t	t	PROPN
ma-55	170	38	;	;	PUNCT
ma-55	170	39	(	(	PUNCT
ma-55	170	40	h1	h1	VERB
ma-55	170	41	0(ω	0(ω	ADJ
ma-55	170	42	)	)	PUNCT
ma-55	170	43	∩h2(ω	∩h2(ω	ADJ
ma-55	170	44	)	)	PUNCT
ma-55	170	45	)	)	PUNCT
ma-55	170	46	)	)	PUNCT
ma-55	170	47	∩	∩	NOUN
ma-55	170	48	l∞(0	l∞(0	NUM
ma-55	170	49	,	,	PUNCT
ma-55	170	50	t	t	NOUN
ma-55	170	51	;	;	PUNCT
ma-55	170	52	h3	h3	NOUN
ma-55	170	53	γ(ω	γ(ω	PROPN
ma-55	170	54	)	)	PUNCT
ma-55	170	55	)	)	PUNCT
ma-55	170	56	,	,	PUNCT
ma-55	170	57	(	(	PUNCT
ma-55	170	58	4.1	4.1	NUM
ma-55	170	59	)	)	PUNCT
ma-55	170	60	ut	ut	PROPN
ma-55	170	61	∈	∈	PROPN
ma-55	170	62	l∞(0	l∞(0	PROPN
ma-55	170	63	,	,	PUNCT
ma-55	170	64	t	t	NOUN
ma-55	170	65	;	;	PUNCT
ma-55	170	66	l2(ω	l2(ω	NOUN
ma-55	170	67	)	)	PUNCT
ma-55	170	68	)	)	PUNCT
ma-55	170	69	∩	∩	ADJ
ma-55	170	70	l2(0	l2(0	NOUN
ma-55	170	71	,	,	PUNCT
ma-55	170	72	t	t	PROPN
ma-55	170	73	;	;	PUNCT
ma-55	170	74	h1	h1	VERB
ma-55	170	75	0(ω	0(ω	NOUN
ma-55	170	76	)	)	PUNCT
ma-55	170	77	∩h2(ω	∩h2(ω	ADJ
ma-55	170	78	)	)	PUNCT
ma-55	170	79	)	)	PUNCT
ma-55	170	80	,	,	PUNCT
ma-55	170	81	(	(	PUNCT
ma-55	170	82	4.2	4.2	NUM
ma-55	170	83	)	)	PUNCT
ma-55	170	84	utt	utt	NOUN
ma-55	170	85	∈	∈	PROPN
ma-55	170	86	l∞(0	l∞(0	PRON
ma-55	170	87	,	,	PUNCT
ma-55	170	88	t	t	PROPN
ma-55	170	89	;	;	PUNCT
ma-55	170	90	h−1(ω	h−1(ω	PROPN
ma-55	170	91	)	)	PUNCT
ma-55	170	92	)	)	PUNCT
ma-55	170	93	,	,	PUNCT
ma-55	170	94	(	(	PUNCT
ma-55	170	95	4.3	4.3	NUM
ma-55	170	96	)	)	PUNCT
ma-55	170	97	ut(t	ut(t	NOUN
ma-55	170	98	)	)	PUNCT
ma-55	171	1	∈	∈	PROPN
ma-55	171	2	k	k	PROPN
ma-55	171	3	a.e	a.e	PROPN
ma-55	171	4	.	.	PROPN
ma-55	172	1	in	in	ADP
ma-55	172	2	[	[	X
ma-55	172	3	0	0	NUM
ma-55	172	4	,	,	PUNCT
ma-55	172	5	t	t	X
ma-55	172	6	]	]	PUNCT
ma-55	172	7	,	,	PUNCT
ma-55	172	8	(	(	PUNCT
ma-55	172	9	4.4	4.4	NUM
ma-55	172	10	)	)	PUNCT
ma-55	172	11	∫	∫	PROPN
ma-55	172	12	t	t	NOUN
ma-55	172	13	0	0	NUM
ma-55	172	14	[	[	PUNCT
ma-55	172	15	〈	〈	PROPN
ma-55	172	16	utt	utt	NOUN
ma-55	172	17	,	,	PUNCT
ma-55	172	18	v	v	ADP
ma-55	172	19	−	−	NOUN
ma-55	172	20	ut〉+	ut〉+	ADJ
ma-55	172	21	(	(	PUNCT
ma-55	172	22	∆2u	∆2u	NOUN
ma-55	172	23	,	,	PUNCT
ma-55	172	24	v	v	ADP
ma-55	172	25	−	−	PROPN
ma-55	172	26	ut)−	ut)−	PROPN
ma-55	172	27	(	(	PUNCT
ma-55	172	28	∆pu	∆pu	PROPN
ma-55	172	29	,	,	PUNCT
ma-55	172	30	v	v	ADP
ma-55	172	31	−	−	PROPN
ma-55	172	32	ut	ut	PROPN
ma-55	172	33	)	)	PUNCT
ma-55	172	34	+	+	CCONJ
ma-55	172	35	(	(	PUNCT
ma-55	172	36	∫	∫	PROPN
ma-55	172	37	t	t	PROPN
ma-55	172	38	0	0	NUM
ma-55	172	39	g(t	g(t	PROPN
ma-55	172	40	−	−	PROPN
ma-55	172	41	s)∆u(s)ds	s)∆u(s)ds	PROPN
ma-55	172	42	,	,	PUNCT
ma-55	172	43	v	v	ADP
ma-55	172	44	−	−	PROPN
ma-55	172	45	ut	ut	PROPN
ma-55	172	46	)	)	PUNCT
ma-55	172	47	−	−	PROPN
ma-55	172	48	(	(	PUNCT
ma-55	172	49	∆ut	∆ut	NOUN
ma-55	172	50	,	,	PUNCT
ma-55	172	51	v	v	ADP
ma-55	172	52	−	−	PROPN
ma-55	172	53	ut	ut	PROPN
ma-55	172	54	)	)	PUNCT
ma-55	172	55	−	−	PROPN
ma-55	172	56	(	(	PUNCT
ma-55	172	57	|u|r−2u	|u|r−2u	PROPN
ma-55	172	58	ln	ln	PROPN
ma-55	172	59	|u|	|u|	PROPN
ma-55	172	60	,	,	PUNCT
ma-55	172	61	v	v	ADP
ma-55	172	62	−	−	PROPN
ma-55	172	63	ut	ut	PROPN
ma-55	172	64	)	)	PUNCT
ma-55	172	65	]	]	PUNCT
ma-55	172	66	≥	≥	NOUN
ma-55	172	67	0	0	NUM
ma-55	172	68	,	,	PUNCT
ma-55	172	69	(	(	PUNCT
ma-55	172	70	4.5	4.5	NUM
ma-55	172	71	)	)	PUNCT
ma-55	172	72	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	172	73	eur	eur	PROPN
ma-55	172	74	.	.	PUNCT
ma-55	173	1	j.	j.	PROPN
ma-55	173	2	math	math	PROPN
ma-55	173	3	.	.	PUNCT
ma-55	174	1	anal	anal	PROPN
ma-55	174	2	.	.	PUNCT
ma-55	175	1	10.28924	10.28924	NUM
ma-55	175	2	/	/	SYM
ma-55	175	3	ada	ada	PROPN
ma-55	175	4	/	/	SYM
ma-55	175	5	ma.2.5	ma.2.5	PROPN
ma-55	175	6	8	8	NUM
ma-55	175	7	for	for	ADP
ma-55	175	8	all	all	DET
ma-55	175	9	v	v	ADP
ma-55	175	10	∈	∈	PROPN
ma-55	175	11	l2(0	l2(0	NOUN
ma-55	175	12	,	,	PUNCT
ma-55	175	13	t	t	PROPN
ma-55	175	14	;	;	PUNCT
ma-55	175	15	h1	h1	NOUN
ma-55	175	16	0(ω	0(ω	ADJ
ma-55	175	17	)	)	PUNCT
ma-55	175	18	)	)	PUNCT
ma-55	176	1	,	,	PUNCT
ma-55	176	2	v(t	v(t	NOUN
ma-55	176	3	)	)	PUNCT
ma-55	176	4	∈	∈	PROPN
ma-55	176	5	k	k	PROPN
ma-55	176	6	a.e	a.e	PROPN
ma-55	176	7	.	.	PROPN
ma-55	176	8	in	in	ADP
ma-55	176	9	t	t	PROPN
ma-55	176	10	and	and	CCONJ
ma-55	176	11	initial	initial	ADJ
ma-55	176	12	data	datum	NOUN
ma-55	176	13	u(0	u(0	PROPN
ma-55	176	14	)	)	PUNCT
ma-55	176	15	=	=	PUNCT
ma-55	177	1	u0	u0	PROPN
ma-55	177	2	,	,	PUNCT
ma-55	177	3	ut(0	ut(0	PROPN
ma-55	177	4	)	)	PUNCT
ma-55	177	5	=	=	SYM
ma-55	177	6	u1	u1	NOUN
ma-55	177	7	.	.	PUNCT
ma-55	178	1	the	the	DET
ma-55	178	2	proof	proof	NOUN
ma-55	178	3	of	of	ADP
ma-55	178	4	theorem	theorem	ADJ
ma-55	178	5	4.1	4.1	NUM
ma-55	178	6	is	be	AUX
ma-55	178	7	given	give	VERB
ma-55	178	8	in	in	ADP
ma-55	178	9	section	section	NOUN
ma-55	178	10	5	5	NUM
ma-55	178	11	by	by	ADP
ma-55	178	12	the	the	DET
ma-55	178	13	penalization	penalization	NOUN
ma-55	178	14	method	method	NOUN
ma-55	178	15	.	.	PUNCT
ma-55	179	1	it	it	PRON
ma-55	179	2	consists	consist	VERB
ma-55	179	3	in	in	ADP
ma-55	179	4	con	con	NOUN
ma-55	179	5	-	-	PUNCT
ma-55	179	6	sidering	sidere	VERB
ma-55	179	7	a	a	DET
ma-55	179	8	perturbation	perturbation	NOUN
ma-55	179	9	of	of	ADP
ma-55	179	10	the	the	DET
ma-55	179	11	problem	problem	NOUN
ma-55	179	12	(	(	PUNCT
ma-55	179	13	1.5	1.5	NUM
ma-55	179	14	)	)	PUNCT
ma-55	179	15	adding	add	VERB
ma-55	179	16	a	a	DET
ma-55	179	17	singular	singular	ADJ
ma-55	179	18	term	term	NOUN
ma-55	179	19	called	call	VERB
ma-55	179	20	penalization	penalization	NOUN
ma-55	179	21	,	,	PUNCT
ma-55	179	22	dependingon	dependingon	NOUN
ma-55	179	23	a	a	DET
ma-55	179	24	parameter	parameter	NOUN
ma-55	179	25	ε	ε	PROPN
ma-55	179	26	>	>	X
ma-55	179	27	0	0	X
ma-55	179	28	.	.	PUNCT
ma-55	180	1	we	we	PRON
ma-55	180	2	solve	solve	VERB
ma-55	180	3	the	the	DET
ma-55	180	4	mixed	mixed	ADJ
ma-55	180	5	problem	problem	NOUN
ma-55	180	6	in	in	ADP
ma-55	180	7	q	q	NOUN
ma-55	180	8	for	for	ADP
ma-55	180	9	the	the	DET
ma-55	180	10	penalization	penalization	NOUN
ma-55	180	11	operator	operator	NOUN
ma-55	180	12	and	and	CCONJ
ma-55	180	13	theestimates	theestimate	NOUN
ma-55	180	14	obtained	obtain	VERB
ma-55	180	15	for	for	ADP
ma-55	180	16	the	the	DET
ma-55	180	17	local	local	ADJ
ma-55	180	18	solution	solution	NOUN
ma-55	180	19	of	of	ADP
ma-55	180	20	the	the	DET
ma-55	180	21	penalized	penalize	VERB
ma-55	180	22	equation	equation	NOUN
ma-55	180	23	,	,	PUNCT
ma-55	180	24	allow	allow	VERB
ma-55	180	25	to	to	PART
ma-55	180	26	pass	pass	VERB
ma-55	180	27	to	to	ADP
ma-55	180	28	limits	limit	NOUN
ma-55	180	29	,	,	PUNCT
ma-55	180	30	when	when	SCONJ
ma-55	180	31	εgoes	εgoe	NOUN
ma-55	180	32	to	to	ADP
ma-55	180	33	zero	zero	NUM
ma-55	180	34	,	,	PUNCT
ma-55	180	35	in	in	ADP
ma-55	180	36	order	order	NOUN
ma-55	180	37	to	to	PART
ma-55	180	38	obtain	obtain	VERB
ma-55	180	39	a	a	DET
ma-55	180	40	function	function	NOUN
ma-55	180	41	u	u	NOUN
ma-55	180	42	which	which	PRON
ma-55	180	43	is	be	AUX
ma-55	180	44	the	the	DET
ma-55	180	45	solution	solution	NOUN
ma-55	180	46	of	of	ADP
ma-55	180	47	our	our	PRON
ma-55	180	48	problem	problem	NOUN
ma-55	180	49	.	.	PUNCT
ma-55	181	1	first	first	ADV
ma-55	181	2	of	of	ADP
ma-55	181	3	all	all	PRON
ma-55	181	4	,	,	PUNCT
ma-55	181	5	letus	letus	PROPN
ma-55	181	6	consider	consider	VERB
ma-55	181	7	the	the	DET
ma-55	181	8	penalization	penalization	NOUN
ma-55	181	9	operator	operator	NOUN
ma-55	181	10	β	β	NOUN
ma-55	181	11	:	:	PUNCT
ma-55	181	12	h1	h1	VERB
ma-55	181	13	0(ω	0(ω	NOUN
ma-55	181	14	)	)	PUNCT
ma-55	181	15	−→	−→	NOUN
ma-55	181	16	h−1(ω	h−1(ω	PROPN
ma-55	181	17	)	)	PUNCT
ma-55	181	18	associated	associate	VERB
ma-55	181	19	to	to	ADP
ma-55	181	20	the	the	DET
ma-55	181	21	closed	closed	ADJ
ma-55	181	22	convex	convex	NOUN
ma-55	181	23	set	set	VERB
ma-55	181	24	k	k	PROPN
ma-55	181	25	,	,	PUNCT
ma-55	181	26	cf	cf	NOUN
ma-55	181	27	.	.	PUNCT
ma-55	182	1	lions	lion	NOUN
ma-55	183	1	[	[	X
ma-55	183	2	16	16	NUM
ma-55	183	3	]	]	PUNCT
ma-55	183	4	,	,	PUNCT
ma-55	183	5	p.	p.	NOUN
ma-55	183	6	370	370	NUM
ma-55	183	7	.	.	PUNCT
ma-55	184	1	the	the	DET
ma-55	184	2	operator	operator	NOUN
ma-55	184	3	β	β	X
ma-55	184	4	is	be	AUX
ma-55	184	5	monotonous	monotonous	ADJ
ma-55	184	6	,	,	PUNCT
ma-55	184	7	hemicontinuous	hemicontinuous	ADJ
ma-55	184	8	,	,	PUNCT
ma-55	184	9	takes	take	VERB
ma-55	184	10	bounded	bounded	ADJ
ma-55	184	11	sets	set	NOUN
ma-55	184	12	of	of	ADP
ma-55	184	13	h1	h1	NOUN
ma-55	184	14	0(ω	0(ω	NOUN
ma-55	184	15	)	)	PUNCT
ma-55	184	16	into	into	ADP
ma-55	184	17	bounded	bounded	ADJ
ma-55	184	18	sets	set	NOUN
ma-55	184	19	of	of	ADP
ma-55	184	20	h−1(ω	h−1(ω	PROPN
ma-55	184	21	)	)	PUNCT
ma-55	184	22	,	,	PUNCT
ma-55	184	23	its	its	PRON
ma-55	184	24	kernel	kernel	NOUN
ma-55	184	25	is	be	AUX
ma-55	184	26	k	k	PROPN
ma-55	184	27	and	and	CCONJ
ma-55	184	28	β	β	ADJ
ma-55	184	29	:	:	PUNCT
ma-55	184	30	l2(0	l2(0	PROPN
ma-55	184	31	,	,	PUNCT
ma-55	184	32	t	t	PROPN
ma-55	184	33	;	;	PUNCT
ma-55	184	34	h1	h1	NOUN
ma-55	184	35	0(ω	0(ω	ADJ
ma-55	184	36	)	)	PUNCT
ma-55	184	37	)	)	PUNCT
ma-55	185	1	−→	−→	NOUN
ma-55	185	2	l2(0	l2(0	NOUN
ma-55	185	3	,	,	PUNCT
ma-55	185	4	t	t	NOUN
ma-55	185	5	;	;	PUNCT
ma-55	185	6	(	(	PUNCT
ma-55	185	7	h−1(ω	h−1(ω	PROPN
ma-55	185	8	)	)	PUNCT
ma-55	185	9	)	)	PUNCT
ma-55	185	10	is	be	AUX
ma-55	185	11	monotone	monotone	ADJ
ma-55	185	12	and	and	CCONJ
ma-55	185	13	hemicontinous	hemicontinous	ADJ
ma-55	185	14	.	.	PUNCT
ma-55	186	1	the	the	DET
ma-55	186	2	penalized	penalize	VERB
ma-55	186	3	problem	problem	NOUN
ma-55	186	4	associated	associate	VERB
ma-55	186	5	with	with	ADP
ma-55	186	6	the	the	DET
ma-55	186	7	variational	variational	PROPN
ma-55	186	8	inequality(1.5)-(1.7	inequality(1.5)-(1.7	PROPN
ma-55	186	9	)	)	PUNCT
ma-55	186	10	,	,	PUNCT
ma-55	186	11	consists	consist	VERB
ma-55	186	12	in	in	ADP
ma-55	186	13	given	give	VERB
ma-55	186	14	0	0	NUM
ma-55	186	15	<	<	X
ma-55	186	16	ε	ε	X
ma-55	186	17	<	<	X
ma-55	186	18	1	1	NUM
ma-55	186	19	,	,	PUNCT
ma-55	186	20	find	find	VERB
ma-55	186	21	uε	uε	INTJ
ma-55	186	22	satisfying	satisfy	VERB
ma-55	186	23	uεtt	uεtt	PROPN
ma-55	187	1	+	+	CCONJ
ma-55	187	2	∆2uε	∆2uε	ADJ
ma-55	188	1	−	−	NOUN
ma-55	188	2	∆pu	∆pu	NOUN
ma-55	188	3	ε	ε	PROPN
ma-55	188	4	+	+	CCONJ
ma-55	188	5	∫	∫	PROPN
ma-55	188	6	t	t	PROPN
ma-55	188	7	0	0	NUM
ma-55	188	8	g(t	g(t	PROPN
ma-55	188	9	−	−	PROPN
ma-55	188	10	s)∆uε(s)ds	s)∆uε(s)ds	PROPN
ma-55	189	1	−	−	PROPN
ma-55	189	2	∆uεt	∆uεt	NOUN
ma-55	190	1	+	+	CCONJ
ma-55	190	2	1	1	NUM
ma-55	190	3	ε	ε	PROPN
ma-55	190	4	(	(	PUNCT
ma-55	190	5	β(uεt	β(uεt	NOUN
ma-55	190	6	)	)	PUNCT
ma-55	190	7	)	)	PUNCT
ma-55	190	8	−	−	PROPN
ma-55	191	1	|uε|r−2uε	|uε|r−2uε	PROPN
ma-55	191	2	ln	ln	ADJ
ma-55	191	3	|u|	|u|	PROPN
ma-55	191	4	=	=	SYM
ma-55	191	5	0	0	NUM
ma-55	191	6	,	,	PUNCT
ma-55	191	7	in	in	ADP
ma-55	191	8	q	q	PROPN
ma-55	191	9	(	(	PUNCT
ma-55	191	10	4.6	4.6	NUM
ma-55	191	11	)	)	PUNCT
ma-55	191	12	and	and	CCONJ
ma-55	191	13	uε(x	uε(x	NOUN
ma-55	191	14	,	,	PUNCT
ma-55	191	15	0	0	NUM
ma-55	191	16	)	)	PUNCT
ma-55	191	17	=	=	SYM
ma-55	191	18	uε0(x	uε0(x	NOUN
ma-55	191	19	)	)	PUNCT
ma-55	191	20	,	,	PUNCT
ma-55	191	21	uεt	uεt	INTJ
ma-55	191	22	(	(	PUNCT
ma-55	191	23	x	x	NOUN
ma-55	191	24	,	,	PUNCT
ma-55	191	25	0	0	NUM
ma-55	191	26	)	)	PUNCT
ma-55	191	27	=	=	SYM
ma-55	191	28	uε1(x	uε1(x	NOUN
ma-55	191	29	)	)	PUNCT
ma-55	191	30	in	in	ADP
ma-55	191	31	ω	ω	PROPN
ma-55	191	32	.	.	PUNCT
ma-55	192	1	uε(x.t	uε(x.t	PROPN
ma-55	192	2	)	)	PUNCT
ma-55	193	1	=	=	PUNCT
ma-55	193	2	∆uε(x	∆uε(x	PROPN
ma-55	193	3	,	,	PUNCT
ma-55	193	4	t	t	NOUN
ma-55	193	5	)	)	PUNCT
ma-55	193	6	=	=	SYM
ma-55	193	7	0	0	NUM
ma-55	193	8	on	on	ADP
ma-55	193	9	∂ω×	∂ω×	NOUN
ma-55	193	10	r+	r+	PROPN
ma-55	193	11	.	.	PUNCT
ma-55	194	1	(	(	PUNCT
ma-55	194	2	4.7	4.7	NUM
ma-55	194	3	)	)	PUNCT
ma-55	194	4	definition	definition	NOUN
ma-55	194	5	4.2	4.2	NUM
ma-55	194	6	.	.	PUNCT
ma-55	194	7	suppose	suppose	VERB
ma-55	194	8	that	that	SCONJ
ma-55	194	9	uε0	uε0	PROPN
ma-55	194	10	∈	∈	PROPN
ma-55	194	11	w1	w1	NOUN
ma-55	194	12	,	,	PUNCT
ma-55	194	13	j(uε0	j(uε0	PROPN
ma-55	194	14	)	)	PUNCT
ma-55	194	15	<	<	X
ma-55	194	16	d	d	X
ma-55	194	17	,	,	PUNCT
ma-55	194	18	uε1	uε1	PROPN
ma-55	194	19	∈	∈	PROPN
ma-55	194	20	h1	h1	NOUN
ma-55	194	21	0(ω	0(ω	ADV
ma-55	194	22	)	)	PUNCT
ma-55	194	23	and	and	CCONJ
ma-55	194	24	hypothesis	hypothesis	NOUN
ma-55	194	25	(	(	PUNCT
ma-55	194	26	h1	h1	PROPN
ma-55	194	27	)	)	PUNCT
ma-55	194	28	−	−	PROPN
ma-55	195	1	(	(	PUNCT
ma-55	195	2	h4	h4	PROPN
ma-55	195	3	)	)	PUNCT
ma-55	195	4	holds	hold	VERB
ma-55	195	5	.	.	PUNCT
ma-55	196	1	a	a	DET
ma-55	196	2	strong	strong	ADJ
ma-55	196	3	solution	solution	NOUN
ma-55	196	4	to	to	ADP
ma-55	196	5	the	the	DET
ma-55	196	6	boundary	boundary	ADJ
ma-55	196	7	value	value	NOUN
ma-55	196	8	problem	problem	NOUN
ma-55	196	9	(	(	PUNCT
ma-55	196	10	4.6)-(4.7	4.6)-(4.7	NUM
ma-55	196	11	)	)	PUNCT
ma-55	196	12	is	be	AUX
ma-55	196	13	a	a	DET
ma-55	196	14	function	function	NOUN
ma-55	196	15	uε	uε	ADP
ma-55	196	16	such	such	ADJ
ma-55	196	17	that	that	SCONJ
ma-55	196	18	uε	uε	PROPN
ma-55	196	19	∈	∈	PROPN
ma-55	196	20	l∞(0	l∞(0	PROPN
ma-55	196	21	,	,	PUNCT
ma-55	196	22	t	t	PROPN
ma-55	196	23	;	;	PUNCT
ma-55	196	24	h1	h1	VERB
ma-55	196	25	0(ω	0(ω	NOUN
ma-55	196	26	)	)	PUNCT
ma-55	196	27	∩h2(ω	∩h2(ω	ADJ
ma-55	196	28	)	)	PUNCT
ma-55	196	29	)	)	PUNCT
ma-55	196	30	,	,	PUNCT
ma-55	196	31	uεt	uεt	NOUN
ma-55	196	32	∈	∈	PROPN
ma-55	196	33	l∞(0	l∞(0	NOUN
ma-55	196	34	,	,	PUNCT
ma-55	196	35	t	t	NOUN
ma-55	196	36	;	;	PUNCT
ma-55	196	37	l2(ω	l2(ω	NOUN
ma-55	196	38	)	)	PUNCT
ma-55	196	39	)	)	PUNCT
ma-55	196	40	∩	∩	ADJ
ma-55	196	41	l2(0	l2(0	NOUN
ma-55	196	42	,	,	PUNCT
ma-55	196	43	t	t	PROPN
ma-55	196	44	;	;	PUNCT
ma-55	196	45	h1	h1	NOUN
ma-55	196	46	0(ω	0(ω	ADJ
ma-55	196	47	)	)	PUNCT
ma-55	196	48	)	)	PUNCT
ma-55	196	49	,	,	PUNCT
ma-55	196	50	uεtt	uεtt	PROPN
ma-55	196	51	∈	∈	PROPN
ma-55	196	52	l2(0	l2(0	NOUN
ma-55	196	53	,	,	PUNCT
ma-55	196	54	t	t	NOUN
ma-55	196	55	;	;	PUNCT
ma-55	196	56	(	(	PUNCT
ma-55	196	57	h1	h1	VERB
ma-55	196	58	0(ω	0(ω	ADJ
ma-55	196	59	)	)	PUNCT
ma-55	196	60	∩h2(ω))′	∩h2(ω))′	PUNCT
ma-55	196	61	)	)	PUNCT
ma-55	196	62	satisfying	satisfy	VERB
ma-55	196	63	for	for	ADP
ma-55	196	64	all	all	DET
ma-55	196	65	w	w	PROPN
ma-55	196	66	∈	∈	PROPN
ma-55	196	67	h1	h1	NOUN
ma-55	196	68	0(ω	0(ω	NOUN
ma-55	196	69	)	)	PUNCT
ma-55	196	70	∩h2(ω	∩h2(ω	ADJ
ma-55	196	71	)	)	PUNCT
ma-55	196	72	d	d	X
ma-55	196	73	dt	dt	X
ma-55	196	74	(	(	PUNCT
ma-55	196	75	uεt	uεt	PROPN
ma-55	196	76	(	(	PUNCT
ma-55	196	77	t	t	NOUN
ma-55	196	78	)	)	PUNCT
ma-55	196	79	,	,	PUNCT
ma-55	196	80	w	w	NOUN
ma-55	196	81	)	)	PUNCT
ma-55	196	82	+	+	CCONJ
ma-55	196	83	(	(	PUNCT
ma-55	196	84	∆uε(t),∆w	∆uε(t),∆w	NOUN
ma-55	196	85	)	)	PUNCT
ma-55	196	86	+	+	CCONJ
ma-55	196	87	(	(	PUNCT
ma-55	196	88	−∆pu	−∆pu	NUM
ma-55	196	89	ε(t	ε(t	NOUN
ma-55	196	90	)	)	PUNCT
ma-55	196	91	,	,	PUNCT
ma-55	196	92	w	w	X
ma-55	196	93	)	)	PUNCT
ma-55	197	1	+	+	CCONJ
ma-55	197	2	∫	∫	PROPN
ma-55	197	3	t	t	PROPN
ma-55	197	4	0	0	NUM
ma-55	197	5	g(t	g(t	PROPN
ma-55	197	6	−	−	PROPN
ma-55	197	7	s)(∆uε(s	s)(∆uε(s	NOUN
ma-55	197	8	)	)	PUNCT
ma-55	197	9	,	,	PUNCT
ma-55	197	10	w)ds	w)ds	PROPN
ma-55	197	11	+	+	PROPN
ma-55	197	12	(	(	PUNCT
ma-55	197	13	∇uεt	∇uεt	PROPN
ma-55	197	14	(	(	PUNCT
ma-55	197	15	t),∇w	t),∇w	NUM
ma-55	197	16	)	)	PUNCT
ma-55	198	1	+	+	CCONJ
ma-55	198	2	1	1	NUM
ma-55	198	3	ε	ε	PROPN
ma-55	198	4	(	(	PUNCT
ma-55	198	5	β(uεt	β(uεt	PROPN
ma-55	198	6	(	(	PUNCT
ma-55	198	7	t	t	PROPN
ma-55	198	8	)	)	PUNCT
ma-55	198	9	)	)	PUNCT
ma-55	198	10	,	,	PUNCT
ma-55	198	11	w)−(|uε(t)|r−2uε(t	w)−(|uε(t)|r−2uε(t	PROPN
ma-55	198	12	)	)	PUNCT
ma-55	198	13	ln	ln	ADJ
ma-55	198	14	|uε(t)|	|uε(t)|	NOUN
ma-55	198	15	)	)	PUNCT
ma-55	198	16	,	,	PUNCT
ma-55	198	17	w	w	NOUN
ma-55	198	18	)	)	PUNCT
ma-55	198	19	=	=	SYM
ma-55	198	20	0	0	NUM
ma-55	198	21	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	198	22	eur	eur	PROPN
ma-55	198	23	.	.	PUNCT
ma-55	199	1	j.	j.	PROPN
ma-55	199	2	math	math	PROPN
ma-55	199	3	.	.	PUNCT
ma-55	200	1	anal	anal	PROPN
ma-55	200	2	.	.	PUNCT
ma-55	201	1	10.28924	10.28924	NUM
ma-55	201	2	/	/	SYM
ma-55	201	3	ada	ada	PROPN
ma-55	201	4	/	/	SYM
ma-55	201	5	ma.2.5	ma.2.5	PROPN
ma-55	201	6	9	9	NUM
ma-55	201	7	and	and	CCONJ
ma-55	201	8	initial	initial	ADJ
ma-55	201	9	data	datum	NOUN
ma-55	201	10	uε(0	uε(0	NOUN
ma-55	201	11	)	)	PUNCT
ma-55	201	12	=	=	SYM
ma-55	201	13	uε0	uε0	PROPN
ma-55	201	14	,	,	PUNCT
ma-55	201	15	u	u	PROPN
ma-55	201	16	ε	ε	PROPN
ma-55	201	17	t	t	PROPN
ma-55	201	18	(	(	PUNCT
ma-55	201	19	0	0	NUM
ma-55	201	20	)	)	PUNCT
ma-55	201	21	=	=	SYM
ma-55	201	22	uε1	uε1	NOUN
ma-55	201	23	.	.	PUNCT
ma-55	202	1	the	the	DET
ma-55	202	2	solution	solution	NOUN
ma-55	202	3	of	of	ADP
ma-55	202	4	problem	problem	NOUN
ma-55	202	5	(	(	PUNCT
ma-55	202	6	4.6)-(4.7	4.6)-(4.7	NUM
ma-55	202	7	)	)	PUNCT
ma-55	202	8	is	be	AUX
ma-55	202	9	given	give	VERB
ma-55	202	10	by	by	ADP
ma-55	202	11	the	the	DET
ma-55	202	12	following	follow	VERB
ma-55	202	13	theorem	theorem	NOUN
ma-55	202	14	:	:	PUNCT
ma-55	202	15	theorem	theorem	NOUN
ma-55	202	16	4.3	4.3	NUM
ma-55	202	17	.	.	PUNCT
ma-55	203	1	assume	assume	VERB
ma-55	203	2	that	that	SCONJ
ma-55	203	3	hypotheses	hypothesis	NOUN
ma-55	203	4	(	(	PUNCT
ma-55	203	5	h1)−	h1)−	PROPN
ma-55	203	6	(	(	PUNCT
ma-55	203	7	h4	h4	PROPN
ma-55	203	8	)	)	PUNCT
ma-55	203	9	holds	hold	NOUN
ma-55	203	10	,	,	PUNCT
ma-55	203	11	uε0	uε0	NOUN
ma-55	203	12	∈	∈	PROPN
ma-55	203	13	w1	w1	NOUN
ma-55	203	14	,	,	PUNCT
ma-55	203	15	j(uε0	j(uε0	PROPN
ma-55	203	16	)	)	PUNCT
ma-55	203	17	<	<	X
ma-55	204	1	d	d	PROPN
ma-55	204	2	and	and	CCONJ
ma-55	204	3	uε1	uε1	PROPN
ma-55	204	4	∈	∈	PROPN
ma-55	204	5	h1	h1	NOUN
ma-55	204	6	0(ω	0(ω	NUM
ma-55	204	7	)	)	PUNCT
ma-55	204	8	,	,	PUNCT
ma-55	204	9	(	(	PUNCT
ma-55	204	10	4.8	4.8	NUM
ma-55	204	11	)	)	PUNCT
ma-55	204	12	then	then	ADV
ma-55	204	13	,	,	PUNCT
ma-55	204	14	for	for	ADP
ma-55	204	15	each	each	DET
ma-55	204	16	0	0	NUM
ma-55	204	17	<	<	X
ma-55	204	18	ε	ε	X
ma-55	204	19	<	<	X
ma-55	204	20	1	1	NUM
ma-55	204	21	,	,	PUNCT
ma-55	204	22	there	there	PRON
ma-55	204	23	exists	exist	VERB
ma-55	204	24	a	a	DET
ma-55	204	25	function	function	NOUN
ma-55	204	26	uε	uε	ADP
ma-55	204	27	strong	strong	ADJ
ma-55	204	28	solution	solution	NOUN
ma-55	204	29	of	of	ADP
ma-55	204	30	(	(	PUNCT
ma-55	204	31	4.6)-(4.7	4.6)-(4.7	NUM
ma-55	204	32	)	)	PUNCT
ma-55	204	33	.	.	PUNCT
ma-55	205	1	5	5	X
ma-55	205	2	.	.	X
ma-55	205	3	penalization	penalization	NOUN
ma-55	205	4	method	method	NOUN
ma-55	205	5	in	in	ADP
ma-55	205	6	order	order	NOUN
ma-55	205	7	to	to	PART
ma-55	205	8	prove	prove	VERB
ma-55	205	9	theorem	theorem	VERB
ma-55	205	10	4.1	4.1	NUM
ma-55	205	11	,	,	PUNCT
ma-55	205	12	we	we	PRON
ma-55	205	13	first	first	ADV
ma-55	205	14	prove	prove	VERB
ma-55	205	15	the	the	DET
ma-55	205	16	penalized	penalize	VERB
ma-55	205	17	theorem	theorem	VERB
ma-55	205	18	4.3	4.3	NUM
ma-55	205	19	.	.	PUNCT
ma-55	206	1	the	the	DET
ma-55	206	2	existence	existence	NOUN
ma-55	206	3	ofglobal	ofglobal	ADJ
ma-55	206	4	solutions	solution	NOUN
ma-55	206	5	will	will	AUX
ma-55	206	6	be	be	AUX
ma-55	206	7	given	give	VERB
ma-55	206	8	by	by	ADP
ma-55	206	9	using	use	VERB
ma-55	206	10	faedo	faedo	ADJ
ma-55	206	11	-	-	PUNCT
ma-55	206	12	galerkin	galerkin	ADJ
ma-55	206	13	method	method	NOUN
ma-55	206	14	.	.	PUNCT
ma-55	207	1	first	first	ADV
ma-55	207	2	we	we	PRON
ma-55	207	3	consider	consider	VERB
ma-55	207	4	the	the	DET
ma-55	207	5	approximateproblem	approximateproblem	NOUN
ma-55	207	6	.	.	PUNCT
ma-55	208	1	then	then	ADV
ma-55	208	2	we	we	PRON
ma-55	208	3	obtain	obtain	VERB
ma-55	208	4	the	the	DET
ma-55	208	5	a	a	DET
ma-55	208	6	priori	priori	ADJ
ma-55	208	7	estimates	estimate	NOUN
ma-55	208	8	needed	need	VERB
ma-55	208	9	to	to	PART
ma-55	208	10	passage	passage	VERB
ma-55	208	11	to	to	ADP
ma-55	208	12	the	the	DET
ma-55	208	13	limit	limit	NOUN
ma-55	208	14	in	in	ADP
ma-55	208	15	the	the	DET
ma-55	208	16	approximatesolutions	approximatesolution	NOUN
ma-55	208	17	.	.	PUNCT
ma-55	209	1	5.1	5.1	NUM
ma-55	209	2	.	.	PUNCT
ma-55	209	3	approximate	approximate	ADJ
ma-55	209	4	problem	problem	NOUN
ma-55	209	5	.	.	PUNCT
ma-55	210	1	let	let	VERB
ma-55	210	2	{	{	PUNCT
ma-55	210	3	wj	wj	NOUN
ma-55	210	4	}	}	PUNCT
ma-55	210	5	be	be	AUX
ma-55	210	6	the	the	DET
ma-55	210	7	galerkin	galerkin	ADJ
ma-55	210	8	basis	basis	NOUN
ma-55	210	9	given	give	VERB
ma-55	210	10	by	by	ADP
ma-55	210	11	eigenfunctions	eigenfunction	NOUN
ma-55	210	12	of	of	ADP
ma-55	210	13	∆2	∆2	PROPN
ma-55	211	1	withboundary	withboundary	PROPN
ma-55	211	2	condition	condition	NOUN
ma-55	211	3	u	u	NOUN
ma-55	211	4	=	=	PUNCT
ma-55	211	5	∆u	∆u	PROPN
ma-55	211	6	=	=	SYM
ma-55	211	7	0	0	NUM
ma-55	211	8	on	on	ADP
ma-55	211	9	γ	γ	X
ma-55	211	10	×	×	NOUN
ma-55	211	11	r+	r+	NOUN
ma-55	212	1	and	and	CCONJ
ma-55	212	2	let	let	VERB
ma-55	212	3	vm	vm	PROPN
ma-55	212	4	⊂	⊂	PROPN
ma-55	212	5	n	n	PRON
ma-55	212	6	be	be	AUX
ma-55	212	7	the	the	DET
ma-55	212	8	subspace	subspace	NOUN
ma-55	212	9	spanned	span	VERB
ma-55	212	10	by	by	ADP
ma-55	212	11	thevectors	thevector	NOUN
ma-55	212	12	w1	w1	NOUN
ma-55	212	13	,	,	PUNCT
ma-55	212	14	w2	w2	NOUN
ma-55	212	15	,	,	PUNCT
ma-55	212	16	...	...	PUNCT
ma-55	212	17	,	,	PUNCT
ma-55	212	18	wm	wm	AUX
ma-55	212	19	..	..	PUNCT
ma-55	212	20	consider	consider	VERB
ma-55	212	21	uεm(t	uεm(t	PRON
ma-55	212	22	)	)	PUNCT
ma-55	213	1	=	=	PUNCT
ma-55	213	2	m∑	m∑	ADV
ma-55	213	3	j=1	j=1	ADJ
ma-55	213	4	gεjm(t)wj	gεjm(t)wj	PROPN
ma-55	213	5	solution	solution	NOUN
ma-55	213	6	of	of	ADP
ma-55	213	7	approximate	approximate	ADJ
ma-55	213	8	problem	problem	NOUN
ma-55	213	9	(	(	PUNCT
ma-55	213	10	uεmtt	uεmtt	NOUN
ma-55	213	11	(	(	PUNCT
ma-55	213	12	t	t	PROPN
ma-55	213	13	)	)	PUNCT
ma-55	213	14	,	,	PUNCT
ma-55	213	15	w)+(∆uεm(t),∆w)+(−∆pu	w)+(∆uεm(t),∆w)+(−∆pu	PROPN
ma-55	213	16	εm(t	εm(t	ADJ
ma-55	213	17	)	)	PUNCT
ma-55	213	18	,	,	PUNCT
ma-55	213	19	w)+	w)+	PROPN
ma-55	213	20	∫	∫	PROPN
ma-55	213	21	t	t	PROPN
ma-55	213	22	0	0	NUM
ma-55	213	23	g(t	g(t	PROPN
ma-55	213	24	−	−	PROPN
ma-55	213	25	s)(∆uεm(t	s)(∆uεm(t	PROPN
ma-55	213	26	)	)	PUNCT
ma-55	213	27	,	,	PUNCT
ma-55	213	28	w)ds	w)ds	PROPN
ma-55	213	29	−(|uεm(t)|r−2uεm(t	−(|uεm(t)|r−2uεm(t	NOUN
ma-55	213	30	)	)	PUNCT
ma-55	213	31	ln	ln	ADJ
ma-55	213	32	|uεm|	|uεm|	NOUN
ma-55	213	33	,	,	PUNCT
ma-55	213	34	w	w	NOUN
ma-55	213	35	)	)	PUNCT
ma-55	213	36	+	+	CCONJ
ma-55	213	37	(	(	PUNCT
ma-55	213	38	∇uεm(t),∇w	∇uεm(t),∇w	ADJ
ma-55	213	39	)	)	PUNCT
ma-55	213	40	+	+	CCONJ
ma-55	213	41	1	1	NUM
ma-55	213	42	ε	ε	PROPN
ma-55	213	43	(	(	PUNCT
ma-55	213	44	β(uεmt	β(uεmt	PROPN
ma-55	213	45	)	)	PUNCT
ma-55	213	46	(	(	PUNCT
ma-55	213	47	t	t	PROPN
ma-55	213	48	)	)	PUNCT
ma-55	213	49	,	,	PUNCT
ma-55	213	50	w	w	X
ma-55	213	51	)	)	PUNCT
ma-55	213	52	=	=	SYM
ma-55	213	53	0	0	NUM
ma-55	213	54	(	(	PUNCT
ma-55	213	55	5.1	5.1	NUM
ma-55	213	56	)	)	PUNCT
ma-55	213	57	with	with	ADP
ma-55	213	58	initial	initial	ADJ
ma-55	213	59	conditions	condition	NOUN
ma-55	213	60	uεm(0	uεm(0	NOUN
ma-55	213	61	)	)	PUNCT
ma-55	213	62	=	=	PUNCT
ma-55	213	63	uε0	uε0	NOUN
ma-55	213	64	m	m	PROPN
ma-55	213	65	→	→	SYM
ma-55	213	66	uε0	uε0	NOUN
ma-55	213	67	strongly	strongly	ADV
ma-55	213	68	in	in	ADP
ma-55	213	69	h2(ω	h2(ω	NOUN
ma-55	213	70	)	)	PUNCT
ma-55	213	71	∩h1	∩h1	NOUN
ma-55	213	72	0(ω	0(ω	NUM
ma-55	213	73	)	)	PUNCT
ma-55	213	74	,	,	PUNCT
ma-55	213	75	(	(	PUNCT
ma-55	213	76	5.2	5.2	X
ma-55	213	77	)	)	PUNCT
ma-55	213	78	uεmt	uεmt	NOUN
ma-55	213	79	(	(	PUNCT
ma-55	213	80	0	0	NUM
ma-55	213	81	)	)	PUNCT
ma-55	213	82	=	=	VERB
ma-55	214	1	uε1	uε1	X
ma-55	214	2	m	m	PROPN
ma-55	214	3	→	→	NOUN
ma-55	214	4	uε1	uε1	X
ma-55	214	5	strongly	strongly	ADV
ma-55	214	6	in	in	ADP
ma-55	214	7	l2(ω	l2(ω	NOUN
ma-55	214	8	)	)	PUNCT
ma-55	214	9	.	.	PUNCT
ma-55	215	1	(	(	PUNCT
ma-55	215	2	5.3	5.3	NUM
ma-55	215	3	)	)	PUNCT
ma-55	215	4	the	the	DET
ma-55	215	5	system	system	NOUN
ma-55	215	6	of	of	ADP
ma-55	215	7	ordinary	ordinary	ADJ
ma-55	215	8	differential	differential	ADJ
ma-55	215	9	equation	equation	NOUN
ma-55	215	10	(	(	PUNCT
ma-55	215	11	5.1	5.1	NUM
ma-55	215	12	)	)	PUNCT
ma-55	215	13	in	in	ADP
ma-55	215	14	the	the	DET
ma-55	215	15	variable	variable	ADJ
ma-55	215	16	t	t	PROPN
ma-55	215	17	has	have	VERB
ma-55	215	18	a	a	DET
ma-55	215	19	local	local	ADJ
ma-55	215	20	solution	solution	NOUN
ma-55	215	21	uεm(t)defined	uεm(t)define	VERB
ma-55	215	22	in	in	ADP
ma-55	215	23	[	[	X
ma-55	215	24	0	0	NUM
ma-55	215	25	,	,	PUNCT
ma-55	215	26	tm	tm	NOUN
ma-55	215	27	[	[	X
ma-55	215	28	,	,	PUNCT
ma-55	215	29	0	0	PUNCT
ma-55	215	30	<	<	X
ma-55	215	31	tm	tm	PROPN
ma-55	215	32	≤	≤	PROPN
ma-55	215	33	t	t	PROPN
ma-55	215	34	.	.	PUNCT
ma-55	216	1	in	in	ADP
ma-55	216	2	the	the	DET
ma-55	216	3	next	next	ADJ
ma-55	216	4	step	step	NOUN
ma-55	216	5	obtain	obtain	VERB
ma-55	216	6	priori	priori	ADJ
ma-55	216	7	estimates	estimate	NOUN
ma-55	216	8	for	for	ADP
ma-55	216	9	the	the	DET
ma-55	216	10	solution	solution	NOUN
ma-55	216	11	uεm(t	uεm(t	NOUN
ma-55	216	12	)	)	PUNCT
ma-55	216	13	thatpermits	thatpermit	VERB
ma-55	216	14	us	we	PRON
ma-55	216	15	to	to	PART
ma-55	216	16	extend	extend	VERB
ma-55	216	17	this	this	DET
ma-55	216	18	solution	solution	NOUN
ma-55	216	19	to	to	ADP
ma-55	216	20	the	the	DET
ma-55	216	21	whole	whole	ADJ
ma-55	216	22	interval	interval	NOUN
ma-55	217	1	[	[	X
ma-55	217	2	0	0	NUM
ma-55	217	3	,	,	PUNCT
ma-55	217	4	t	t	X
ma-55	217	5	]	]	PUNCT
ma-55	217	6	.	.	PUNCT
ma-55	218	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	218	2	eur	eur	PROPN
ma-55	218	3	.	.	PUNCT
ma-55	219	1	j.	j.	PROPN
ma-55	219	2	math	math	PROPN
ma-55	219	3	.	.	PUNCT
ma-55	220	1	anal	anal	PROPN
ma-55	220	2	.	.	PUNCT
ma-55	221	1	10.28924	10.28924	NUM
ma-55	221	2	/	/	SYM
ma-55	221	3	ada	ada	PROPN
ma-55	221	4	/	/	SYM
ma-55	221	5	ma.2.5	ma.2.5	PROPN
ma-55	221	6	105.2	105.2	NUM
ma-55	221	7	.	.	PUNCT
ma-55	222	1	first	first	ADJ
ma-55	222	2	estimate	estimate	NOUN
ma-55	222	3	.	.	PUNCT
ma-55	223	1	we	we	PRON
ma-55	223	2	consider	consider	VERB
ma-55	223	3	w	w	NOUN
ma-55	223	4	=	=	NOUN
ma-55	223	5	uεmt	uεmt	PROPN
ma-55	223	6	in	in	ADP
ma-55	223	7	(	(	PUNCT
ma-55	223	8	5.1	5.1	NUM
ma-55	223	9	)	)	PUNCT
ma-55	223	10	to	to	PART
ma-55	223	11	obtain	obtain	VERB
ma-55	223	12	d	d	X
ma-55	223	13	dt	dt	X
ma-55	223	14	[	[	PUNCT
ma-55	223	15	1	1	NUM
ma-55	223	16	2	2	NUM
ma-55	223	17	|uεmt	|uεmt	NOUN
ma-55	223	18	(	(	PUNCT
ma-55	223	19	t)|2	t)|2	NOUN
ma-55	223	20	+	+	CCONJ
ma-55	223	21	1	1	NUM
ma-55	223	22	2	2	NUM
ma-55	223	23	|∆uεm(t)|2	|∆uεm(t)|2	NOUN
ma-55	223	24	+	+	CCONJ
ma-55	223	25	1	1	NUM
ma-55	223	26	p	p	NOUN
ma-55	223	27	|∇uεm(t)|pp	|∇uεm(t)|pp	PROPN
ma-55	223	28	+	+	CCONJ
ma-55	223	29	1	1	NUM
ma-55	223	30	r2	r2	PROPN
ma-55	223	31	|uεm(t)|pp	|uεm(t)|pp	PROPN
ma-55	224	1	−	−	NOUN
ma-55	224	2	1	1	NUM
ma-55	224	3	r	r	NOUN
ma-55	224	4	∫	∫	PROPN
ma-55	224	5	ω	ω	NUM
ma-55	224	6	|uεm(t)|r	|uεm(t)|r	PROPN
ma-55	224	7	ln	ln	X
ma-55	224	8	|uεm(t)|dx	|uεm(t)|dx	PROPN
ma-55	224	9	]	]	PUNCT
ma-55	225	1	+	+	NUM
ma-55	225	2	|∇uεmt	|∇uεmt	X
ma-55	225	3	(	(	PUNCT
ma-55	225	4	t)|2	t)|2	NOUN
ma-55	225	5	+	+	CCONJ
ma-55	225	6	1	1	NUM
ma-55	225	7	ε	ε	PROPN
ma-55	225	8	(	(	PUNCT
ma-55	225	9	β(uεmt	β(uεmt	PROPN
ma-55	225	10	(	(	PUNCT
ma-55	225	11	t	t	PROPN
ma-55	225	12	)	)	PUNCT
ma-55	225	13	)	)	PUNCT
ma-55	225	14	,	,	PUNCT
ma-55	225	15	uεmt	uεmt	PROPN
ma-55	225	16	(	(	PUNCT
ma-55	225	17	t	t	PROPN
ma-55	225	18	)	)	PUNCT
ma-55	225	19	)	)	PUNCT
ma-55	226	1	=	=	SYM
ma-55	226	2	∫	∫	PROPN
ma-55	226	3	t	t	PROPN
ma-55	226	4	0	0	NUM
ma-55	226	5	g(t	g(t	PROPN
ma-55	226	6	−	−	PROPN
ma-55	226	7	s)(∇uεm(s),∇uεmt	s)(∇uεm(s),∇uεmt	X
ma-55	226	8	(	(	PUNCT
ma-55	226	9	t))ds	t))ds	NOUN
ma-55	226	10	.	.	PUNCT
ma-55	227	1	(	(	PUNCT
ma-55	227	2	5.4	5.4	NUM
ma-55	227	3	)	)	PUNCT
ma-55	227	4	we	we	PRON
ma-55	227	5	have	have	VERB
ma-55	227	6	(	(	PUNCT
ma-55	227	7	β(uεmt	β(uεmt	X
ma-55	227	8	(	(	PUNCT
ma-55	227	9	t	t	PROPN
ma-55	227	10	)	)	PUNCT
ma-55	227	11	)	)	PUNCT
ma-55	227	12	,	,	PUNCT
ma-55	227	13	uεmt	uεmt	PROPN
ma-55	227	14	(	(	PUNCT
ma-55	227	15	t	t	PROPN
ma-55	227	16	)	)	PUNCT
ma-55	227	17	)	)	PUNCT
ma-55	227	18	≥	≥	NOUN
ma-55	227	19	0	0	NUM
ma-55	227	20	.	.	PUNCT
ma-55	228	1	then	then	ADV
ma-55	228	2	from	from	ADP
ma-55	228	3	lemma	lemma	PROPN
ma-55	228	4	2.2	2.2	NUM
ma-55	228	5	and	and	CCONJ
ma-55	228	6	(	(	PUNCT
ma-55	228	7	h4	h4	PROPN
ma-55	228	8	)	)	PUNCT
ma-55	228	9	1	1	NUM
ma-55	228	10	2	2	NUM
ma-55	228	11	d	d	NOUN
ma-55	228	12	dt	dt	X
ma-55	228	13	[	[	PUNCT
ma-55	228	14	|uεmt	|uεmt	X
ma-55	228	15	(	(	PUNCT
ma-55	228	16	t)|2	t)|2	NOUN
ma-55	228	17	+	+	X
ma-55	228	18	|∆uεm(t)|2	|∆uεm(t)|2	X
ma-55	228	19	+	+	CCONJ
ma-55	228	20	2	2	NUM
ma-55	228	21	p	p	NOUN
ma-55	228	22	|∇uεm(t)|pp	|∇uεm(t)|pp	PROPN
ma-55	228	23	+	+	CCONJ
ma-55	228	24	(	(	PUNCT
ma-55	228	25	g	g	PROPN
ma-55	228	26	�	�	PROPN
ma-55	228	27	∇uεm)(t	∇uεm)(t	PROPN
ma-55	228	28	)	)	PUNCT
ma-55	228	29	−	−	PROPN
ma-55	228	30	(	(	PUNCT
ma-55	228	31	∫	∫	PROPN
ma-55	228	32	t	t	PROPN
ma-55	228	33	0	0	NUM
ma-55	228	34	g(s)ds	g(s)ds	PROPN
ma-55	228	35	)	)	PUNCT
ma-55	228	36	|∇uεm(t)|2	|∇uεm(t)|2	NOUN
ma-55	229	1	+	+	CCONJ
ma-55	229	2	2	2	NUM
ma-55	229	3	r2	r2	NOUN
ma-55	229	4	|uεm(t)|rr	|uεm(t)|rr	CCONJ
ma-55	229	5	−	−	NUM
ma-55	229	6	2	2	NUM
ma-55	229	7	r	r	NOUN
ma-55	229	8	∫	∫	PROPN
ma-55	229	9	ω	ω	NUM
ma-55	229	10	|uεm(t)|r	|uεm(t)|r	PROPN
ma-55	229	11	ln	ln	X
ma-55	229	12	|uεm(t)|dx	|uεm(t)|dx	PROPN
ma-55	229	13	]	]	PUNCT
ma-55	230	1	+	+	NUM
ma-55	230	2	|∇uεmt	|∇uεmt	X
ma-55	230	3	(	(	PUNCT
ma-55	230	4	t)|2	t)|2	X
ma-55	230	5	+	+	X
ma-55	230	6	≤	≤	NUM
ma-55	230	7	1	1	NUM
ma-55	230	8	2	2	NUM
ma-55	230	9	(	(	PUNCT
ma-55	230	10	g′	g′	NOUN
ma-55	230	11	�	�	PROPN
ma-55	230	12	∇uεm)(t)−	∇uεm)(t)−	PROPN
ma-55	230	13	1	1	NUM
ma-55	230	14	2	2	NUM
ma-55	230	15	g(t)|∇uεm(t)|2	g(t)|∇uεm(t)|2	NOUN
ma-55	230	16	≤	≤	X
ma-55	230	17	0	0	NUM
ma-55	230	18	.	.	PUNCT
ma-55	231	1	(	(	PUNCT
ma-55	231	2	5.5	5.5	NUM
ma-55	231	3	)	)	PUNCT
ma-55	231	4	let	let	VERB
ma-55	231	5	eεm(t	eεm(t	NOUN
ma-55	231	6	)	)	PUNCT
ma-55	231	7	=	=	SYM
ma-55	231	8	1	1	NUM
ma-55	231	9	2	2	NUM
ma-55	231	10	[	[	PUNCT
ma-55	231	11	|uεmt	|uεmt	X
ma-55	231	12	(	(	PUNCT
ma-55	231	13	t)|2	t)|2	NOUN
ma-55	231	14	+	+	X
ma-55	231	15	|∆uεm(t)|2	|∆uεm(t)|2	X
ma-55	232	1	+	+	CCONJ
ma-55	232	2	2	2	NUM
ma-55	232	3	p	p	NOUN
ma-55	232	4	|∇uεm(t)|pp	|∇uεm(t)|pp	PROPN
ma-55	232	5	+	+	CCONJ
ma-55	232	6	(	(	PUNCT
ma-55	232	7	g	g	PROPN
ma-55	232	8	�	�	PROPN
ma-55	232	9	∇uεm)(t	∇uεm)(t	PROPN
ma-55	232	10	)	)	PUNCT
ma-55	232	11	−	−	PROPN
ma-55	233	1	(	(	PUNCT
ma-55	233	2	∫	∫	PROPN
ma-55	233	3	t	t	PROPN
ma-55	233	4	0	0	NUM
ma-55	233	5	g(s)ds	g(s)ds	PROPN
ma-55	233	6	)	)	PUNCT
ma-55	233	7	|∇uεm(t)|2	|∇uεm(t)|2	NOUN
ma-55	233	8	+	+	CCONJ
ma-55	233	9	2	2	NUM
ma-55	233	10	r2	r2	NOUN
ma-55	233	11	|uεm(t)|rr	|uεm(t)|rr	CCONJ
ma-55	233	12	−	−	NUM
ma-55	233	13	2	2	NUM
ma-55	233	14	r	r	NOUN
ma-55	233	15	∫	∫	PROPN
ma-55	233	16	ω	ω	NUM
ma-55	233	17	|uεm(t)|r	|uεm(t)|r	PROPN
ma-55	233	18	ln	ln	X
ma-55	233	19	|uεm(t)|dx	|uεm(t)|dx	PROPN
ma-55	233	20	]	]	PUNCT
ma-55	233	21	.	.	PUNCT
ma-55	234	1	(	(	PUNCT
ma-55	234	2	5.6	5.6	NUM
ma-55	234	3	)	)	PUNCT
ma-55	234	4	so	so	ADV
ma-55	234	5	,	,	PUNCT
ma-55	234	6	by	by	ADP
ma-55	234	7	(	(	PUNCT
ma-55	234	8	5.5	5.5	NUM
ma-55	234	9	)	)	PUNCT
ma-55	234	10	and	and	CCONJ
ma-55	234	11	(	(	PUNCT
ma-55	234	12	5.8	5.8	NUM
ma-55	234	13	)	)	PUNCT
ma-55	234	14	,	,	PUNCT
ma-55	234	15	we	we	PRON
ma-55	234	16	have	have	VERB
ma-55	234	17	d	d	NOUN
ma-55	234	18	dt	dt	NOUN
ma-55	234	19	eεm(t	eεm(t	NOUN
ma-55	234	20	)	)	PUNCT
ma-55	234	21	≤	≤	NOUN
ma-55	235	1	−|∇uεmt	−|∇uεmt	PROPN
ma-55	235	2	(	(	PUNCT
ma-55	235	3	t)|2	t)|2	ADV
ma-55	235	4	.	.	PUNCT
ma-55	235	5	integrating	integrate	VERB
ma-55	235	6	from	from	ADP
ma-55	235	7	0	0	NUM
ma-55	235	8	to	to	ADP
ma-55	235	9	t	t	PROPN
ma-55	235	10	,	,	PUNCT
ma-55	235	11	t	t	PROPN
ma-55	235	12	≤	≤	PROPN
ma-55	235	13	tm	tm	PROPN
ma-55	235	14	,	,	PUNCT
ma-55	235	15	we	we	PRON
ma-55	235	16	obtain	obtain	VERB
ma-55	235	17	eεm(t	eεm(t	NOUN
ma-55	235	18	)	)	PUNCT
ma-55	236	1	+	+	CCONJ
ma-55	236	2	∫	∫	PROPN
ma-55	236	3	t	t	PROPN
ma-55	236	4	0	0	NUM
ma-55	236	5	|∇uεmt	|∇uεmt	X
ma-55	236	6	(	(	PUNCT
ma-55	236	7	t)|2	t)|2	PROPN
ma-55	236	8	≤	≤	ADJ
ma-55	236	9	eεm(0	eεm(0	NOUN
ma-55	236	10	)	)	PUNCT
ma-55	236	11	.	.	PUNCT
ma-55	237	1	(	(	PUNCT
ma-55	237	2	5.7	5.7	NUM
ma-55	237	3	)	)	PUNCT
ma-55	237	4	by	by	ADP
ma-55	237	5	(	(	PUNCT
ma-55	237	6	h3	h3	NOUN
ma-55	237	7	)	)	PUNCT
ma-55	237	8	,	,	PUNCT
ma-55	237	9	it	it	PRON
ma-55	237	10	follows	follow	VERB
ma-55	237	11	1	1	NUM
ma-55	237	12	2	2	NUM
ma-55	237	13	[	[	PUNCT
ma-55	237	14	|uεmt	|uεmt	X
ma-55	237	15	(	(	PUNCT
ma-55	237	16	t)|2	t)|2	PROPN
ma-55	237	17	+	+	CCONJ
ma-55	237	18	(	(	PUNCT
ma-55	237	19	1−	1−	NUM
ma-55	237	20	µ	µ	NUM
ma-55	237	21	∫	∫	PROPN
ma-55	237	22	t	t	PROPN
ma-55	237	23	0	0	NUM
ma-55	237	24	g(s)ds	g(s)ds	PROPN
ma-55	237	25	)	)	PUNCT
ma-55	237	26	|∆uεm(t)|2	|∆uεm(t)|2	NOUN
ma-55	238	1	+	+	PROPN
ma-55	238	2	(	(	PUNCT
ma-55	238	3	g	g	PROPN
ma-55	238	4	�	�	PROPN
ma-55	238	5	∇uεm)(t)+	∇uεm)(t)+	NOUN
ma-55	238	6	2	2	NUM
ma-55	238	7	p	p	NOUN
ma-55	238	8	|∇uεm(t)|pp	|∇uεm(t)|pp	PROPN
ma-55	238	9	+	+	CCONJ
ma-55	238	10	2	2	NUM
ma-55	238	11	r2	r2	NOUN
ma-55	238	12	|uεm(t)|rr	|uεm(t)|rr	CCONJ
ma-55	238	13	−	−	NUM
ma-55	238	14	2	2	NUM
ma-55	238	15	r	r	NOUN
ma-55	238	16	∫	∫	PROPN
ma-55	238	17	ω	ω	NUM
ma-55	238	18	|uεm(t)|r	|uεm(t)|r	PROPN
ma-55	238	19	ln	ln	X
ma-55	238	20	|uεm(t)|dx	|uεm(t)|dx	PROPN
ma-55	238	21	]	]	PUNCT
ma-55	239	1	+	+	NUM
ma-55	239	2	∫	∫	PROPN
ma-55	239	3	t	t	PROPN
ma-55	239	4	0	0	NUM
ma-55	239	5	|∇uεmt	|∇uεmt	X
ma-55	239	6	|2ds	|2ds	NOUN
ma-55	239	7	≤	≤	ADJ
ma-55	239	8	eεm(t	eεm(t	NOUN
ma-55	239	9	)	)	PUNCT
ma-55	239	10	≤	≤	NOUN
ma-55	239	11	eεm(0	eεm(0	NOUN
ma-55	239	12	)	)	PUNCT
ma-55	240	1	=	=	SYM
ma-55	240	2	1	1	NUM
ma-55	240	3	2	2	NUM
ma-55	240	4	|uε1m|2	|uε1m|2	NOUN
ma-55	240	5	+	+	CCONJ
ma-55	240	6	c1j(uε0	c1j(uε0	PROPN
ma-55	240	7	m	m	NOUN
ma-55	240	8	)	)	PUNCT
ma-55	240	9	,	,	PUNCT
ma-55	240	10	(	(	PUNCT
ma-55	240	11	5.8	5.8	NUM
ma-55	240	12	)	)	PUNCT
ma-55	240	13	where	where	SCONJ
ma-55	240	14	c1	c1	PROPN
ma-55	240	15	>	>	X
ma-55	240	16	0	0	PUNCT
ma-55	240	17	is	be	AUX
ma-55	240	18	a	a	DET
ma-55	240	19	positive	positive	ADJ
ma-55	240	20	constant	constant	ADJ
ma-55	240	21	,	,	PUNCT
ma-55	240	22	independent	independent	ADJ
ma-55	240	23	of	of	ADP
ma-55	240	24	m	m	PROPN
ma-55	240	25	and	and	CCONJ
ma-55	240	26	t	t	PROPN
ma-55	240	27	.	.	PUNCT
ma-55	241	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	241	2	eur	eur	PROPN
ma-55	241	3	.	.	PUNCT
ma-55	242	1	j.	j.	PROPN
ma-55	242	2	math	math	PROPN
ma-55	242	3	.	.	PUNCT
ma-55	243	1	anal	anal	PROPN
ma-55	243	2	.	.	PUNCT
ma-55	244	1	10.28924	10.28924	NUM
ma-55	244	2	/	/	SYM
ma-55	244	3	ada	ada	PROPN
ma-55	244	4	/	/	SYM
ma-55	244	5	ma.2.5	ma.2.5	PROPN
ma-55	244	6	11we	11we	ADJ
ma-55	244	7	have	have	AUX
ma-55	244	8	j(u0εm	j(u0εm	NOUN
ma-55	244	9	)	)	PUNCT
ma-55	245	1	<	<	X
ma-55	245	2	d	d	X
ma-55	245	3	and	and	CCONJ
ma-55	245	4	by	by	ADP
ma-55	245	5	(	(	PUNCT
ma-55	245	6	5.3	5.3	NUM
ma-55	245	7	)	)	PUNCT
ma-55	245	8	,	,	PUNCT
ma-55	245	9	there	there	PRON
ma-55	245	10	exists	exist	VERB
ma-55	245	11	a	a	DET
ma-55	245	12	constant	constant	ADJ
ma-55	245	13	c2	c2	PROPN
ma-55	245	14	>	>	X
ma-55	245	15	0	0	NUM
ma-55	246	1	such	such	ADJ
ma-55	246	2	that	that	SCONJ
ma-55	246	3	|uεmt	|uεmt	PROPN
ma-55	246	4	(	(	PUNCT
ma-55	246	5	t)|2	t)|2	PROPN
ma-55	246	6	+	+	CCONJ
ma-55	246	7	(	(	PUNCT
ma-55	246	8	1−	1−	NUM
ma-55	246	9	µ	µ	NUM
ma-55	246	10	∫	∫	PROPN
ma-55	246	11	t	t	PROPN
ma-55	246	12	0	0	NUM
ma-55	246	13	g(s)ds	g(s)ds	PROPN
ma-55	246	14	)	)	PUNCT
ma-55	246	15	|∆uεm(t)|2	|∆uεm(t)|2	NOUN
ma-55	247	1	+	+	PROPN
ma-55	247	2	(	(	PUNCT
ma-55	247	3	g	g	PROPN
ma-55	247	4	�	�	PROPN
ma-55	247	5	∇uεm)(t)+	∇uεm)(t)+	NOUN
ma-55	247	6	2	2	NUM
ma-55	247	7	p	p	NOUN
ma-55	247	8	|∇uεm(t)|pp	|∇uεm(t)|pp	PROPN
ma-55	247	9	+	+	CCONJ
ma-55	247	10	2	2	NUM
ma-55	247	11	r2	r2	NOUN
ma-55	247	12	|uεm(t)|rr	|uεm(t)|rr	CCONJ
ma-55	247	13	−	−	NUM
ma-55	247	14	2	2	NUM
ma-55	247	15	r	r	NOUN
ma-55	247	16	∫	∫	PROPN
ma-55	247	17	ω	ω	NUM
ma-55	247	18	|uεm(t)|r	|uεm(t)|r	PROPN
ma-55	247	19	ln	ln	X
ma-55	247	20	|uεm(t)|dx	|uεm(t)|dx	PROPN
ma-55	248	1	+	+	CCONJ
ma-55	248	2	∫	∫	PROPN
ma-55	248	3	t	t	PROPN
ma-55	248	4	0	0	NUM
ma-55	248	5	|∇uεmt	|∇uεmt	X
ma-55	248	6	|2ds	|2ds	NOUN
ma-55	248	7	≤	≤	PROPN
ma-55	248	8	c2	c2	PROPN
ma-55	248	9	.	.	PUNCT
ma-55	249	1	(	(	PUNCT
ma-55	249	2	5.9	5.9	NUM
ma-55	249	3	)	)	PUNCT
ma-55	249	4	from	from	ADP
ma-55	249	5	(	(	PUNCT
ma-55	249	6	3.7	3.7	NUM
ma-55	249	7	)	)	PUNCT
ma-55	249	8	and	and	CCONJ
ma-55	249	9	(	(	PUNCT
ma-55	249	10	5.9	5.9	NUM
ma-55	249	11	)	)	PUNCT
ma-55	249	12	we	we	PRON
ma-55	249	13	get	get	VERB
ma-55	249	14	∆uεm	∆uεm	ADJ
ma-55	249	15	⇀	⇀	X
ma-55	249	16	∆uε	∆uε	X
ma-55	249	17	in	in	ADP
ma-55	249	18	l∞(0	l∞(0	ADJ
ma-55	249	19	,	,	PUNCT
ma-55	249	20	t	t	NOUN
ma-55	249	21	;	;	PUNCT
ma-55	249	22	l2(ω	l2(ω	NUM
ma-55	249	23	)	)	PUNCT
ma-55	249	24	)	)	PUNCT
ma-55	249	25	,	,	PUNCT
ma-55	249	26	(	(	PUNCT
ma-55	249	27	5.10	5.10	NUM
ma-55	249	28	)	)	PUNCT
ma-55	249	29	uεm	uεm	ADV
ma-55	250	1	⇀	⇀	X
ma-55	250	2	uε	uε	NOUN
ma-55	250	3	in	in	ADP
ma-55	250	4	l∞(0	l∞(0	PRON
ma-55	250	5	,	,	PUNCT
ma-55	250	6	t	t	PROPN
ma-55	250	7	;	;	PUNCT
ma-55	250	8	h1	h1	VERB
ma-55	250	9	0(ω	0(ω	NOUN
ma-55	250	10	)	)	PUNCT
ma-55	250	11	∩h2(ω	∩h2(ω	ADJ
ma-55	250	12	)	)	PUNCT
ma-55	250	13	)	)	PUNCT
ma-55	250	14	,	,	PUNCT
ma-55	250	15	(	(	PUNCT
ma-55	250	16	5.11	5.11	NUM
ma-55	250	17	)	)	PUNCT
ma-55	250	18	−∆pu	−∆pu	VERB
ma-55	251	1	εm	εm	NOUN
ma-55	251	2	⇀	⇀	PROPN
ma-55	251	3	χ	χ	NOUN
ma-55	251	4	in	in	ADP
ma-55	251	5	l2(0	l2(0	PROPN
ma-55	251	6	,	,	PUNCT
ma-55	251	7	t	t	PROPN
ma-55	251	8	;	;	PUNCT
ma-55	251	9	h−1(ω	h−1(ω	PROPN
ma-55	251	10	)	)	PUNCT
ma-55	251	11	)	)	PUNCT
ma-55	252	1	,	,	PUNCT
ma-55	252	2	(	(	PUNCT
ma-55	252	3	5.12	5.12	NUM
ma-55	252	4	)	)	PUNCT
ma-55	252	5	uεmt	uεmt	NOUN
ma-55	252	6	⇀	⇀	NUM
ma-55	252	7	uεt	uεt	NOUN
ma-55	252	8	in	in	ADP
ma-55	252	9	l∞(0	l∞(0	ADJ
ma-55	252	10	,	,	PUNCT
ma-55	252	11	t	t	NOUN
ma-55	252	12	;	;	PUNCT
ma-55	252	13	l2(ω	l2(ω	NOUN
ma-55	252	14	)	)	PUNCT
ma-55	252	15	)	)	PUNCT
ma-55	252	16	∩	∩	ADJ
ma-55	252	17	l2(0	l2(0	NOUN
ma-55	252	18	,	,	PUNCT
ma-55	252	19	t	t	PROPN
ma-55	252	20	;	;	PUNCT
ma-55	252	21	h1	h1	NOUN
ma-55	252	22	0(ω	0(ω	ADJ
ma-55	252	23	)	)	PUNCT
ma-55	252	24	)	)	PUNCT
ma-55	252	25	,	,	PUNCT
ma-55	252	26	(	(	PUNCT
ma-55	252	27	5.13	5.13	NUM
ma-55	252	28	)	)	PUNCT
ma-55	252	29	β(uεmt	β(uεmt	NOUN
ma-55	252	30	)	)	PUNCT
ma-55	253	1	⇀	⇀	PUNCT
ma-55	254	1	ψ	ψ	NOUN
ma-55	254	2	in	in	ADP
ma-55	254	3	l2(0	l2(0	NOUN
ma-55	254	4	,	,	PUNCT
ma-55	254	5	t	t	PROPN
ma-55	254	6	;	;	PUNCT
ma-55	254	7	h−1(ω	h−1(ω	PROPN
ma-55	254	8	)	)	PUNCT
ma-55	254	9	)	)	PUNCT
ma-55	254	10	.	.	PUNCT
ma-55	255	1	(	(	PUNCT
ma-55	255	2	5.14	5.14	NUM
ma-55	255	3	)	)	PUNCT
ma-55	255	4	follows	follow	VERB
ma-55	255	5	from(5.11	from(5.11	NOUN
ma-55	255	6	)	)	PUNCT
ma-55	255	7	,	,	PUNCT
ma-55	255	8	(	(	PUNCT
ma-55	255	9	5.13	5.13	NUM
ma-55	255	10	)	)	PUNCT
ma-55	256	1	and	and	CCONJ
ma-55	256	2	aubin	aubin	PROPN
ma-55	256	3	-	-	PUNCT
ma-55	256	4	lions	lion	NOUN
ma-55	256	5	theorem	theorem	VERB
ma-55	256	6	,	,	PUNCT
ma-55	256	7	for	for	ADP
ma-55	256	8	any	any	DET
ma-55	256	9	t	t	NOUN
ma-55	256	10	>	>	X
ma-55	256	11	0	0	NUM
ma-55	256	12	,	,	PUNCT
ma-55	256	13	uεm	uεm	ADV
ma-55	256	14	→	→	SYM
ma-55	256	15	uε	uε	NOUN
ma-55	256	16	in	in	ADP
ma-55	256	17	l2(0	l2(0	PROPN
ma-55	256	18	,	,	PUNCT
ma-55	256	19	t	t	PROPN
ma-55	256	20	;	;	PUNCT
ma-55	256	21	h1	h1	NOUN
ma-55	256	22	0(ω	0(ω	ADJ
ma-55	256	23	)	)	PUNCT
ma-55	256	24	)	)	PUNCT
ma-55	256	25	,	,	PUNCT
ma-55	256	26	strong	strong	ADJ
ma-55	256	27	and	and	CCONJ
ma-55	256	28	a.e	a.e	PROPN
ma-55	256	29	.	.	PROPN
ma-55	257	1	in	in	ADP
ma-55	257	2	q.	q.	PROPN
ma-55	257	3	(	(	PUNCT
ma-55	257	4	5.15	5.15	NUM
ma-55	257	5	)	)	PUNCT
ma-55	257	6	now	now	ADV
ma-55	257	7	,	,	PUNCT
ma-55	257	8	we	we	PRON
ma-55	257	9	prove	prove	VERB
ma-55	257	10	that	that	SCONJ
ma-55	257	11	χ(t	χ(t	NOUN
ma-55	257	12	)	)	PUNCT
ma-55	257	13	=	=	PUNCT
ma-55	258	1	−∆pu	−∆pu	VERB
ma-55	258	2	ε(t	ε(t	NOUN
ma-55	258	3	)	)	PUNCT
ma-55	258	4	.	.	PUNCT
ma-55	259	1	we	we	PRON
ma-55	259	2	consider	consider	VERB
ma-55	259	3	x	x	PRON
ma-55	259	4	,	,	PUNCT
ma-55	259	5	y	y	PROPN
ma-55	259	6	∈	∈	PROPN
ma-55	259	7	r	r	NOUN
ma-55	259	8	,	,	PUNCT
ma-55	259	9	p	p	PRON
ma-55	259	10	≥	≥	NUM
ma-55	259	11	2	2	NUM
ma-55	259	12	.	.	PUNCT
ma-55	260	1	then	then	ADV
ma-55	260	2	the	the	DET
ma-55	260	3	elementaryinequality	elementaryinequality	NOUN
ma-55	260	4	∣∣|x	∣∣|x	NOUN
ma-55	260	5	|p−2x	|p−2x	NOUN
ma-55	260	6	−	−	NOUN
ma-55	260	7	|y	|y	NOUN
ma-55	260	8	|p−2y	|p−2y	X
ma-55	260	9	∣∣	∣∣	NUM
ma-55	260	10	≤	≤	PROPN
ma-55	260	11	c	c	PROPN
ma-55	260	12	(	(	PUNCT
ma-55	260	13	|x	|x	X
ma-55	260	14	|p−2	|p−2	PROPN
ma-55	260	15	+	+	CCONJ
ma-55	260	16	|y	|y	ADJ
ma-55	260	17	|p−2	|p−2	PROPN
ma-55	260	18	)	)	PUNCT
ma-55	260	19	|x	|x	NOUN
ma-55	261	1	−	−	PROPN
ma-55	262	1	y	y	PROPN
ma-55	263	1	|	|	ADV
ma-55	263	2	(	(	PUNCT
ma-55	263	3	5.16)is	5.16)is	DET
ma-55	263	4	a	a	DET
ma-55	263	5	consequence	consequence	NOUN
ma-55	263	6	of	of	ADP
ma-55	263	7	the	the	DET
ma-55	263	8	mean	mean	ADJ
ma-55	263	9	value	value	NOUN
ma-55	263	10	theorem	theorem	VERB
ma-55	263	11	.	.	PUNCT
ma-55	264	1	using	use	VERB
ma-55	264	2	(	(	PUNCT
ma-55	264	3	5.16	5.16	NUM
ma-55	264	4	)	)	PUNCT
ma-55	264	5	and	and	CCONJ
ma-55	264	6	hölder	hölder	VERB
ma-55	264	7	generalized	generalized	ADJ
ma-55	264	8	inequality	inequality	NOUN
ma-55	264	9	with	with	ADP
ma-55	264	10	p	p	NOUN
ma-55	264	11	−	−	PROPN
ma-55	264	12	2	2	NUM
ma-55	264	13	2(p	2(p	NUM
ma-55	264	14	−	−	PROPN
ma-55	264	15	1	1	NUM
ma-55	264	16	)	)	PUNCT
ma-55	264	17	+	+	CCONJ
ma-55	264	18	1	1	NUM
ma-55	264	19	2	2	NUM
ma-55	264	20	+	+	CCONJ
ma-55	264	21	1	1	NUM
ma-55	264	22	2(p	2(p	NUM
ma-55	264	23	−	−	NUM
ma-55	264	24	1	1	NUM
ma-55	264	25	)	)	PUNCT
ma-55	264	26	=	=	SYM
ma-55	264	27	1	1	NUM
ma-55	264	28	,	,	PUNCT
ma-55	264	29	we	we	PRON
ma-55	264	30	deduce	deduce	VERB
ma-55	264	31	for	for	ADP
ma-55	264	32	θ	θ	PROPN
ma-55	264	33	∈	∈	PROPN
ma-55	264	34	d(0	d(0	PROPN
ma-55	264	35	,	,	PUNCT
ma-55	264	36	t	t	PROPN
ma-55	264	37	)	)	PUNCT
ma-55	264	38	and	and	CCONJ
ma-55	264	39	v	v	ADP
ma-55	264	40	∈	∈	PROPN
ma-55	264	41	vm,∣∣∣∣∫	vm,∣∣∣∣∫	PROPN
ma-55	264	42	t	t	PROPN
ma-55	264	43	0	0	PUNCT
ma-55	265	1	〈	〈	PROPN
ma-55	265	2	(	(	PUNCT
ma-55	265	3	−∆uεmp	−∆uεmp	X
ma-55	265	4	(	(	PUNCT
ma-55	265	5	t))−	t))−	NOUN
ma-55	265	6	(	(	PUNCT
ma-55	265	7	−∆uεp(t	−∆uεp(t	NOUN
ma-55	265	8	)	)	PUNCT
ma-55	265	9	)	)	PUNCT
ma-55	265	10	,	,	PUNCT
ma-55	265	11	v〉pθ(t)dt	v〉pθ(t)dt	PROPN
ma-55	265	12	∣∣∣∣	∣∣∣∣	PROPN
ma-55	265	13	=	=	SYM
ma-55	265	14	∣∣∣∣∫	∣∣∣∣∫	NOUN
ma-55	265	15	t	t	PROPN
ma-55	265	16	0	0	NUM
ma-55	266	1	∫	∫	PROPN
ma-55	267	1	ω	ω	PROPN
ma-55	268	1	(	(	PUNCT
ma-55	268	2	|∇uεm(t)|p−2∇uεm(t)−	|∇uεm(t)|p−2∇uεm(t)−	PROPN
ma-55	268	3	|∇uε(t)|p−2∇uε(t	|∇uε(t)|p−2∇uε(t	NUM
ma-55	268	4	)	)	PUNCT
ma-55	268	5	)	)	PUNCT
ma-55	269	1	∇v	∇v	ADV
ma-55	269	2	dx	dx	PROPN
ma-55	269	3	θ(t	θ(t	PROPN
ma-55	269	4	)	)	PUNCT
ma-55	270	1	dt	dt	PROPN
ma-55	270	2	∣∣∣∣	∣∣∣∣	PROPN
ma-55	270	3	≤	≤	PROPN
ma-55	271	1	c|θ|∞	c|θ|∞	PROPN
ma-55	271	2	∫	∫	PROPN
ma-55	271	3	t	t	PROPN
ma-55	271	4	0	0	NUM
ma-55	272	1	∫	∫	PROPN
ma-55	272	2	ω	ω	PROPN
ma-55	272	3	(	(	PUNCT
ma-55	272	4	|∇uεm(t)|p−2	|∇uεm(t)|p−2	VERB
ma-55	272	5	+	+	NUM
ma-55	272	6	|∇uε(t)|p−2	|∇uε(t)|p−2	X
ma-55	272	7	)	)	PUNCT
ma-55	272	8	|∇uεm(t)−∇uε(t)||∇v	|∇uεm(t)−∇uε(t)||∇v	NOUN
ma-55	272	9	|	|	NOUN
ma-55	272	10	dx	dx	PROPN
ma-55	272	11	dt	dt	PROPN
ma-55	272	12	≤	≤	PROPN
ma-55	272	13	c1	c1	PROPN
ma-55	272	14	∫	∫	PROPN
ma-55	272	15	t	t	PROPN
ma-55	272	16	0	0	NUM
ma-55	273	1	(	(	PUNCT
ma-55	273	2	|∇uεm(t)|p−2	|∇uεm(t)|p−2	PROPN
ma-55	273	3	2(p−1	2(p−1	NOUN
ma-55	273	4	)	)	PUNCT
ma-55	274	1	+	+	CCONJ
ma-55	274	2	|∇uε(t)|p−2	|∇uε(t)|p−2	NUM
ma-55	274	3	2(p−1	2(p−1	ADJ
ma-55	274	4	)	)	PUNCT
ma-55	274	5	)	)	PUNCT
ma-55	274	6	|∇uεm(t)−∇uε(t)||∇v	|∇uεm(t)−∇uε(t)||∇v	NOUN
ma-55	274	7	|2(p−1	|2(p−1	NOUN
ma-55	274	8	)	)	PUNCT
ma-55	274	9	dt	dt	PART
ma-55	275	1	≤	≤	PROPN
ma-55	275	2	c2	c2	PROPN
ma-55	275	3	∫	∫	PROPN
ma-55	275	4	t	t	PROPN
ma-55	275	5	0	0	NUM
ma-55	275	6	|∇uεm(t)−∇uε(t)|	|∇uεm(t)−∇uε(t)|	NOUN
ma-55	275	7	dt	dt	X
ma-55	275	8	(	(	PUNCT
ma-55	275	9	5.17	5.17	NUM
ma-55	275	10	)	)	PUNCT
ma-55	275	11	where	where	SCONJ
ma-55	275	12	c1	c1	PROPN
ma-55	275	13	and	and	CCONJ
ma-55	275	14	c2	c2	PROPN
ma-55	275	15	are	be	AUX
ma-55	275	16	positive	positive	ADJ
ma-55	275	17	constants	constant	NOUN
ma-55	275	18	independent	independent	ADJ
ma-55	275	19	of	of	ADP
ma-55	275	20	m	m	PROPN
ma-55	275	21	and	and	CCONJ
ma-55	275	22	t	t	PROPN
ma-55	275	23	.	.	PUNCT
ma-55	276	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	276	2	eur	eur	PROPN
ma-55	276	3	.	.	PUNCT
ma-55	277	1	j.	j.	PROPN
ma-55	277	2	math	math	PROPN
ma-55	277	3	.	.	PUNCT
ma-55	278	1	anal	anal	PROPN
ma-55	278	2	.	.	PUNCT
ma-55	279	1	10.28924	10.28924	NUM
ma-55	279	2	/	/	SYM
ma-55	279	3	ada	ada	PROPN
ma-55	279	4	/	/	SYM
ma-55	279	5	ma.2.5	ma.2.5	PROPN
ma-55	279	6	12now	12now	INTJ
ma-55	279	7	,	,	PUNCT
ma-55	279	8	from	from	ADP
ma-55	279	9	estimate	estimate	NOUN
ma-55	279	10	(	(	PUNCT
ma-55	279	11	5.10	5.10	NUM
ma-55	279	12	)	)	PUNCT
ma-55	279	13	and	and	CCONJ
ma-55	279	14	(	(	PUNCT
ma-55	279	15	5.11	5.11	NUM
ma-55	279	16	)	)	PUNCT
ma-55	279	17	,	,	PUNCT
ma-55	279	18	we	we	PRON
ma-55	279	19	have	have	VERB
ma-55	279	20	d	d	NOUN
ma-55	279	21	dt	dt	ADP
ma-55	279	22	|∇uεm(t)−∇uε(t)|2	|∇uεm(t)−∇uε(t)|2	PROPN
ma-55	279	23	≤	≤	NUM
ma-55	279	24	2|∆(uεm(t)−	2|∆(uεm(t)−	NUM
ma-55	279	25	uε(t))||∇(uεmt	uε(t))||∇(uεmt	NOUN
ma-55	279	26	(	(	PUNCT
ma-55	279	27	t)−	t)−	PROPN
ma-55	279	28	uεt	uεt	NOUN
ma-55	279	29	(	(	PUNCT
ma-55	279	30	t))|	t))|	PROPN
ma-55	279	31	≤	≤	PROPN
ma-55	279	32	c3	c3	PROPN
ma-55	279	33	,	,	PUNCT
ma-55	279	34	where	where	SCONJ
ma-55	279	35	c3	c3	PROPN
ma-55	279	36	is	be	AUX
ma-55	279	37	a	a	DET
ma-55	279	38	constant	constant	ADJ
ma-55	279	39	independent	independent	NOUN
ma-55	279	40	of	of	ADP
ma-55	279	41	m	m	PROPN
ma-55	279	42	and	and	CCONJ
ma-55	279	43	t	t	PROPN
ma-55	279	44	.	.	PUNCT
ma-55	280	1	so	so	ADV
ma-55	280	2	,	,	PUNCT
ma-55	280	3	|uεm(t)−	|uεm(t)−	PRON
ma-55	280	4	uε(t)|h1	uε(t)|h1	ADJ
ma-55	280	5	0(ω	0(ω	ADJ
ma-55	280	6	)	)	PUNCT
ma-55	280	7	∈	∈	PROPN
ma-55	280	8	h1[0	h1[0	PROPN
ma-55	280	9	,	,	PUNCT
ma-55	280	10	t	t	X
ma-55	280	11	]	]	PUNCT
ma-55	280	12	↪	↪	PROPN
ma-55	280	13	→	→	SYM
ma-55	280	14	c[0	c[0	PROPN
ma-55	280	15	,	,	PUNCT
ma-55	280	16	t	t	X
ma-55	280	17	]	]	PUNCT
ma-55	280	18	,	,	PUNCT
ma-55	281	1	whence	whence	ADP
ma-55	281	2	∇uεm(t)→	∇uεm(t)→	X
ma-55	281	3	∇uε(t	∇uε(t	NOUN
ma-55	281	4	)	)	PUNCT
ma-55	281	5	a.	a.	NOUN
ma-55	281	6	e.	e.	PROPN
ma-55	281	7	in	in	ADP
ma-55	281	8	[	[	X
ma-55	281	9	0	0	NUM
ma-55	281	10	,	,	PUNCT
ma-55	281	11	t	t	X
ma-55	281	12	]	]	PUNCT
ma-55	281	13	.	.	PUNCT
ma-55	282	1	therefore	therefore	ADV
ma-55	282	2	,	,	PUNCT
ma-55	282	3	χ	χ	X
ma-55	282	4	=	=	PUNCT
ma-55	282	5	−∆pu	−∆pu	PROPN
ma-55	282	6	ε	ε	PROPN
ma-55	282	7	.	.	PUNCT
ma-55	283	1	now	now	ADV
ma-55	283	2	,	,	PUNCT
ma-55	283	3	we	we	PRON
ma-55	283	4	observe	observe	VERB
ma-55	283	5	that	that	SCONJ
ma-55	283	6	sobolev	sobolev	NOUN
ma-55	283	7	inequality∫	inequality∫	PROPN
ma-55	283	8	ω	ω	PROPN
ma-55	283	9	||uεm(t)|r−2uεm(t	||uεm(t)|r−2uεm(t	ADJ
ma-55	283	10	)	)	PUNCT
ma-55	284	1	ln	ln	PROPN
ma-55	284	2	|uεm(t)||2dx	|uεm(t)||2dx	PROPN
ma-55	284	3	≤	≤	NUM
ma-55	284	4	|uεm(t)|2r2r	|uεm(t)|2r2r	NOUN
ma-55	284	5	≤	≤	NUM
ma-55	284	6	c2r	c2r	NOUN
ma-55	284	7	|∇uεm(t)|2r	|∇uεm(t)|2r	NOUN
ma-55	284	8	≤	≤	NUM
ma-55	284	9	µrc2r	µrc2r	PUNCT
ma-55	284	10	|∆uεm(t)|r	|∆uεm(t)|r	PROPN
ma-55	284	11	≤	≤	NUM
ma-55	284	12	c4	c4	NOUN
ma-55	284	13	,	,	PUNCT
ma-55	284	14	where	where	SCONJ
ma-55	284	15	c4	c4	NOUN
ma-55	284	16	is	be	AUX
ma-55	284	17	a	a	DET
ma-55	284	18	constant	constant	ADJ
ma-55	284	19	independent	independent	NOUN
ma-55	284	20	of	of	ADP
ma-55	284	21	m	m	PRON
ma-55	284	22	and	and	CCONJ
ma-55	284	23	t.then	t.then	PRON
ma-55	284	24	(	(	PUNCT
ma-55	284	25	|uεm|r−2uεm	|uεm|r−2uεm	PROPN
ma-55	284	26	ln	ln	ADJ
ma-55	284	27	|uεm|	|uεm|	NOUN
ma-55	284	28	)	)	PUNCT
ma-55	284	29	is	be	AUX
ma-55	284	30	bounded	bound	VERB
ma-55	284	31	in	in	ADP
ma-55	284	32	l2(0	l2(0	PROPN
ma-55	284	33	,	,	PUNCT
ma-55	284	34	t	t	NOUN
ma-55	284	35	;	;	PUNCT
ma-55	284	36	l2(ω	l2(ω	NUM
ma-55	284	37	)	)	PUNCT
ma-55	284	38	)	)	PUNCT
ma-55	285	1	=	=	SYM
ma-55	285	2	l2(q	l2(q	PROPN
ma-55	285	3	)	)	PUNCT
ma-55	285	4	.	.	PUNCT
ma-55	286	1	(	(	PUNCT
ma-55	286	2	5.18	5.18	NUM
ma-55	286	3	)	)	PUNCT
ma-55	286	4	using	use	VERB
ma-55	286	5	continuity	continuity	NOUN
ma-55	286	6	of	of	ADP
ma-55	286	7	function	function	NOUN
ma-55	286	8	s	s	PART
ma-55	286	9	→	→	PUNCT
ma-55	286	10	|s|r−2s	|s|r−2s	PROPN
ma-55	286	11	ln	ln	ADJ
ma-55	286	12	|s|	|s|	PROPN
ma-55	286	13	and	and	CCONJ
ma-55	286	14	(	(	PUNCT
ma-55	286	15	5.15	5.15	NUM
ma-55	286	16	)	)	PUNCT
ma-55	286	17	we	we	PRON
ma-55	286	18	have	have	VERB
ma-55	286	19	|uεm|r−2uεm	|uεm|r−2uεm	PROPN
ma-55	286	20	ln	ln	ADJ
ma-55	286	21	|uεm|	|uεm|	NOUN
ma-55	286	22	→	→	SYM
ma-55	286	23	|uε|r−2uε	|uε|r−2uε	PRON
ma-55	286	24	ln	ln	ADJ
ma-55	286	25	|uε|	|uε|	PROPN
ma-55	287	1	a.e	a.e	PROPN
ma-55	288	1	.	.	PROPN
ma-55	289	1	in	in	ADP
ma-55	289	2	q.	q.	PROPN
ma-55	289	3	(	(	PUNCT
ma-55	289	4	5.19	5.19	NUM
ma-55	289	5	)	)	PUNCT
ma-55	289	6	by	by	ADP
ma-55	289	7	(	(	PUNCT
ma-55	289	8	5.18	5.18	NUM
ma-55	289	9	)	)	PUNCT
ma-55	289	10	,	,	PUNCT
ma-55	289	11	(	(	PUNCT
ma-55	289	12	5.19	5.19	NUM
ma-55	289	13	)	)	PUNCT
ma-55	289	14	and	and	CCONJ
ma-55	289	15	applying	apply	VERB
ma-55	289	16	lions	lion	NOUN
ma-55	289	17	lemma	lemma	PROPN
ma-55	289	18	(	(	PUNCT
ma-55	289	19	lemma	lemma	PROPN
ma-55	289	20	1.3	1.3	NUM
ma-55	289	21	,	,	PUNCT
ma-55	289	22	page	page	NOUN
ma-55	289	23	12	12	NUM
ma-55	289	24	,	,	PUNCT
ma-55	289	25	[	[	X
ma-55	289	26	16	16	NUM
ma-55	289	27	]	]	PUNCT
ma-55	289	28	)	)	PUNCT
ma-55	289	29	,	,	PUNCT
ma-55	289	30	we	we	PRON
ma-55	289	31	get	get	VERB
ma-55	289	32	|uεm|r−2uεm	|uεm|r−2uεm	PROPN
ma-55	289	33	ln	ln	ADJ
ma-55	289	34	|uεm|	|uεm|	NOUN
ma-55	289	35	⇀	⇀	PRON
ma-55	289	36	|uε|r−2uε	|uε|r−2uε	PRON
ma-55	289	37	ln	ln	ADJ
ma-55	289	38	|uε|	|uε|	VERB
ma-55	289	39	weakly	weakly	ADV
ma-55	289	40	in	in	ADP
ma-55	289	41	l2(0	l2(0	NOUN
ma-55	289	42	,	,	PUNCT
ma-55	289	43	t	t	NOUN
ma-55	289	44	;	;	PUNCT
ma-55	289	45	l2(ω	l2(ω	NUM
ma-55	289	46	)	)	PUNCT
ma-55	289	47	)	)	PUNCT
ma-55	289	48	.	.	PUNCT
ma-55	290	1	(	(	PUNCT
ma-55	290	2	5.20	5.20	NUM
ma-55	290	3	)	)	PUNCT
ma-55	290	4	5.3	5.3	NUM
ma-55	290	5	.	.	PUNCT
ma-55	290	6	second	second	ADJ
ma-55	290	7	estimate	estimate	NOUN
ma-55	290	8	.	.	PUNCT
ma-55	291	1	let	let	VERB
ma-55	291	2	us	we	PRON
ma-55	291	3	consider	consider	VERB
ma-55	291	4	the	the	DET
ma-55	291	5	initial	initial	ADJ
ma-55	291	6	data	datum	NOUN
ma-55	291	7	uε0	uε0	NOUN
ma-55	291	8	∈	∈	PROPN
ma-55	291	9	h3	h3	NOUN
ma-55	291	10	γ(ω	γ(ω	PROPN
ma-55	291	11	)	)	PUNCT
ma-55	291	12	,	,	PUNCT
ma-55	291	13	uε1	uε1	PROPN
ma-55	291	14	∈	∈	PROPN
ma-55	291	15	h1	h1	NOUN
ma-55	291	16	0(ω	0(ω	NOUN
ma-55	291	17	)	)	PUNCT
ma-55	291	18	and	and	CCONJ
ma-55	291	19	uεm0	uεm0	NOUN
ma-55	291	20	=	=	SYM
ma-55	291	21	∆uεm0	∆uεm0	PROPN
ma-55	291	22	=	=	NOUN
ma-55	291	23	0	0	NUM
ma-55	291	24	on	on	ADP
ma-55	291	25	γ	γ	PROPN
ma-55	291	26	.	.	PROPN
ma-55	292	1	(	(	PUNCT
ma-55	292	2	5.21	5.21	NUM
ma-55	292	3	)	)	PUNCT
ma-55	292	4	we	we	PRON
ma-55	292	5	consider	consider	VERB
ma-55	292	6	w	w	NOUN
ma-55	292	7	=	=	PUNCT
ma-55	292	8	−∆uεmt	−∆uεmt	PROPN
ma-55	292	9	in	in	ADP
ma-55	292	10	approximate	approximate	ADJ
ma-55	292	11	equation	equation	NOUN
ma-55	292	12	(	(	PUNCT
ma-55	292	13	5.1).then	5.1).then	ADJ
ma-55	292	14	we	we	PRON
ma-55	292	15	have	have	AUX
ma-55	292	16	d	d	ADJ
ma-55	292	17	dt	dt	X
ma-55	292	18	{	{	PUNCT
ma-55	292	19	1	1	NUM
ma-55	292	20	2	2	NUM
ma-55	292	21	|∇uεmt	|∇uεmt	X
ma-55	292	22	(	(	PUNCT
ma-55	292	23	t)|2	t)|2	NOUN
ma-55	292	24	+	+	CCONJ
ma-55	292	25	1	1	NUM
ma-55	292	26	2	2	NUM
ma-55	292	27	|∇∆uεm(t)|2	|∇∆uεm(t)|2	NOUN
ma-55	292	28	}	}	PUNCT
ma-55	293	1	+	+	CCONJ
ma-55	293	2	〈	〈	PROPN
ma-55	293	3	∆puεmt	∆puεmt	NOUN
ma-55	293	4	(	(	PUNCT
ma-55	293	5	t),∆uεmt	t),∆uεmt	X
ma-55	293	6	(	(	PUNCT
ma-55	293	7	t	t	PROPN
ma-55	293	8	)	)	PUNCT
ma-55	293	9	〉	〉	PROPN
ma-55	293	10	+	+	NOUN
ma-55	293	11	|∆uεmt	|∆uεmt	X
ma-55	293	12	(	(	PUNCT
ma-55	293	13	t)|2	t)|2	NOUN
ma-55	293	14	+	+	CCONJ
ma-55	293	15	1	1	NUM
ma-55	293	16	ε	ε	PROPN
ma-55	293	17	(	(	PUNCT
ma-55	293	18	β(uεmt	β(uεmt	X
ma-55	293	19	(	(	PUNCT
ma-55	293	20	t),−∆uεmt	t),−∆uεmt	X
ma-55	293	21	(	(	PUNCT
ma-55	293	22	t	t	NOUN
ma-55	293	23	)	)	PUNCT
ma-55	293	24	)	)	PUNCT
ma-55	294	1	=	=	PUNCT
ma-55	295	1	(	(	PUNCT
ma-55	295	2	|uεm(t)|r−2uεm	|uεm(t)|r−2uεm	NOUN
ma-55	295	3	ln	ln	PROPN
ma-55	295	4	|uεm(t)|,−∆uεmt	|uεm(t)|,−∆uεmt	PROPN
ma-55	295	5	(	(	PUNCT
ma-55	295	6	t	t	PROPN
ma-55	295	7	)	)	PUNCT
ma-55	295	8	)	)	PUNCT
ma-55	296	1	+	+	CCONJ
ma-55	297	1	∫	∫	PROPN
ma-55	297	2	t	t	PROPN
ma-55	297	3	0	0	NUM
ma-55	297	4	g(t	g(t	PROPN
ma-55	297	5	−	−	PROPN
ma-55	298	1	s)(∆uεm(s),∆uεmt	s)(∆uεm(s),∆uεmt	PROPN
ma-55	298	2	(	(	PUNCT
ma-55	298	3	t))ds	t))ds	PROPN
ma-55	298	4	.	.	PUNCT
ma-55	299	1	now	now	ADV
ma-55	299	2	,	,	PUNCT
ma-55	299	3	〈	〈	PROPN
ma-55	299	4	∆puεm(t),∆uεmt	∆puεm(t),∆uεmt	PROPN
ma-55	299	5	(	(	PUNCT
ma-55	299	6	t	t	NOUN
ma-55	299	7	)	)	PUNCT
ma-55	299	8	〉	〉	NOUN
ma-55	299	9	=	=	SYM
ma-55	300	1	d	d	NOUN
ma-55	300	2	dt	dt	X
ma-55	300	3	〈	〈	PROPN
ma-55	300	4	∆puεm(t),∆uεm(t	∆puεm(t),∆uεm(t	NOUN
ma-55	300	5	)	)	PUNCT
ma-55	300	6	〉	〉	PROPN
ma-55	300	7	−	−	PROPN
ma-55	300	8	j1	j1	PROPN
ma-55	300	9	,	,	PUNCT
ma-55	300	10	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	300	11	eur	eur	PROPN
ma-55	300	12	.	.	PUNCT
ma-55	301	1	j.	j.	PROPN
ma-55	301	2	math	math	PROPN
ma-55	301	3	.	.	PUNCT
ma-55	302	1	anal	anal	PROPN
ma-55	302	2	.	.	PUNCT
ma-55	303	1	10.28924	10.28924	NUM
ma-55	303	2	/	/	SYM
ma-55	303	3	ada	ada	PROPN
ma-55	303	4	/	/	SYM
ma-55	303	5	ma.2.5	ma.2.5	PROPN
ma-55	303	6	13where	13where	X
ma-55	303	7	j1	j1	PROPN
ma-55	303	8	=	=	SYM
ma-55	303	9	∫	∫	PROPN
ma-55	303	10	ω	ω	PROPN
ma-55	303	11	{	{	PUNCT
ma-55	303	12	(	(	PUNCT
ma-55	303	13	p	p	NOUN
ma-55	303	14	−	−	PROPN
ma-55	303	15	2)|∇uεm(t)|p−4(∇uεm(t	2)|∇uεm(t)|p−4(∇uεm(t	NUM
ma-55	303	16	)	)	PUNCT
ma-55	303	17	·	·	PUNCT
ma-55	303	18	∇uεmt	∇uεmt	X
ma-55	303	19	(	(	PUNCT
ma-55	303	20	t))∇uεm(t	t))∇uεm(t	NUM
ma-55	303	21	)	)	PUNCT
ma-55	304	1	+	+	ADJ
ma-55	304	2	|∇uεm(t)|p−2∇uεmt	|∇uεm(t)|p−2∇uεmt	X
ma-55	304	3	(	(	PUNCT
ma-55	304	4	t	t	NOUN
ma-55	304	5	)	)	PUNCT
ma-55	304	6	}	}	PUNCT
ma-55	304	7	·	·	PUNCT
ma-55	304	8	∇∆uεm(t)dx	∇∆uεm(t)dx	NOUN
ma-55	304	9	.	.	PUNCT
ma-55	305	1	then	then	ADV
ma-55	305	2	d	d	X
ma-55	305	3	dt	dt	X
ma-55	305	4	{	{	PUNCT
ma-55	305	5	1	1	NUM
ma-55	305	6	2	2	NUM
ma-55	305	7	|∇uεmt	|∇uεmt	X
ma-55	305	8	(	(	PUNCT
ma-55	305	9	t)|2	t)|2	NOUN
ma-55	305	10	+	+	CCONJ
ma-55	305	11	1	1	NUM
ma-55	305	12	2	2	NUM
ma-55	305	13	|∇∆uεm(t)|2	|∇∆uεm(t)|2	NOUN
ma-55	305	14	+	+	CCONJ
ma-55	305	15	〈	〈	PROPN
ma-55	305	16	∆puεm(t),∆uεm(t	∆puεm(t),∆uεm(t	ADJ
ma-55	305	17	)	)	PUNCT
ma-55	305	18	〉	〉	NOUN
ma-55	305	19	}	}	PUNCT
ma-55	306	1	+	+	PROPN
ma-55	306	2	|∆uεmt	|∆uεmt	X
ma-55	306	3	(	(	PUNCT
ma-55	306	4	t)|2	t)|2	NOUN
ma-55	306	5	+	+	CCONJ
ma-55	306	6	1	1	NUM
ma-55	306	7	ε	ε	PROPN
ma-55	306	8	(	(	PUNCT
ma-55	306	9	β(uεmt	β(uεmt	X
ma-55	306	10	(	(	PUNCT
ma-55	306	11	t)),−∆uεmt	t)),−∆uεmt	NOUN
ma-55	306	12	(	(	PUNCT
ma-55	306	13	t	t	NOUN
ma-55	306	14	)	)	PUNCT
ma-55	306	15	)	)	PUNCT
ma-55	307	1	=	=	SYM
ma-55	307	2	j1	j1	PROPN
ma-55	307	3	+	+	CCONJ
ma-55	307	4	j2	j2	PROPN
ma-55	307	5	+	+	CCONJ
ma-55	307	6	j3	j3	PROPN
ma-55	307	7	.	.	PUNCT
ma-55	308	1	(	(	PUNCT
ma-55	308	2	5.22	5.22	NUM
ma-55	308	3	)	)	PUNCT
ma-55	308	4	where	where	SCONJ
ma-55	308	5	j2	j2	PROPN
ma-55	308	6	=	=	SYM
ma-55	308	7	∫	∫	PROPN
ma-55	309	1	ω	ω	PROPN
ma-55	309	2	|uεm(t)|r−2uεm	|uεm(t)|r−2uεm	NOUN
ma-55	309	3	ln	ln	PROPN
ma-55	309	4	|uεm(t)|∆uεmt	|uεm(t)|∆uεmt	X
ma-55	309	5	(	(	PUNCT
ma-55	309	6	t	t	PROPN
ma-55	309	7	)	)	PUNCT
ma-55	309	8	and	and	CCONJ
ma-55	309	9	j3	j3	PROPN
ma-55	309	10	=	=	SYM
ma-55	309	11	∫	∫	PROPN
ma-55	310	1	t	t	PROPN
ma-55	310	2	0	0	NUM
ma-55	310	3	g(t	g(t	PROPN
ma-55	310	4	−	−	PROPN
ma-55	311	1	s)(∆uεm(s),∆uεmt	s)(∆uεm(s),∆uεmt	PROPN
ma-55	311	2	(	(	PUNCT
ma-55	311	3	t))ds	t))ds	PROPN
ma-55	311	4	.	.	PUNCT
ma-55	311	5	let	let	VERB
ma-55	311	6	us	we	PRON
ma-55	311	7	the	the	DET
ma-55	311	8	right	right	ADJ
ma-55	311	9	hand	hand	NOUN
ma-55	311	10	side	side	NOUN
ma-55	311	11	of	of	ADP
ma-55	311	12	(	(	PUNCT
ma-55	311	13	5.22	5.22	NUM
ma-55	311	14	)	)	PUNCT
ma-55	311	15	.	.	PUNCT
ma-55	312	1	we	we	PRON
ma-55	312	2	denote	denote	VERB
ma-55	312	3	by	by	ADP
ma-55	312	4	c	c	PROPN
ma-55	312	5	a	a	DET
ma-55	312	6	generic	generic	ADJ
ma-55	312	7	positive	positive	ADJ
ma-55	312	8	constant	constant	NOUN
ma-55	312	9	not	not	PART
ma-55	312	10	dependingon	dependingon	PROPN
ma-55	312	11	m	m	PROPN
ma-55	312	12	,	,	PUNCT
ma-55	312	13	t	t	PROPN
ma-55	312	14	.	.	PUNCT
ma-55	313	1	by	by	ADP
ma-55	313	2	estimate	estimate	NOUN
ma-55	313	3	(	(	PUNCT
ma-55	313	4	5.9	5.9	NUM
ma-55	313	5	)	)	PUNCT
ma-55	313	6	and	and	CCONJ
ma-55	313	7	p	p	NOUN
ma-55	313	8	−	−	PROPN
ma-55	313	9	2	2	NUM
ma-55	313	10	2(p	2(p	NUM
ma-55	313	11	−	−	PROPN
ma-55	313	12	1	1	NUM
ma-55	313	13	)	)	PUNCT
ma-55	313	14	+	+	CCONJ
ma-55	314	1	1	1	NUM
ma-55	314	2	2(p	2(p	NUM
ma-55	314	3	−	−	NUM
ma-55	314	4	1	1	NUM
ma-55	314	5	)	)	PUNCT
ma-55	314	6	+	+	CCONJ
ma-55	314	7	1	1	NUM
ma-55	314	8	2	2	NUM
ma-55	314	9	=	=	SYM
ma-55	314	10	1	1	NUM
ma-55	314	11	,	,	PUNCT
ma-55	314	12	|j1|	|j1|	NOUN
ma-55	314	13	≤	≤	NOUN
ma-55	314	14	(	(	PUNCT
ma-55	314	15	p	p	NOUN
ma-55	314	16	−	−	PROPN
ma-55	314	17	1	1	NUM
ma-55	314	18	)	)	PUNCT
ma-55	314	19	∫	∫	PROPN
ma-55	315	1	ω	ω	NUM
ma-55	315	2	|∇uεm(t)|p−2|∇uεmt	|∇uεm(t)|p−2|∇uεmt	PROPN
ma-55	315	3	(	(	PUNCT
ma-55	315	4	t)||∇∆uεm(t)|dx	t)||∇∆uεm(t)|dx	VERB
ma-55	315	5	≤	≤	NOUN
ma-55	315	6	(	(	PUNCT
ma-55	315	7	p	p	NOUN
ma-55	315	8	−	−	PROPN
ma-55	315	9	1)|∇uεm(t)|p−2	1)|∇uεm(t)|p−2	PROPN
ma-55	315	10	2(p−1	2(p−1	NOUN
ma-55	315	11	)	)	PUNCT
ma-55	316	1	|∇uεmt	|∇uεmt	X
ma-55	316	2	(	(	PUNCT
ma-55	316	3	t)|2(p−1)|∇∆uεm(t)|	t)|2(p−1)|∇∆uεm(t)|	NOUN
ma-55	316	4	≤	≤	NUM
ma-55	316	5	c|∇uεmt	c|∇uεmt	X
ma-55	316	6	(	(	PUNCT
ma-55	316	7	t)|2(p−1)|∇∆uεm(t)|	t)|2(p−1)|∇∆uεm(t)|	NOUN
ma-55	316	8	.	.	PUNCT
ma-55	317	1	how	how	SCONJ
ma-55	317	2	h1	h1	VERB
ma-55	317	3	0(ω	0(ω	ADV
ma-55	317	4	)	)	PUNCT
ma-55	317	5	∩h2(ω	∩h2(ω	ADJ
ma-55	317	6	)	)	PUNCT
ma-55	317	7	↪	↪	PROPN
ma-55	317	8	→	→	SYM
ma-55	317	9	w	w	NOUN
ma-55	317	10	1,2	1,2	NUM
ma-55	317	11	0	0	NUM
ma-55	317	12	(	(	PUNCT
ma-55	317	13	ω	ω	NOUN
ma-55	317	14	)	)	PUNCT
ma-55	317	15	,	,	PUNCT
ma-55	317	16	we	we	PRON
ma-55	317	17	have	have	VERB
ma-55	317	18	|∇uεmt	|∇uεmt	X
ma-55	317	19	(	(	PUNCT
ma-55	317	20	t)|22(p−1	t)|22(p−1	PROPN
ma-55	317	21	)	)	PUNCT
ma-55	317	22	≤	≤	PUNCT
ma-55	318	1	µ2|∆uεmt	µ2|∆uεmt	PROPN
ma-55	318	2	(	(	PUNCT
ma-55	318	3	t)|2	t)|2	PROPN
ma-55	318	4	,	,	PUNCT
ma-55	318	5	where	where	SCONJ
ma-55	318	6	µ2	µ2	PROPN
ma-55	318	7	>	>	X
ma-55	318	8	0	0	NUM
ma-55	318	9	is	be	AUX
ma-55	318	10	the	the	DET
ma-55	318	11	corresponding	corresponding	ADJ
ma-55	318	12	embedding	embed	VERB
ma-55	318	13	constant	constant	ADJ
ma-55	318	14	.	.	PUNCT
ma-55	319	1	then	then	ADV
ma-55	319	2	|j1|	|j1|	VERB
ma-55	319	3	≤	≤	ADV
ma-55	319	4	1	1	NUM
ma-55	319	5	2	2	NUM
ma-55	319	6	|∆uεmt	|∆uεmt	X
ma-55	319	7	(	(	PUNCT
ma-55	319	8	t)|2	t)|2	NOUN
ma-55	319	9	+	+	CCONJ
ma-55	319	10	c|∇∆uεm(t)|2	c|∇∆uεm(t)|2	PROPN
ma-55	319	11	.	.	PUNCT
ma-55	320	1	(	(	PUNCT
ma-55	320	2	5.23	5.23	NUM
ma-55	320	3	)	)	PUNCT
ma-55	320	4	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	320	5	eur	eur	PROPN
ma-55	320	6	.	.	PUNCT
ma-55	321	1	j.	j.	PROPN
ma-55	321	2	math	math	PROPN
ma-55	321	3	.	.	PUNCT
ma-55	322	1	anal	anal	PROPN
ma-55	322	2	.	.	PUNCT
ma-55	323	1	10.28924	10.28924	NUM
ma-55	323	2	/	/	SYM
ma-55	323	3	ada	ada	PROPN
ma-55	323	4	/	/	SYM
ma-55	323	5	ma.2.5	ma.2.5	PROPN
ma-55	323	6	14let	14let	ADJ
ma-55	323	7	ω1	ω1	PROPN
ma-55	323	8	=	=	PUNCT
ma-55	323	9	{	{	PUNCT
ma-55	323	10	x	x	PUNCT
ma-55	323	11	∈	∈	PROPN
ma-55	323	12	ω	ω	NOUN
ma-55	323	13	:	:	PUNCT
ma-55	323	14	|uεm(t)|	|uεm(t)|	ADP
ma-55	323	15	<	<	X
ma-55	323	16	1	1	NUM
ma-55	323	17	}	}	PUNCT
ma-55	323	18	and	and	CCONJ
ma-55	323	19	ω2	ω2	NOUN
ma-55	323	20	=	=	SYM
ma-55	323	21	{	{	PUNCT
ma-55	323	22	x	x	PUNCT
ma-55	323	23	∈	∈	PROPN
ma-55	323	24	ω	ω	NOUN
ma-55	323	25	:	:	PUNCT
ma-55	323	26	|uεm(t)|	|uεm(t)|	NUM
ma-55	323	27	≥	≥	NOUN
ma-55	323	28	1	1	NUM
ma-55	323	29	}	}	PUNCT
ma-55	323	30	.	.	PUNCT
ma-55	324	1	by	by	ADP
ma-55	324	2	(	(	PUNCT
ma-55	324	3	5.9	5.9	NUM
ma-55	324	4	)	)	PUNCT
ma-55	324	5	and	and	CCONJ
ma-55	324	6	sobolevinequality	sobolevinequality	NOUN
ma-55	324	7	|j2|	|j2|	PROPN
ma-55	324	8	≤	≤	NUM
ma-55	324	9	∫	∫	PROPN
ma-55	324	10	ω1	ω1	PROPN
ma-55	324	11	||uεm(t)|r−2uεm	||uεm(t)|r−2uεm	PROPN
ma-55	324	12	ln	ln	PROPN
ma-55	324	13	|uεm(t)|∆uεmt	|uεm(t)|∆uεmt	X
ma-55	324	14	(	(	PUNCT
ma-55	324	15	t)|dx	t)|dx	PROPN
ma-55	324	16	+	+	CCONJ
ma-55	324	17	∫	∫	PROPN
ma-55	324	18	ω2	ω2	PROPN
ma-55	324	19	||uεm(t)|r−2uεm	||uεm(t)|r−2uεm	PROPN
ma-55	324	20	ln	ln	PROPN
ma-55	324	21	|uεm(t)|∆uεmt	|uεm(t)|∆uεmt	X
ma-55	324	22	(	(	PUNCT
ma-55	324	23	t)|dx	t)|dx	PROPN
ma-55	324	24	≤	≤	NOUN
ma-55	324	25	(	(	PUNCT
ma-55	324	26	e(r	e(r	CCONJ
ma-55	324	27	−	−	PROPN
ma-55	324	28	1))−1	1))−1	NUM
ma-55	324	29	∫	∫	PROPN
ma-55	324	30	ω	ω	X
ma-55	324	31	|∆uεmt	|∆uεmt	X
ma-55	324	32	(	(	PUNCT
ma-55	324	33	t)|dx	t)|dx	NOUN
ma-55	324	34	+	+	CCONJ
ma-55	324	35	(	(	PUNCT
ma-55	324	36	e(r	e(r	CCONJ
ma-55	324	37	−	−	PROPN
ma-55	324	38	1))−1	1))−1	NUM
ma-55	324	39	∫	∫	PROPN
ma-55	324	40	ω	ω	PROPN
ma-55	324	41	|uεm(t)|r−1|∆uεmt	|uεm(t)|r−1|∆uεmt	ADV
ma-55	324	42	(	(	PUNCT
ma-55	324	43	t)|dx	t)|dx	PROPN
ma-55	324	44	≤	≤	NOUN
ma-55	324	45	2(e(r	2(e(r	NUM
ma-55	324	46	−	−	NUM
ma-55	324	47	1))−2	1))−2	NUM
ma-55	325	1	+	+	CCONJ
ma-55	325	2	1	1	NUM
ma-55	325	3	8	8	NUM
ma-55	325	4	|∆uεmt	|∆uεmt	X
ma-55	325	5	(	(	PUNCT
ma-55	325	6	t)|2	t)|2	NOUN
ma-55	325	7	+	+	CCONJ
ma-55	325	8	2(e(r	2(e(r	NUM
ma-55	325	9	−	−	NUM
ma-55	325	10	1))−2|uεm(t)|2(r−1	1))−2|uεm(t)|2(r−1	NUM
ma-55	325	11	)	)	PUNCT
ma-55	325	12	2(r−1	2(r−1	NOUN
ma-55	325	13	)	)	PUNCT
ma-55	326	1	+	+	CCONJ
ma-55	326	2	1	1	NUM
ma-55	326	3	8	8	NUM
ma-55	326	4	|∆uεmt	|∆uεmt	X
ma-55	326	5	(	(	PUNCT
ma-55	326	6	t)|2	t)|2	PROPN
ma-55	326	7	≤	≤	NOUN
ma-55	326	8	2(e(r	2(e(r	NUM
ma-55	326	9	−	−	NUM
ma-55	326	10	1))−2	1))−2	NUM
ma-55	327	1	+	+	CCONJ
ma-55	327	2	1	1	NUM
ma-55	327	3	4	4	NUM
ma-55	327	4	|∆uεmt	|∆uεmt	X
ma-55	327	5	(	(	PUNCT
ma-55	327	6	t)|2	t)|2	NOUN
ma-55	327	7	+	+	X
ma-55	327	8	2c(e(r	2c(e(r	NUM
ma-55	327	9	−	−	NUM
ma-55	327	10	1))−2|∇uεm(t)|2(r−1	1))−2|∇uεm(t)|2(r−1	NUM
ma-55	327	11	)	)	PUNCT
ma-55	327	12	≤	≤	NOUN
ma-55	328	1	c	c	NOUN
ma-55	329	1	+	+	NOUN
ma-55	329	2	1	1	NUM
ma-55	329	3	4	4	NUM
ma-55	329	4	|∆uεmt	|∆uεmt	X
ma-55	329	5	(	(	PUNCT
ma-55	329	6	t)|2	t)|2	NOUN
ma-55	329	7	(	(	PUNCT
ma-55	329	8	5.24	5.24	NUM
ma-55	329	9	)	)	PUNCT
ma-55	329	10	where	where	SCONJ
ma-55	329	11	we	we	PRON
ma-55	329	12	have	have	AUX
ma-55	329	13	used	use	VERB
ma-55	329	14	|x	|x	NOUN
ma-55	330	1	r−1	r−1	PROPN
ma-55	330	2	ln	ln	NOUN
ma-55	330	3	x	x	INTJ
ma-55	331	1	|	|	ADV
ma-55	331	2	≤	≤	NUM
ma-55	331	3	(	(	PUNCT
ma-55	331	4	e(r	e(r	CCONJ
ma-55	331	5	−	−	PROPN
ma-55	331	6	1))−1	1))−1	NUM
ma-55	331	7	for	for	ADP
ma-55	331	8	0	0	NUM
ma-55	331	9	<	<	X
ma-55	331	10	x	x	X
ma-55	331	11	<	<	X
ma-55	331	12	1	1	NUM
ma-55	331	13	and	and	CCONJ
ma-55	331	14	ln	ln	ADJ
ma-55	331	15	x	x	SYM
ma-55	331	16	≤	≤	X
ma-55	331	17	(	(	PUNCT
ma-55	331	18	e(r	e(r	CCONJ
ma-55	331	19	−	−	PROPN
ma-55	331	20	1))−1x	1))−1x	NUM
ma-55	331	21	r−1	r−1	PROPN
ma-55	331	22	,	,	PUNCT
ma-55	331	23	if	if	SCONJ
ma-55	331	24	x	x	PRON
ma-55	331	25	≥	≥	NUM
ma-55	331	26	1	1	NUM
ma-55	331	27	.	.	PUNCT
ma-55	331	28	remark	remark	VERB
ma-55	331	29	5.1	5.1	NUM
ma-55	331	30	.	.	PUNCT
ma-55	332	1	we	we	PRON
ma-55	332	2	note	note	VERB
ma-55	332	3	from	from	ADP
ma-55	332	4	the	the	DET
ma-55	332	5	cauchy	cauchy	PROPN
ma-55	332	6	-	-	PUNCT
ma-55	332	7	schwarz	schwarz	PROPN
ma-55	332	8	inequality	inequality	NOUN
ma-55	332	9	and	and	CCONJ
ma-55	332	10	fubini	fubini	NOUN
ma-55	332	11	’s	’s	PART
ma-55	332	12	theorem	theorem	NOUN
ma-55	332	13	follows	follow	VERB
ma-55	332	14	‖g	‖g	PROPN
ma-55	332	15	�	�	PROPN
ma-55	332	16	∇u‖l2(q	∇u‖l2(q	PROPN
ma-55	332	17	)	)	PUNCT
ma-55	332	18	≤	≤	NOUN
ma-55	333	1	‖g‖l1(0,∞)‖∇u‖l2(q	‖g‖l1(0,∞)‖∇u‖l2(q	NOUN
ma-55	333	2	)	)	PUNCT
ma-55	333	3	again	again	ADV
ma-55	333	4	from	from	ADP
ma-55	333	5	estimate	estimate	NOUN
ma-55	333	6	(	(	PUNCT
ma-55	333	7	5.9	5.9	NUM
ma-55	333	8	)	)	PUNCT
ma-55	333	9	and	and	CCONJ
ma-55	333	10	remark	remark	VERB
ma-55	333	11	5.1	5.1	NUM
ma-55	333	12	|j3|	|j3|	ADJ
ma-55	333	13	≤	≤	PROPN
ma-55	333	14	(	(	PUNCT
ma-55	333	15	∫	∫	PROPN
ma-55	333	16	t	t	PROPN
ma-55	333	17	0	0	NUM
ma-55	333	18	g(t	g(t	PROPN
ma-55	333	19	−	−	PROPN
ma-55	333	20	s)|∆uεm(t)|ds	s)|∆uεm(t)|ds	NOUN
ma-55	333	21	)	)	PUNCT
ma-55	333	22	|∆uεmt	|∆uεmt	X
ma-55	334	1	(	(	PUNCT
ma-55	334	2	t)|	t)|	INTJ
ma-55	334	3	(	(	PUNCT
ma-55	334	4	5.25	5.25	NUM
ma-55	334	5	)	)	PUNCT
ma-55	334	6	≤	≤	NOUN
ma-55	334	7	c‖g‖l1(r+)|∆uεmt	c‖g‖l1(r+)|∆uεmt	VERB
ma-55	334	8	(	(	PUNCT
ma-55	334	9	t)|	t)|	ADV
ma-55	334	10	≤	≤	ADJ
ma-55	334	11	c	c	NOUN
ma-55	334	12	+	+	NOUN
ma-55	334	13	1	1	NUM
ma-55	334	14	4	4	NUM
ma-55	334	15	|∆uεmt	|∆uεmt	X
ma-55	334	16	(	(	PUNCT
ma-55	334	17	t)|2	t)|2	X
ma-55	334	18	.	.	PROPN
ma-55	334	19	follows	follow	VERB
ma-55	334	20	from	from	ADP
ma-55	334	21	(	(	PUNCT
ma-55	334	22	5.22)-(5.25	5.22)-(5.25	NUM
ma-55	334	23	)	)	PUNCT
ma-55	334	24	that	that	SCONJ
ma-55	335	1	d	d	NOUN
ma-55	335	2	dt	dt	X
ma-55	336	1	[	[	PUNCT
ma-55	336	2	1	1	NUM
ma-55	336	3	2	2	NUM
ma-55	336	4	|∇uεmt	|∇uεmt	X
ma-55	336	5	(	(	PUNCT
ma-55	336	6	t)|2	t)|2	NOUN
ma-55	336	7	+	+	CCONJ
ma-55	336	8	1	1	NUM
ma-55	336	9	2	2	NUM
ma-55	336	10	|∇∆uεm(t)|2	|∇∆uεm(t)|2	NOUN
ma-55	336	11	+	+	CCONJ
ma-55	336	12	〈	〈	PROPN
ma-55	336	13	∆puεm(t),∆uεm(t	∆puεm(t),∆uεm(t	NOUN
ma-55	336	14	)	)	PUNCT
ma-55	336	15	〉	〉	NOUN
ma-55	336	16	]	]	PUNCT
ma-55	337	1	+	+	CCONJ
ma-55	337	2	1	1	NUM
ma-55	337	3	2	2	NUM
ma-55	337	4	|∆uεmt	|∆uεmt	X
ma-55	337	5	(	(	PUNCT
ma-55	337	6	t)|2	t)|2	NOUN
ma-55	337	7	+	+	CCONJ
ma-55	337	8	1	1	NUM
ma-55	337	9	ε	ε	PROPN
ma-55	337	10	(	(	PUNCT
ma-55	337	11	β(uεmt	β(uεmt	X
ma-55	337	12	(	(	PUNCT
ma-55	337	13	t)),−∆uεmt	t)),−∆uεmt	NOUN
ma-55	337	14	(	(	PUNCT
ma-55	337	15	t	t	NOUN
ma-55	337	16	)	)	PUNCT
ma-55	337	17	)	)	PUNCT
ma-55	337	18	≤	≤	NUM
ma-55	338	1	c	c	X
ma-55	338	2	+	+	CCONJ
ma-55	338	3	c|∇∆uεm(t)|2	c|∇∆uεm(t)|2	PROPN
ma-55	338	4	.	.	PUNCT
ma-55	339	1	(	(	PUNCT
ma-55	339	2	5.26	5.26	NUM
ma-55	339	3	)	)	PUNCT
ma-55	339	4	now	now	ADV
ma-55	339	5	,	,	PUNCT
ma-55	339	6	observe	observe	VERB
ma-55	339	7	that	that	SCONJ
ma-55	339	8	|〈∆puεm(t),∆uεm(t)〉|	|〈∆puεm(t),∆uεm(t)〉|	ADJ
ma-55	339	9	≤	≤	NUM
ma-55	339	10	∫	∫	PROPN
ma-55	339	11	ω	ω	PROPN
ma-55	339	12	|∇uεm(t)|p−1|∇∆uεm(t)|dx	|∇uεm(t)|p−1|∇∆uεm(t)|dx	PROPN
ma-55	339	13	≤	≤	X
ma-55	339	14	|∆uεm(t)|p−1	|∆uεm(t)|p−1	CCONJ
ma-55	339	15	2(p−1	2(p−1	ADJ
ma-55	339	16	)	)	PUNCT
ma-55	339	17	|∇∆uεm(t)|	|∇∆uεm(t)|	ADJ
ma-55	339	18	(	(	PUNCT
ma-55	339	19	5.27	5.27	NUM
ma-55	339	20	)	)	PUNCT
ma-55	339	21	≤	≤	NOUN
ma-55	340	1	c	c	NOUN
ma-55	340	2	+	+	CCONJ
ma-55	340	3	|∇∆uεm(t)|2	|∇∆uεm(t)|2	NOUN
ma-55	340	4	,	,	PUNCT
ma-55	340	5	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	340	6	eur	eur	PROPN
ma-55	340	7	.	.	PUNCT
ma-55	341	1	j.	j.	PROPN
ma-55	341	2	math	math	PROPN
ma-55	341	3	.	.	PUNCT
ma-55	342	1	anal	anal	PROPN
ma-55	342	2	.	.	PUNCT
ma-55	343	1	10.28924	10.28924	NUM
ma-55	343	2	/	/	SYM
ma-55	343	3	ada	ada	PROPN
ma-55	343	4	/	/	SYM
ma-55	343	5	ma.2.5	ma.2.5	PROPN
ma-55	343	6	15and	15and	INTJ
ma-55	343	7	then	then	ADV
ma-55	344	1	c	c	AUX
ma-55	344	2	+	+	CCONJ
ma-55	344	3	|∇∆uεm(t)|2	|∇∆uεm(t)|2	X
ma-55	345	1	+	+	CCONJ
ma-55	345	2	〈	〈	PROPN
ma-55	345	3	∆puεm(t),∆uεm(t	∆puεm(t),∆uεm(t	NOUN
ma-55	345	4	)	)	PUNCT
ma-55	345	5	〉	〉	PROPN
ma-55	345	6	≥	≥	NOUN
ma-55	345	7	0	0	NUM
ma-55	345	8	.	.	PUNCT
ma-55	346	1	therefore	therefore	ADV
ma-55	346	2	,	,	PUNCT
ma-55	346	3	there	there	PRON
ma-55	346	4	exists	exist	VERB
ma-55	346	5	c0	c0	PROPN
ma-55	346	6	>	>	X
ma-55	346	7	0	0	PUNCT
ma-55	347	1	such	such	ADJ
ma-55	347	2	that	that	SCONJ
ma-55	347	3	d	d	NOUN
ma-55	347	4	dt	dt	X
ma-55	347	5	[	[	PUNCT
ma-55	347	6	1	1	NUM
ma-55	347	7	2	2	NUM
ma-55	347	8	|∇uεmt	|∇uεmt	X
ma-55	347	9	(	(	PUNCT
ma-55	347	10	t)|2	t)|2	NOUN
ma-55	347	11	+	+	CCONJ
ma-55	347	12	1	1	NUM
ma-55	347	13	2	2	NUM
ma-55	347	14	|∇∆uεm(t)|2	|∇∆uεm(t)|2	NOUN
ma-55	347	15	+	+	CCONJ
ma-55	347	16	〈	〈	PROPN
ma-55	347	17	∆puεm(t),∆uεm(t	∆puεm(t),∆uεm(t	NOUN
ma-55	347	18	)	)	PUNCT
ma-55	347	19	〉	〉	NOUN
ma-55	347	20	]	]	PUNCT
ma-55	348	1	+	+	CCONJ
ma-55	348	2	1	1	NUM
ma-55	348	3	2	2	NUM
ma-55	348	4	|∆uεmt	|∆uεmt	X
ma-55	348	5	(	(	PUNCT
ma-55	348	6	t)|2	t)|2	NOUN
ma-55	348	7	+	+	CCONJ
ma-55	348	8	1	1	NUM
ma-55	348	9	ε	ε	PROPN
ma-55	348	10	(	(	PUNCT
ma-55	348	11	β(uεmt	β(uεmt	X
ma-55	348	12	(	(	PUNCT
ma-55	348	13	t)),−∆uεmt	t)),−∆uεmt	NOUN
ma-55	348	14	(	(	PUNCT
ma-55	348	15	t	t	NOUN
ma-55	348	16	)	)	PUNCT
ma-55	348	17	)	)	PUNCT
ma-55	348	18	≤	≤	NUM
ma-55	348	19	c0	c0	NOUN
ma-55	348	20	+	+	CCONJ
ma-55	348	21	c0|∇∆uεm(t)|2	c0|∇∆uεm(t)|2	NOUN
ma-55	348	22	+	+	CCONJ
ma-55	348	23	〈	〈	PROPN
ma-55	348	24	∆puεm(t),∆uεm(t	∆puεm(t),∆uεm(t	NOUN
ma-55	348	25	)	)	PUNCT
ma-55	348	26	〉	〉	PROPN
ma-55	348	27	.	.	PUNCT
ma-55	349	1	(	(	PUNCT
ma-55	349	2	5.28	5.28	NUM
ma-55	349	3	)	)	PUNCT
ma-55	349	4	taking	take	VERB
ma-55	349	5	into	into	ADP
ma-55	349	6	account	account	NOUN
ma-55	349	7	that	that	SCONJ
ma-55	349	8	(	(	PUNCT
ma-55	349	9	β(uεmt	β(uεmt	X
ma-55	349	10	(	(	PUNCT
ma-55	349	11	t),−∆uεmt	t),−∆uεmt	X
ma-55	349	12	(	(	PUNCT
ma-55	349	13	t	t	PROPN
ma-55	349	14	)	)	PUNCT
ma-55	349	15	)	)	PUNCT
ma-55	349	16	≥	≥	NOUN
ma-55	349	17	0	0	NUM
ma-55	349	18	,	,	PUNCT
ma-55	349	19	(	(	PUNCT
ma-55	349	20	5.21	5.21	NUM
ma-55	349	21	)	)	PUNCT
ma-55	349	22	,	,	PUNCT
ma-55	349	23	integrating	integrate	VERB
ma-55	349	24	from	from	ADP
ma-55	349	25	0	0	NUM
ma-55	349	26	to	to	ADP
ma-55	349	27	t	t	PROPN
ma-55	349	28	and	and	CCONJ
ma-55	349	29	applyinggronwall	applyinggronwall	PROPN
ma-55	349	30	inequality	inequality	NOUN
ma-55	349	31	,	,	PUNCT
ma-55	349	32	we	we	PRON
ma-55	349	33	obtain	obtain	VERB
ma-55	349	34	|∇uεmt	|∇uεmt	X
ma-55	349	35	(	(	PUNCT
ma-55	349	36	t)|2	t)|2	NOUN
ma-55	349	37	+	+	CCONJ
ma-55	349	38	|∇∆uεm(t)|2	|∇∆uεm(t)|2	NOUN
ma-55	349	39	+	+	NUM
ma-55	349	40	∫	∫	PROPN
ma-55	349	41	t	t	PROPN
ma-55	349	42	0	0	NUM
ma-55	349	43	|∆uεmt	|∆uεmt	X
ma-55	349	44	(	(	PUNCT
ma-55	349	45	t)|2	t)|2	PROPN
ma-55	349	46	≤	≤	NUM
ma-55	349	47	c	c	NOUN
ma-55	349	48	,	,	PUNCT
ma-55	349	49	(	(	PUNCT
ma-55	349	50	5.29	5.29	NUM
ma-55	349	51	)	)	PUNCT
ma-55	349	52	then	then	ADV
ma-55	349	53	uεm	uεm	ADV
ma-55	350	1	⇀	⇀	X
ma-55	350	2	uε	uε	NOUN
ma-55	350	3	in	in	ADP
ma-55	350	4	l∞(0	l∞(0	PRON
ma-55	350	5	,	,	PUNCT
ma-55	350	6	t	t	PROPN
ma-55	350	7	;	;	PUNCT
ma-55	350	8	h3	h3	NOUN
ma-55	350	9	γ(ω	γ(ω	PROPN
ma-55	350	10	)	)	PUNCT
ma-55	350	11	)	)	PUNCT
ma-55	350	12	,	,	PUNCT
ma-55	350	13	weakly	weakly	ADJ
ma-55	350	14	star	star	NOUN
ma-55	350	15	.	.	PUNCT
ma-55	351	1	(	(	PUNCT
ma-55	351	2	5.30	5.30	NUM
ma-55	351	3	)	)	PUNCT
ma-55	351	4	uεmt	uεmt	NOUN
ma-55	351	5	⇀	⇀	NUM
ma-55	352	1	uεt	uεt	NOUN
ma-55	352	2	in	in	ADP
ma-55	352	3	l2(0	l2(0	NOUN
ma-55	352	4	,	,	PUNCT
ma-55	352	5	t	t	PROPN
ma-55	352	6	;	;	PUNCT
ma-55	352	7	h1	h1	VERB
ma-55	352	8	0(ω	0(ω	NOUN
ma-55	352	9	)	)	PUNCT
ma-55	352	10	∩h2(ω	∩h2(ω	ADJ
ma-55	352	11	)	)	PUNCT
ma-55	352	12	)	)	PUNCT
ma-55	352	13	,	,	PUNCT
ma-55	352	14	weakly	weakly	ADJ
ma-55	352	15	(	(	PUNCT
ma-55	352	16	5.31	5.31	NUM
ma-55	352	17	)	)	PUNCT
ma-55	352	18	∆uεm	∆uεm	ADJ
ma-55	352	19	⇀	⇀	PROPN
ma-55	352	20	∆uε	∆uε	X
ma-55	352	21	in	in	ADP
ma-55	352	22	l∞(0	l∞(0	ADJ
ma-55	352	23	,	,	PUNCT
ma-55	352	24	t	t	PROPN
ma-55	352	25	;	;	PUNCT
ma-55	352	26	h1	h1	NOUN
ma-55	352	27	0(ω	0(ω	ADJ
ma-55	352	28	)	)	PUNCT
ma-55	352	29	)	)	PUNCT
ma-55	352	30	,	,	PUNCT
ma-55	352	31	weakly	weakly	ADJ
ma-55	352	32	star	star	NOUN
ma-55	352	33	.	.	PUNCT
ma-55	353	1	(	(	PUNCT
ma-55	353	2	5.32	5.32	NUM
ma-55	353	3	)	)	PUNCT
ma-55	353	4	5.4	5.4	NUM
ma-55	353	5	.	.	PUNCT
ma-55	353	6	third	third	ADJ
ma-55	353	7	estimate	estimate	NOUN
ma-55	353	8	.	.	PUNCT
ma-55	354	1	let	let	VERB
ma-55	354	2	pm	pm	NOUN
ma-55	354	3	be	be	AUX
ma-55	354	4	the	the	DET
ma-55	354	5	ortogonal	ortogonal	ADJ
ma-55	354	6	projection	projection	NOUN
ma-55	354	7	pm	pm	NOUN
ma-55	354	8	:	:	PUNCT
ma-55	354	9	l2(ω)→	l2(ω)→	PROPN
ma-55	354	10	vm	vm	PROPN
ma-55	354	11	,	,	PUNCT
ma-55	354	12	that	that	PRON
ma-55	354	13	is	be	AUX
ma-55	354	14	pmφ	pmφ	NOUN
ma-55	354	15	=	=	PUNCT
ma-55	355	1	m∑	m∑	NOUN
ma-55	355	2	n=1	n=1	PROPN
ma-55	355	3	(	(	PUNCT
ma-55	355	4	φ	φ	PROPN
ma-55	355	5	,	,	PUNCT
ma-55	355	6	wj)wj	wj)wj	PROPN
ma-55	355	7	,	,	PUNCT
ma-55	355	8	φ	φ	PROPN
ma-55	355	9	∈	∈	PROPN
ma-55	355	10	l2(ω	l2(ω	PROPN
ma-55	355	11	)	)	PUNCT
ma-55	355	12	.	.	PUNCT
ma-55	356	1	remark	remark	VERB
ma-55	356	2	5.2	5.2	NUM
ma-55	356	3	.	.	PUNCT
ma-55	357	1	by	by	ADP
ma-55	357	2	remark	remark	NOUN
ma-55	357	3	5.1	5.1	NUM
ma-55	357	4	,	,	PUNCT
ma-55	357	5	we	we	PRON
ma-55	357	6	observe	observe	VERB
ma-55	357	7	that	that	SCONJ
ma-55	357	8	if	if	SCONJ
ma-55	357	9	ψ	ψ	ADP
ma-55	357	10	∈	∈	PROPN
ma-55	357	11	l2(0	l2(0	NOUN
ma-55	357	12	,	,	PUNCT
ma-55	357	13	t	t	NOUN
ma-55	357	14	;	;	PUNCT
ma-55	357	15	h1	h1	NOUN
ma-55	357	16	0(ω	0(ω	ADJ
ma-55	357	17	)	)	PUNCT
ma-55	357	18	)	)	PUNCT
ma-55	358	1	then	then	ADV
ma-55	358	2	∫	∫	PROPN
ma-55	358	3	t	t	PROPN
ma-55	358	4	0	0	NUM
ma-55	359	1	g(t	g(t	PROPN
ma-55	359	2	−	−	PROPN
ma-55	359	3	s)ψ(s)ds	s)ψ(s)ds	PROPN
ma-55	359	4	∈	∈	PROPN
ma-55	359	5	l2(0	l2(0	NOUN
ma-55	359	6	,	,	PUNCT
ma-55	359	7	t	t	PROPN
ma-55	359	8	;	;	PUNCT
ma-55	359	9	h−1(ω	h−1(ω	PROPN
ma-55	359	10	)	)	PUNCT
ma-55	359	11	)	)	PUNCT
ma-55	359	12	and	and	CCONJ
ma-55	359	13	by	by	ADP
ma-55	359	14	(	(	PUNCT
ma-55	359	15	5.12	5.12	NUM
ma-55	359	16	)	)	PUNCT
ma-55	359	17	−∆pu	−∆pu	ADP
ma-55	359	18	εm	εm	PRON
ma-55	359	19	∈	∈	PROPN
ma-55	359	20	l2(0	l2(0	NOUN
ma-55	359	21	,	,	PUNCT
ma-55	359	22	t	t	NOUN
ma-55	359	23	;	;	PUNCT
ma-55	359	24	(	(	PUNCT
ma-55	359	25	h−1(ω	h−1(ω	PROPN
ma-55	359	26	)	)	PUNCT
ma-55	359	27	)	)	PUNCT
ma-55	359	28	.	.	PUNCT
ma-55	360	1	we	we	PRON
ma-55	360	2	obtain	obtain	VERB
ma-55	360	3	using	use	VERB
ma-55	360	4	the	the	DET
ma-55	360	5	notation	notation	NOUN
ma-55	360	6	and	and	CCONJ
ma-55	360	7	ideas	idea	NOUN
ma-55	360	8	of	of	ADP
ma-55	360	9	lions	lion	NOUN
ma-55	360	10	[	[	X
ma-55	360	11	16	16	NUM
ma-55	360	12	]	]	PUNCT
ma-55	360	13	,	,	PUNCT
ma-55	360	14	pages	page	NOUN
ma-55	360	15	75	75	NUM
ma-55	360	16	-	-	SYM
ma-55	360	17	76	76	NUM
ma-55	360	18	,	,	PUNCT
ma-55	360	19	remark	remark	VERB
ma-55	360	20	5.2	5.2	NUM
ma-55	360	21	and	and	CCONJ
ma-55	360	22	estimatesabove	estimatesabove	VERB
ma-55	360	23	that	that	DET
ma-55	360	24	uεmtt	uεmtt	NOUN
ma-55	361	1	⇀	⇀	X
ma-55	361	2	uεtt	uεtt	ADJ
ma-55	361	3	in	in	ADP
ma-55	361	4	l2(0	l2(0	PROPN
ma-55	361	5	,	,	PUNCT
ma-55	361	6	t	t	NOUN
ma-55	361	7	;	;	PUNCT
ma-55	361	8	(	(	PUNCT
ma-55	361	9	h−1(ω	h−1(ω	PROPN
ma-55	361	10	)	)	PUNCT
ma-55	361	11	)	)	PUNCT
ma-55	361	12	,	,	PUNCT
ma-55	361	13	weakly	weakly	ADV
ma-55	361	14	.	.	PUNCT
ma-55	362	1	(	(	PUNCT
ma-55	362	2	5.33	5.33	NUM
ma-55	362	3	)	)	PUNCT
ma-55	362	4	(	(	PUNCT
ma-55	362	5	5.31	5.31	NUM
ma-55	362	6	)	)	PUNCT
ma-55	362	7	,	,	PUNCT
ma-55	362	8	(	(	PUNCT
ma-55	362	9	5.33	5.33	NUM
ma-55	362	10	)	)	PUNCT
ma-55	362	11	and	and	CCONJ
ma-55	362	12	aubin	aubin	PROPN
ma-55	362	13	-	-	PUNCT
ma-55	362	14	lions	lion	NOUN
ma-55	362	15	compactness	compactness	NOUN
ma-55	362	16	theorem	theorem	NOUN
ma-55	362	17	imply	imply	VERB
ma-55	362	18	that	that	SCONJ
ma-55	362	19	there	there	PRON
ma-55	362	20	exists	exist	VERB
ma-55	362	21	a	a	DET
ma-55	362	22	subsequence	subsequence	NOUN
ma-55	362	23	from	from	ADP
ma-55	362	24	(	(	PUNCT
ma-55	362	25	uεmt	uεmt	PROPN
ma-55	362	26	)	)	PUNCT
ma-55	362	27	,	,	PUNCT
ma-55	362	28	still	still	ADV
ma-55	362	29	denoted	denote	VERB
ma-55	362	30	by	by	ADP
ma-55	362	31	(	(	PUNCT
ma-55	362	32	uεmt	uεmt	PROPN
ma-55	362	33	)	)	PUNCT
ma-55	362	34	,	,	PUNCT
ma-55	362	35	such	such	ADJ
ma-55	362	36	that	that	DET
ma-55	362	37	uεmt	uεmt	NOUN
ma-55	362	38	→	→	PUNCT
ma-55	362	39	uεt	uεt	NOUN
ma-55	362	40	strongly	strongly	ADV
ma-55	362	41	in	in	ADP
ma-55	362	42	l2(0	l2(0	NOUN
ma-55	362	43	,	,	PUNCT
ma-55	362	44	t	t	PROPN
ma-55	362	45	;	;	PUNCT
ma-55	362	46	h1	h1	NOUN
ma-55	362	47	0(ω	0(ω	ADJ
ma-55	362	48	)	)	PUNCT
ma-55	362	49	)	)	PUNCT
ma-55	362	50	and	and	CCONJ
ma-55	362	51	a.e	a.e	PROPN
ma-55	362	52	.	.	PROPN
ma-55	362	53	in	in	ADP
ma-55	362	54	q.	q.	PROPN
ma-55	362	55	(	(	PUNCT
ma-55	362	56	5.34	5.34	NUM
ma-55	362	57	)	)	PUNCT
ma-55	362	58	now	now	ADV
ma-55	362	59	,	,	PUNCT
ma-55	362	60	we	we	PRON
ma-55	362	61	are	be	AUX
ma-55	362	62	in	in	ADP
ma-55	362	63	position	position	NOUN
ma-55	362	64	to	to	PART
ma-55	362	65	prove	prove	VERB
ma-55	362	66	theorem	theorem	VERB
ma-55	362	67	4.1	4.1	NUM
ma-55	362	68	.	.	PUNCT
ma-55	363	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	363	2	eur	eur	PROPN
ma-55	363	3	.	.	PUNCT
ma-55	364	1	j.	j.	PROPN
ma-55	364	2	math	math	PROPN
ma-55	364	3	.	.	PUNCT
ma-55	365	1	anal	anal	PROPN
ma-55	365	2	.	.	PUNCT
ma-55	366	1	10.28924	10.28924	NUM
ma-55	366	2	/	/	SYM
ma-55	366	3	ada	ada	PROPN
ma-55	366	4	/	/	SYM
ma-55	366	5	ma.2.5	ma.2.5	PROPN
ma-55	366	6	165.5	165.5	NUM
ma-55	366	7	.	.	PUNCT
ma-55	366	8	strong	strong	ADJ
ma-55	366	9	solution	solution	NOUN
ma-55	366	10	.	.	PUNCT
ma-55	367	1	let	let	VERB
ma-55	367	2	v	v	NUM
ma-55	367	3	∈	∈	PROPN
ma-55	367	4	l2(0	l2(0	NOUN
ma-55	367	5	,	,	PUNCT
ma-55	367	6	t	t	PROPN
ma-55	367	7	;	;	PUNCT
ma-55	367	8	h1	h1	NOUN
ma-55	367	9	0(ω	0(ω	ADJ
ma-55	367	10	)	)	PUNCT
ma-55	367	11	)	)	PUNCT
ma-55	367	12	be	be	AUX
ma-55	367	13	v(t	v(t	NOUN
ma-55	367	14	)	)	PUNCT
ma-55	367	15	∈	∈	PROPN
ma-55	367	16	k	k	PROPN
ma-55	367	17	a.	a.	PROPN
ma-55	367	18	e.	e.	PROPN
ma-55	367	19	for	for	ADP
ma-55	367	20	t	t	PROPN
ma-55	367	21	∈	∈	PROPN
ma-55	367	22	(	(	PUNCT
ma-55	367	23	0	0	NUM
ma-55	367	24	,	,	PUNCT
ma-55	367	25	t	t	NOUN
ma-55	367	26	)	)	PUNCT
ma-55	367	27	.	.	PUNCT
ma-55	368	1	from	from	ADP
ma-55	368	2	(	(	PUNCT
ma-55	368	3	4.6)1follows	4.6)1follow	NOUN
ma-55	368	4	that	that	PRON
ma-55	368	5	∫	∫	PROPN
ma-55	368	6	t	t	NOUN
ma-55	368	7	0	0	NUM
ma-55	368	8	(	(	PUNCT
ma-55	368	9	uεtt	uεtt	ADJ
ma-55	368	10	,	,	PUNCT
ma-55	368	11	v	v	ADP
ma-55	368	12	−	−	PROPN
ma-55	368	13	uεt	uεt	NOUN
ma-55	368	14	)	)	PUNCT
ma-55	368	15	dt	dt	PUNCT
ma-55	369	1	+	+	CCONJ
ma-55	369	2	∫	∫	PROPN
ma-55	369	3	t	t	PROPN
ma-55	369	4	0	0	NUM
ma-55	369	5	(	(	PUNCT
ma-55	369	6	∆2uε	∆2uε	PROPN
ma-55	369	7	,	,	PUNCT
ma-55	369	8	v	v	ADP
ma-55	369	9	−	−	PROPN
ma-55	369	10	uεt	uεt	NOUN
ma-55	369	11	)	)	PUNCT
ma-55	369	12	dt	dt	PUNCT
ma-55	370	1	+	+	CCONJ
ma-55	370	2	∫	∫	PROPN
ma-55	370	3	t	t	PROPN
ma-55	370	4	0	0	NUM
ma-55	370	5	(	(	PUNCT
ma-55	370	6	−∆pu	−∆pu	PROPN
ma-55	370	7	ε	ε	PROPN
ma-55	370	8	,	,	PUNCT
ma-55	370	9	v	v	ADP
ma-55	370	10	−	−	PROPN
ma-55	370	11	uεt	uεt	NOUN
ma-55	370	12	)	)	PUNCT
ma-55	370	13	dt	dt	PUNCT
ma-55	371	1	+	+	CCONJ
ma-55	371	2	∫	∫	PROPN
ma-55	371	3	t	t	PROPN
ma-55	371	4	0	0	NUM
ma-55	372	1	(	(	PUNCT
ma-55	372	2	∫	∫	PROPN
ma-55	372	3	t	t	PROPN
ma-55	372	4	0	0	NUM
ma-55	372	5	g(t	g(t	PROPN
ma-55	372	6	−	−	PROPN
ma-55	372	7	s)∆uε(s)ds	s)∆uε(s)ds	PROPN
ma-55	372	8	,	,	PUNCT
ma-55	372	9	v	v	ADP
ma-55	372	10	−	−	PROPN
ma-55	372	11	uεt	uεt	NOUN
ma-55	372	12	)	)	PUNCT
ma-55	372	13	dt	dt	PUNCT
ma-55	373	1	+	+	CCONJ
ma-55	373	2	∫	∫	PROPN
ma-55	373	3	t	t	PROPN
ma-55	373	4	0	0	NUM
ma-55	373	5	(	(	PUNCT
ma-55	373	6	−∆uεt	−∆uεt	NOUN
ma-55	373	7	,	,	PUNCT
ma-55	373	8	v	v	ADP
ma-55	373	9	−	−	PROPN
ma-55	373	10	uεt	uεt	NOUN
ma-55	373	11	)	)	PUNCT
ma-55	373	12	dt	dt	PART
ma-55	374	1	−	−	PROPN
ma-55	374	2	∫	∫	PROPN
ma-55	374	3	t	t	PROPN
ma-55	374	4	0	0	NUM
ma-55	374	5	(	(	PUNCT
ma-55	374	6	|uε|r−2uε	|uε|r−2uε	PROPN
ma-55	374	7	ln	ln	ADJ
ma-55	374	8	|uε|	|uε|	PROPN
ma-55	374	9	,	,	PUNCT
ma-55	374	10	v	v	ADP
ma-55	374	11	−	−	PROPN
ma-55	374	12	uεt	uεt	NOUN
ma-55	374	13	)	)	PUNCT
ma-55	374	14	dt	dt	NOUN
ma-55	375	1	=	=	SYM
ma-55	375	2	1	1	NUM
ma-55	375	3	ε	ε	PROPN
ma-55	375	4	∫	∫	PROPN
ma-55	375	5	t	t	PROPN
ma-55	375	6	0	0	NUM
ma-55	375	7	(	(	PUNCT
ma-55	375	8	β(uεt	β(uεt	NOUN
ma-55	375	9	)	)	PUNCT
ma-55	375	10	,	,	PUNCT
ma-55	375	11	uεt	uεt	ADP
ma-55	375	12	−	−	PROPN
ma-55	375	13	v	v	NOUN
ma-55	375	14	)	)	PUNCT
ma-55	375	15	dt	dt	NOUN
ma-55	376	1	=	=	SYM
ma-55	376	2	1	1	NUM
ma-55	376	3	ε	ε	PROPN
ma-55	376	4	∫	∫	PROPN
ma-55	376	5	t	t	PROPN
ma-55	376	6	0	0	NUM
ma-55	376	7	(	(	PUNCT
ma-55	376	8	β(uεt	β(uεt	NOUN
ma-55	376	9	)	)	PUNCT
ma-55	376	10	−	−	PROPN
ma-55	376	11	βv	βv	NOUN
ma-55	376	12	,	,	PUNCT
ma-55	376	13	uεt	uεt	ADP
ma-55	376	14	−	−	PROPN
ma-55	376	15	v	v	NOUN
ma-55	376	16	)	)	PUNCT
ma-55	376	17	dt	dt	X
ma-55	376	18	≥	≥	NOUN
ma-55	376	19	0	0	NUM
ma-55	376	20	,	,	PUNCT
ma-55	376	21	(	(	PUNCT
ma-55	376	22	5.35	5.35	NUM
ma-55	376	23	)	)	PUNCT
ma-55	376	24	because	because	SCONJ
ma-55	376	25	v	v	NUM
ma-55	376	26	∈	∈	PROPN
ma-55	376	27	k	k	X
ma-55	376	28	(	(	PUNCT
ma-55	376	29	β(v	β(v	ADJ
ma-55	376	30	)	)	PUNCT
ma-55	376	31	=	=	SYM
ma-55	376	32	0	0	X
ma-55	376	33	)	)	PUNCT
ma-55	376	34	and	and	CCONJ
ma-55	376	35	β	β	X
ma-55	376	36	is	be	AUX
ma-55	376	37	monotone.from	monotone.from	PROPN
ma-55	376	38	(	(	PUNCT
ma-55	376	39	5.11	5.11	NUM
ma-55	376	40	)	)	PUNCT
ma-55	376	41	,	,	PUNCT
ma-55	376	42	(	(	PUNCT
ma-55	376	43	5.12	5.12	NUM
ma-55	376	44	)	)	PUNCT
ma-55	376	45	,	,	PUNCT
ma-55	376	46	(	(	PUNCT
ma-55	376	47	5.15	5.15	NUM
ma-55	376	48	)	)	PUNCT
ma-55	376	49	,	,	PUNCT
ma-55	376	50	(	(	PUNCT
ma-55	376	51	5.20	5.20	NUM
ma-55	376	52	)	)	PUNCT
ma-55	376	53	,	,	PUNCT
ma-55	376	54	(	(	PUNCT
ma-55	376	55	5.30	5.30	NUM
ma-55	376	56	)	)	PUNCT
ma-55	376	57	,	,	PUNCT
ma-55	376	58	(	(	PUNCT
ma-55	376	59	5.31	5.31	NUM
ma-55	376	60	)	)	PUNCT
ma-55	376	61	,	,	PUNCT
ma-55	376	62	(	(	PUNCT
ma-55	376	63	5.33	5.33	NUM
ma-55	376	64	)	)	PUNCT
ma-55	376	65	,	,	PUNCT
ma-55	376	66	(	(	PUNCT
ma-55	376	67	5.34	5.34	NUM
ma-55	376	68	)	)	PUNCT
ma-55	376	69	and	and	CCONJ
ma-55	376	70	the	the	DET
ma-55	376	71	bannach	bannach	NOUN
ma-55	376	72	-	-	PUNCT
ma-55	376	73	steinhauss	steinhaus	VERB
ma-55	376	74	the	the	DET
ma-55	376	75	-	-	PUNCT
ma-55	376	76	orem	orem	NOUN
ma-55	376	77	,	,	PUNCT
ma-55	376	78	it	it	PRON
ma-55	376	79	follows	follow	VERB
ma-55	376	80	that	that	SCONJ
ma-55	376	81	there	there	PRON
ma-55	376	82	exists	exist	VERB
ma-55	376	83	a	a	DET
ma-55	376	84	subsequence	subsequence	NOUN
ma-55	376	85	(	(	PUNCT
ma-55	376	86	uε)0	uε)0	NOUN
ma-55	376	87	<	<	X
ma-55	376	88	ε<1	ε<1	PROPN
ma-55	376	89	,	,	PUNCT
ma-55	376	90	such	such	ADJ
ma-55	376	91	that	that	SCONJ
ma-55	376	92	it	it	PRON
ma-55	376	93	converge	converge	VERB
ma-55	376	94	to	to	ADP
ma-55	376	95	u	u	PRON
ma-55	376	96	as	as	ADP
ma-55	376	97	ε	ε	PROPN
ma-55	376	98	→	→	PUNCT
ma-55	376	99	0,that	0,that	PRON
ma-55	376	100	is	be	AUX
ma-55	376	101	uε	uε	ADP
ma-55	376	102	⇀	⇀	PROPN
ma-55	376	103	u	u	NOUN
ma-55	376	104	in	in	ADP
ma-55	376	105	l∞(r+;h1	l∞(r+;h1	PROPN
ma-55	376	106	0(ω	0(ω	NUM
ma-55	376	107	)	)	PUNCT
ma-55	376	108	∩h2(ω	∩h2(ω	ADJ
ma-55	376	109	)	)	PUNCT
ma-55	376	110	)	)	PUNCT
ma-55	376	111	,	,	PUNCT
ma-55	376	112	(	(	PUNCT
ma-55	376	113	5.36	5.36	NUM
ma-55	376	114	)	)	PUNCT
ma-55	377	1	−∆puε	−∆puε	NOUN
ma-55	377	2	⇀	⇀	NOUN
ma-55	378	1	−∆pu	−∆pu	VERB
ma-55	378	2	in	in	ADP
ma-55	378	3	l2(0	l2(0	PROPN
ma-55	378	4	,	,	PUNCT
ma-55	378	5	t	t	PROPN
ma-55	378	6	;	;	PUNCT
ma-55	378	7	h−1(ω	h−1(ω	PROPN
ma-55	378	8	)	)	PUNCT
ma-55	378	9	,	,	PUNCT
ma-55	378	10	(	(	PUNCT
ma-55	378	11	5.37	5.37	NUM
ma-55	378	12	)	)	PUNCT
ma-55	378	13	uε	uε	NOUN
ma-55	378	14	→	→	SYM
ma-55	378	15	u	u	PROPN
ma-55	378	16	in	in	ADP
ma-55	378	17	l2(0	l2(0	NOUN
ma-55	378	18	,	,	PUNCT
ma-55	378	19	t	t	PROPN
ma-55	378	20	;	;	PUNCT
ma-55	378	21	h1	h1	PROPN
ma-55	378	22	0(ω))and	0(ω))and	PROPN
ma-55	378	23	a.e	a.e	PROPN
ma-55	378	24	.	.	PROPN
ma-55	379	1	in	in	ADP
ma-55	379	2	q	q	NOUN
ma-55	379	3	,	,	PUNCT
ma-55	379	4	(	(	PUNCT
ma-55	379	5	5.38	5.38	NUM
ma-55	379	6	)	)	PUNCT
ma-55	379	7	uε	uε	NOUN
ma-55	379	8	⇀	⇀	PROPN
ma-55	379	9	u	u	PROPN
ma-55	379	10	in	in	ADP
ma-55	379	11	l∞(0	l∞(0	PRON
ma-55	379	12	,	,	PUNCT
ma-55	379	13	t	t	PROPN
ma-55	379	14	;	;	PUNCT
ma-55	379	15	h3	h3	NOUN
ma-55	379	16	γ(ω	γ(ω	PROPN
ma-55	379	17	)	)	PUNCT
ma-55	379	18	)	)	PUNCT
ma-55	379	19	,	,	PUNCT
ma-55	379	20	(	(	PUNCT
ma-55	379	21	5.39	5.39	NUM
ma-55	379	22	)	)	PUNCT
ma-55	379	23	uεt	uεt	NOUN
ma-55	379	24	⇀	⇀	X
ma-55	379	25	ut	ut	PROPN
ma-55	379	26	in	in	ADP
ma-55	379	27	l2(0	l2(0	PROPN
ma-55	379	28	,	,	PUNCT
ma-55	379	29	t	t	PROPN
ma-55	379	30	;	;	PUNCT
ma-55	379	31	h1	h1	VERB
ma-55	379	32	0(ω	0(ω	NOUN
ma-55	379	33	)	)	PUNCT
ma-55	379	34	∩h2(ω	∩h2(ω	ADJ
ma-55	379	35	)	)	PUNCT
ma-55	379	36	)	)	PUNCT
ma-55	379	37	,	,	PUNCT
ma-55	379	38	(	(	PUNCT
ma-55	379	39	5.40	5.40	NUM
ma-55	379	40	)	)	PUNCT
ma-55	379	41	uεtt	uεtt	ADV
ma-55	379	42	⇀	⇀	PUNCT
ma-55	379	43	utt	utt	NOUN
ma-55	379	44	in	in	ADP
ma-55	379	45	l2(0	l2(0	NOUN
ma-55	379	46	,	,	PUNCT
ma-55	379	47	t	t	PROPN
ma-55	379	48	;	;	PUNCT
ma-55	379	49	h−1(ω	h−1(ω	PROPN
ma-55	379	50	)	)	PUNCT
ma-55	379	51	)	)	PUNCT
ma-55	379	52	,	,	PUNCT
ma-55	379	53	(	(	PUNCT
ma-55	379	54	5.41	5.41	NUM
ma-55	379	55	)	)	PUNCT
ma-55	379	56	|uε|r−2uε	|uε|r−2uε	PROPN
ma-55	379	57	ln	ln	ADJ
ma-55	379	58	|uε|	|uε|	PROPN
ma-55	379	59	⇀	⇀	NUM
ma-55	379	60	|u|r−2u	|u|r−2u	PROPN
ma-55	379	61	ln	ln	ADJ
ma-55	379	62	|u|	|u|	PROPN
ma-55	379	63	in	in	ADP
ma-55	379	64	l2(0	l2(0	PROPN
ma-55	379	65	,	,	PUNCT
ma-55	379	66	t	t	NOUN
ma-55	379	67	;	;	PUNCT
ma-55	379	68	l2(ω	l2(ω	NOUN
ma-55	379	69	)	)	PUNCT
ma-55	379	70	)	)	PUNCT
ma-55	379	71	,	,	PUNCT
ma-55	379	72	(	(	PUNCT
ma-55	379	73	5.42	5.42	NUM
ma-55	379	74	)	)	PUNCT
ma-55	379	75	uεt	uεt	NOUN
ma-55	379	76	→	→	SYM
ma-55	379	77	ut	ut	PROPN
ma-55	379	78	in	in	ADP
ma-55	379	79	l2(0	l2(0	PROPN
ma-55	379	80	,	,	PUNCT
ma-55	379	81	t	t	PROPN
ma-55	379	82	;	;	PUNCT
ma-55	379	83	h1	h1	NOUN
ma-55	379	84	0(ω	0(ω	ADJ
ma-55	379	85	)	)	PUNCT
ma-55	379	86	)	)	PUNCT
ma-55	379	87	and	and	CCONJ
ma-55	379	88	a.e	a.e	PROPN
ma-55	379	89	.	.	PROPN
ma-55	380	1	in	in	ADP
ma-55	380	2	q.	q.	PROPN
ma-55	380	3	(	(	PUNCT
ma-55	380	4	5.43	5.43	NUM
ma-55	380	5	)	)	PUNCT
ma-55	380	6	the	the	DET
ma-55	380	7	convergences	convergence	NOUN
ma-55	380	8	above	above	ADV
ma-55	380	9	are	be	AUX
ma-55	380	10	sufficient	sufficient	ADJ
ma-55	380	11	to	to	PART
ma-55	380	12	pass	pass	VERB
ma-55	380	13	to	to	ADP
ma-55	380	14	the	the	DET
ma-55	380	15	limit	limit	NOUN
ma-55	380	16	in	in	ADP
ma-55	380	17	(	(	PUNCT
ma-55	380	18	5.35	5.35	NUM
ma-55	380	19	)	)	PUNCT
ma-55	380	20	with	with	ADP
ma-55	380	21	ε	ε	PROPN
ma-55	380	22	>	>	X
ma-55	380	23	0	0	PUNCT
ma-55	380	24	to	to	PART
ma-55	380	25	conclude	conclude	VERB
ma-55	380	26	that(4.5	that(4.5	ADV
ma-55	380	27	)	)	PUNCT
ma-55	381	1	is	be	AUX
ma-55	381	2	valid	valid	ADJ
ma-55	381	3	.	.	PUNCT
ma-55	382	1	to	to	PART
ma-55	382	2	complete	complete	VERB
ma-55	382	3	the	the	DET
ma-55	382	4	proof	proof	NOUN
ma-55	382	5	of	of	ADP
ma-55	382	6	theorem	theorem	NOUN
ma-55	382	7	4.1	4.1	NUM
ma-55	382	8	,	,	PUNCT
ma-55	382	9	it	it	PRON
ma-55	382	10	remains	remain	VERB
ma-55	382	11	to	to	PART
ma-55	382	12	show	show	VERB
ma-55	382	13	that	that	SCONJ
ma-55	382	14	ut(t	ut(t	NOUN
ma-55	382	15	)	)	PUNCT
ma-55	382	16	∈	∈	PROPN
ma-55	383	1	k	k	X
ma-55	383	2	a.e.in	a.e.in	VERB
ma-55	383	3	the	the	DET
ma-55	383	4	position	position	NOUN
ma-55	383	5	,	,	PUNCT
ma-55	383	6	we	we	PRON
ma-55	383	7	observe	observe	VERB
ma-55	383	8	that	that	SCONJ
ma-55	383	9	using	use	VERB
ma-55	383	10	convergences	convergence	NOUN
ma-55	383	11	(	(	PUNCT
ma-55	383	12	5.10)-(5.16	5.10)-(5.16	NUM
ma-55	383	13	)	)	PUNCT
ma-55	383	14	and	and	CCONJ
ma-55	383	15	(	(	PUNCT
ma-55	383	16	5.30)-(5.32	5.30)-(5.32	NUM
ma-55	383	17	)	)	PUNCT
ma-55	383	18	,	,	PUNCT
ma-55	383	19	making	make	VERB
ma-55	383	20	m	m	PRON
ma-55	383	21	→	→	SYM
ma-55	383	22	∞	∞	NUM
ma-55	383	23	in	in	ADP
ma-55	383	24	(	(	PUNCT
ma-55	383	25	5.1	5.1	NUM
ma-55	383	26	)	)	PUNCT
ma-55	383	27	,	,	PUNCT
ma-55	383	28	we	we	PRON
ma-55	383	29	can	can	AUX
ma-55	383	30	find	find	VERB
ma-55	383	31	uε	uε	ADP
ma-55	383	32	such	such	ADJ
ma-55	383	33	that	that	SCONJ
ma-55	383	34	uεtt	uεtt	ADJ
ma-55	383	35	+	+	NUM
ma-55	383	36	∆2uε	∆2uε	ADJ
ma-55	383	37	−	−	NOUN
ma-55	383	38	∆pu	∆pu	NOUN
ma-55	383	39	ε	ε	PROPN
ma-55	383	40	+	+	CCONJ
ma-55	383	41	∫	∫	PROPN
ma-55	383	42	t	t	PROPN
ma-55	383	43	0	0	NUM
ma-55	383	44	g(t	g(t	PROPN
ma-55	383	45	−	−	PROPN
ma-55	383	46	s)∆uε(s)ds	s)∆uε(s)ds	PROPN
ma-55	383	47	−	−	PROPN
ma-55	383	48	∆uεt	∆uεt	NOUN
ma-55	383	49	−	−	ADP
ma-55	383	50	|uε|r−2uε	|uε|r−2uε	ADJ
ma-55	383	51	ln	ln	ADJ
ma-55	383	52	|uε|+	|uε|+	NOUN
ma-55	383	53	1	1	NUM
ma-55	383	54	ε	ε	PROPN
ma-55	383	55	β(uεt	β(uεt	NOUN
ma-55	383	56	)	)	PUNCT
ma-55	384	1	=	=	SYM
ma-55	384	2	0	0	NUM
ma-55	384	3	in	in	ADP
ma-55	384	4	l2(0	l2(0	PROPN
ma-55	384	5	,	,	PUNCT
ma-55	384	6	t	t	PROPN
ma-55	384	7	;	;	PUNCT
ma-55	384	8	h−1(ω	h−1(ω	PROPN
ma-55	384	9	)	)	PUNCT
ma-55	384	10	.	.	PUNCT
ma-55	385	1	(	(	PUNCT
ma-55	385	2	5.44	5.44	NUM
ma-55	385	3	)	)	PUNCT
ma-55	385	4	then	then	ADV
ma-55	385	5	,	,	PUNCT
ma-55	385	6	β(uεt	β(uεt	NOUN
ma-55	385	7	)	)	PUNCT
ma-55	386	1	=	=	PUNCT
ma-55	387	1	ε[−uεtt	ε[−uεtt	NOUN
ma-55	387	2	−	−	PROPN
ma-55	387	3	∆2uε	∆2uε	NOUN
ma-55	387	4	+	+	NUM
ma-55	387	5	∆pu	∆pu	NOUN
ma-55	387	6	ε	ε	PROPN
ma-55	387	7	−	−	PROPN
ma-55	387	8	∫	∫	PROPN
ma-55	387	9	t	t	PROPN
ma-55	387	10	0	0	PROPN
ma-55	387	11	g(t	g(t	PROPN
ma-55	387	12	−	−	PROPN
ma-55	387	13	s)∆uε(s)ds	s)∆uε(s)ds	PROPN
ma-55	387	14	+	+	CCONJ
ma-55	387	15	∆uεt	∆uεt	NOUN
ma-55	387	16	+	+	CCONJ
ma-55	387	17	|uε|r−2uε	|uε|r−2uε	DET
ma-55	387	18	ln	ln	ADJ
ma-55	387	19	|uε|	|uε|	NOUN
ma-55	387	20	]	]	PUNCT
ma-55	387	21	.	.	PUNCT
ma-55	388	1	(	(	PUNCT
ma-55	388	2	5.45	5.45	NUM
ma-55	388	3	)	)	PUNCT
ma-55	388	4	so	so	ADV
ma-55	388	5	,	,	PUNCT
ma-55	388	6	β(uεt	β(uεt	NOUN
ma-55	388	7	)	)	PUNCT
ma-55	388	8	→	→	SYM
ma-55	388	9	0	0	NUM
ma-55	388	10	in	in	ADP
ma-55	388	11	d′(0	d′(0	PROPN
ma-55	388	12	,	,	PUNCT
ma-55	388	13	t	t	NOUN
ma-55	388	14	;	;	PUNCT
ma-55	388	15	h−1ω	h−1ω	X
ma-55	388	16	)	)	PUNCT
ma-55	388	17	.	.	PUNCT
ma-55	389	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	389	2	eur	eur	PROPN
ma-55	389	3	.	.	PUNCT
ma-55	390	1	j.	j.	PROPN
ma-55	390	2	math	math	PROPN
ma-55	390	3	.	.	PUNCT
ma-55	391	1	anal	anal	PROPN
ma-55	391	2	.	.	PUNCT
ma-55	392	1	10.28924	10.28924	NUM
ma-55	392	2	/	/	SYM
ma-55	392	3	ada	ada	PROPN
ma-55	392	4	/	/	SYM
ma-55	392	5	ma.2.5	ma.2.5	PROPN
ma-55	392	6	17from	17from	NUM
ma-55	392	7	(	(	PUNCT
ma-55	392	8	5.45	5.45	NUM
ma-55	392	9	)	)	PUNCT
ma-55	392	10	it	it	PRON
ma-55	392	11	follows	follow	VERB
ma-55	392	12	that	that	PRON
ma-55	392	13	β(uεt	β(uεt	NOUN
ma-55	392	14	)	)	PUNCT
ma-55	392	15	is	be	AUX
ma-55	392	16	bounded	bound	VERB
ma-55	392	17	in	in	ADP
ma-55	392	18	l2(0	l2(0	PROPN
ma-55	392	19	,	,	PUNCT
ma-55	392	20	t	t	PROPN
ma-55	392	21	;	;	PUNCT
ma-55	392	22	h−1(ω	h−1(ω	PROPN
ma-55	392	23	)	)	PUNCT
ma-55	392	24	)	)	PUNCT
ma-55	392	25	,	,	PUNCT
ma-55	392	26	therefore	therefore	ADV
ma-55	392	27	β(uεt	β(uεt	NOUN
ma-55	392	28	)	)	PUNCT
ma-55	393	1	⇀	⇀	PUNCT
ma-55	393	2	0	0	NUM
ma-55	393	3	weak	weak	ADJ
ma-55	393	4	in	in	ADP
ma-55	393	5	l2(0	l2(0	NOUN
ma-55	393	6	,	,	PUNCT
ma-55	393	7	t	t	NOUN
ma-55	393	8	;	;	PUNCT
ma-55	393	9	h−1ω	h−1ω	NOUN
ma-55	393	10	)	)	PUNCT
ma-55	393	11	.	.	PUNCT
ma-55	394	1	(	(	PUNCT
ma-55	394	2	5.46	5.46	NUM
ma-55	394	3	)	)	PUNCT
ma-55	394	4	on	on	ADP
ma-55	394	5	the	the	DET
ma-55	394	6	other	other	ADJ
ma-55	394	7	hand	hand	NOUN
ma-55	394	8	we	we	PRON
ma-55	394	9	deduce	deduce	VERB
ma-55	394	10	from	from	ADP
ma-55	394	11	(	(	PUNCT
ma-55	394	12	5.45	5.45	NUM
ma-55	394	13	)	)	PUNCT
ma-55	395	1	that	that	PRON
ma-55	395	2	0	0	NUM
ma-55	395	3	≤	≤	NUM
ma-55	395	4	∫	∫	PROPN
ma-55	395	5	t	t	PROPN
ma-55	395	6	0	0	NUM
ma-55	395	7	(	(	PUNCT
ma-55	395	8	β(uεt	β(uεt	NOUN
ma-55	395	9	)	)	PUNCT
ma-55	395	10	,	,	PUNCT
ma-55	395	11	uεt	uεt	NOUN
ma-55	395	12	)	)	PUNCT
ma-55	395	13	dt	dt	PART
ma-55	395	14	≤	≤	NUM
ma-55	395	15	ε	ε	PROPN
ma-55	395	16	c.	c.	PROPN
ma-55	395	17	(	(	PUNCT
ma-55	395	18	5.47	5.47	NUM
ma-55	395	19	)	)	PUNCT
ma-55	395	20	thus	thus	ADV
ma-55	395	21	∫	∫	PROPN
ma-55	395	22	t	t	PROPN
ma-55	395	23	0	0	NUM
ma-55	395	24	(	(	PUNCT
ma-55	395	25	β(uεt	β(uεt	NOUN
ma-55	395	26	)	)	PUNCT
ma-55	395	27	,	,	PUNCT
ma-55	395	28	uεt	uεt	NOUN
ma-55	395	29	)	)	PUNCT
ma-55	395	30	dt	dt	PROPN
ma-55	396	1	−→	−→	ADJ
ma-55	396	2	0	0	NUM
ma-55	396	3	.	.	PUNCT
ma-55	397	1	(	(	PUNCT
ma-55	397	2	5.48	5.48	NUM
ma-55	397	3	)	)	PUNCT
ma-55	397	4	we	we	PRON
ma-55	397	5	have	have	VERB
ma-55	397	6	that	that	DET
ma-55	397	7	∫	∫	PROPN
ma-55	397	8	t	t	PROPN
ma-55	397	9	0	0	NUM
ma-55	397	10	(	(	PUNCT
ma-55	397	11	β(uεt	β(uεt	NOUN
ma-55	397	12	)	)	PUNCT
ma-55	397	13	−	−	PROPN
ma-55	397	14	β(ϕ	β(ϕ	NUM
ma-55	397	15	)	)	PUNCT
ma-55	397	16	,	,	PUNCT
ma-55	397	17	uεt	uεt	NOUN
ma-55	397	18	−	−	PROPN
ma-55	397	19	ϕ	ϕ	NOUN
ma-55	397	20	)	)	PUNCT
ma-55	397	21	dt	dt	X
ma-55	397	22	≥	≥	PROPN
ma-55	397	23	0	0	NUM
ma-55	397	24	,	,	PUNCT
ma-55	397	25	∀ϕ	∀ϕ	NUM
ma-55	397	26	in	in	ADP
ma-55	397	27	l2(0	l2(0	NOUN
ma-55	397	28	,	,	PUNCT
ma-55	397	29	t	t	PROPN
ma-55	397	30	;	;	PUNCT
ma-55	397	31	h1	h1	NOUN
ma-55	397	32	0(ω	0(ω	ADJ
ma-55	397	33	)	)	PUNCT
ma-55	397	34	)	)	PUNCT
ma-55	397	35	,	,	PUNCT
ma-55	397	36	because	because	SCONJ
ma-55	397	37	β	β	NOUN
ma-55	397	38	is	be	AUX
ma-55	397	39	a	a	DET
ma-55	397	40	monotonous	monotonous	ADJ
ma-55	397	41	operator	operator	NOUN
ma-55	397	42	.	.	PUNCT
ma-55	398	1	thus,∫	thus,∫	PROPN
ma-55	398	2	t	t	PROPN
ma-55	398	3	0	0	NUM
ma-55	398	4	(	(	PUNCT
ma-55	398	5	β(uεt	β(uεt	NOUN
ma-55	398	6	)	)	PUNCT
ma-55	398	7	,	,	PUNCT
ma-55	398	8	uεt	uεt	NOUN
ma-55	398	9	)	)	PUNCT
ma-55	398	10	dt	dt	PUNCT
ma-55	399	1	−	−	PROPN
ma-55	399	2	∫	∫	PROPN
ma-55	399	3	t	t	PROPN
ma-55	399	4	0	0	NUM
ma-55	399	5	(	(	PUNCT
ma-55	399	6	β(uεt	β(uεt	NOUN
ma-55	399	7	)	)	PUNCT
ma-55	399	8	,	,	PUNCT
ma-55	399	9	ϕ	ϕ	NOUN
ma-55	399	10	)	)	PUNCT
ma-55	399	11	dt	dt	PROPN
ma-55	400	1	−	−	PROPN
ma-55	400	2	∫	∫	PROPN
ma-55	400	3	t	t	PROPN
ma-55	400	4	0	0	NUM
ma-55	400	5	(	(	PUNCT
ma-55	400	6	β(ϕ	β(ϕ	NUM
ma-55	400	7	)	)	PUNCT
ma-55	400	8	,	,	PUNCT
ma-55	400	9	uεt	uεt	ADP
ma-55	400	10	−	−	PROPN
ma-55	400	11	ϕ	ϕ	NOUN
ma-55	400	12	)	)	PUNCT
ma-55	400	13	dt	dt	X
ma-55	400	14	≥	≥	NOUN
ma-55	400	15	0	0	NUM
ma-55	400	16	.	.	PUNCT
ma-55	400	17	(	(	PUNCT
ma-55	400	18	5.49	5.49	NUM
ma-55	400	19	)	)	PUNCT
ma-55	400	20	from	from	ADP
ma-55	400	21	(	(	PUNCT
ma-55	400	22	5.40	5.40	NUM
ma-55	400	23	)	)	PUNCT
ma-55	400	24	,	,	PUNCT
ma-55	400	25	(	(	PUNCT
ma-55	400	26	5.46	5.46	NUM
ma-55	400	27	)	)	PUNCT
ma-55	400	28	and	and	CCONJ
ma-55	400	29	(	(	PUNCT
ma-55	400	30	5.48	5.48	NUM
ma-55	400	31	)	)	PUNCT
ma-55	400	32	we	we	PRON
ma-55	400	33	obtain∫	obtain∫	VERB
ma-55	400	34	t	t	PROPN
ma-55	400	35	0	0	NUM
ma-55	400	36	(	(	PUNCT
ma-55	400	37	β(ϕ	β(ϕ	NUM
ma-55	400	38	)	)	PUNCT
ma-55	400	39	,	,	PUNCT
ma-55	400	40	ut(t)−	ut(t)−	PROPN
ma-55	400	41	ϕ	ϕ	NOUN
ma-55	400	42	)	)	PUNCT
ma-55	400	43	dt	dt	PROPN
ma-55	401	1	≤	≤	NUM
ma-55	401	2	0	0	NUM
ma-55	401	3	.	.	PUNCT
ma-55	402	1	(	(	PUNCT
ma-55	402	2	5.50	5.50	NUM
ma-55	402	3	)	)	PUNCT
ma-55	402	4	taking	take	VERB
ma-55	402	5	ϕ	ϕ	NOUN
ma-55	402	6	=	=	PUNCT
ma-55	402	7	ut	ut	PROPN
ma-55	402	8	−	−	NOUN
ma-55	402	9	λv	λv	X
ma-55	402	10	,	,	PUNCT
ma-55	402	11	with	with	ADP
ma-55	402	12	v	v	PROPN
ma-55	402	13	∈	∈	PROPN
ma-55	402	14	l2(0	l2(0	NOUN
ma-55	402	15	,	,	PUNCT
ma-55	402	16	t	t	PROPN
ma-55	402	17	;	;	PUNCT
ma-55	402	18	h1	h1	NOUN
ma-55	402	19	0(ω	0(ω	ADJ
ma-55	402	20	)	)	PUNCT
ma-55	402	21	)	)	PUNCT
ma-55	403	1	and	and	CCONJ
ma-55	403	2	λ	λ	X
ma-55	403	3	>	>	X
ma-55	403	4	0	0	NUM
ma-55	403	5	,	,	PUNCT
ma-55	403	6	we	we	PRON
ma-55	403	7	deduce	deduce	VERB
ma-55	403	8	using	use	VERB
ma-55	403	9	the	the	DET
ma-55	403	10	hemicontinuityof	hemicontinuityof	PROPN
ma-55	403	11	β	β	PROPN
ma-55	403	12	that	that	SCONJ
ma-55	403	13	β(ut(t	β(ut(t	PROPN
ma-55	403	14	)	)	PUNCT
ma-55	403	15	)	)	PUNCT
ma-55	404	1	=	=	SYM
ma-55	404	2	0	0	NUM
ma-55	404	3	,	,	PUNCT
ma-55	404	4	(	(	PUNCT
ma-55	404	5	5.51	5.51	NUM
ma-55	404	6	)	)	PUNCT
ma-55	404	7	and	and	CCONJ
ma-55	404	8	this	this	PRON
ma-55	404	9	implies	imply	VERB
ma-55	404	10	that	that	SCONJ
ma-55	404	11	ut(t	ut(t	NOUN
ma-55	404	12	)	)	PUNCT
ma-55	404	13	∈	∈	PROPN
ma-55	404	14	k	k	PROPN
ma-55	404	15	a.	a.	PROPN
ma-55	404	16	e.	e.	PROPN
ma-55	404	17	6	6	PROPN
ma-55	404	18	.	.	PUNCT
ma-55	405	1	uniqueness	uniqueness	NOUN
ma-55	405	2	let	let	VERB
ma-55	405	3	u1	u1	NOUN
ma-55	405	4	,	,	PUNCT
ma-55	405	5	u2	u2	PROPN
ma-55	405	6	two	two	NUM
ma-55	405	7	solutions	solution	NOUN
ma-55	405	8	of	of	ADP
ma-55	405	9	(	(	PUNCT
ma-55	405	10	4.5	4.5	NUM
ma-55	405	11	)	)	PUNCT
ma-55	405	12	,	,	PUNCT
ma-55	405	13	w	w	NOUN
ma-55	405	14	=	=	VERB
ma-55	405	15	u2−	u2−	NOUN
ma-55	405	16	u1	u1	NOUN
ma-55	405	17	and	and	CCONJ
ma-55	405	18	t	t	NOUN
ma-55	405	19	∈	∈	PROPN
ma-55	405	20	(	(	PUNCT
ma-55	405	21	0	0	NUM
ma-55	405	22	,	,	PUNCT
ma-55	405	23	t	t	NOUN
ma-55	405	24	)	)	PUNCT
ma-55	405	25	.	.	PUNCT
ma-55	406	1	because	because	SCONJ
ma-55	406	2	ut	ut	PROPN
ma-55	406	3	∈	∈	PROPN
ma-55	406	4	l2(0	l2(0	PROPN
ma-55	406	5	,	,	PUNCT
ma-55	406	6	t	t	PROPN
ma-55	406	7	;	;	PUNCT
ma-55	406	8	h1	h1	NOUN
ma-55	406	9	0(ω	0(ω	ADJ
ma-55	406	10	)	)	PUNCT
ma-55	406	11	,	,	PUNCT
ma-55	406	12	wecan	wecan	VERB
ma-55	406	13	talking	talking	NOUN
ma-55	406	14	u1	u1	PROPN
ma-55	406	15	t	t	PROPN
ma-55	406	16	(	(	PUNCT
ma-55	406	17	resp	resp	NOUN
ma-55	406	18	.	.	PUNCT
ma-55	407	1	u2	u2	PROPN
ma-55	407	2	t	t	PROPN
ma-55	407	3	)	)	PUNCT
ma-55	407	4	in	in	ADP
ma-55	407	5	the	the	DET
ma-55	407	6	inequality	inequality	NOUN
ma-55	407	7	(	(	PUNCT
ma-55	407	8	4.5	4.5	NUM
ma-55	407	9	)	)	PUNCT
ma-55	407	10	relative	relative	ADJ
ma-55	407	11	to	to	ADP
ma-55	407	12	v2	v2	PROPN
ma-55	407	13	(	(	PUNCT
ma-55	407	14	resp	resp	NOUN
ma-55	407	15	.	.	PUNCT
ma-55	407	16	v1	v1	PROPN
ma-55	407	17	)	)	PUNCT
ma-55	408	1	and	and	CCONJ
ma-55	408	2	adding	add	VERB
ma-55	408	3	up	up	ADP
ma-55	408	4	the	the	DET
ma-55	408	5	resultswe	resultswe	NOUN
ma-55	408	6	obtain	obtain	VERB
ma-55	408	7	−	−	PROPN
ma-55	408	8	∫	∫	PROPN
ma-55	408	9	t	t	PROPN
ma-55	408	10	0	0	NUM
ma-55	409	1	(	(	PUNCT
ma-55	409	2	wtt	wtt	PROPN
ma-55	409	3	,	,	PUNCT
ma-55	409	4	wt)ds	wt)ds	PROPN
ma-55	410	1	−	−	PROPN
ma-55	410	2	∫	∫	PROPN
ma-55	410	3	t	t	PROPN
ma-55	410	4	0	0	NUM
ma-55	410	5	(	(	PUNCT
ma-55	410	6	∆2w	∆2w	NOUN
ma-55	410	7	,	,	PUNCT
ma-55	410	8	wt)ds	wt)ds	X
ma-55	410	9	+	+	CCONJ
ma-55	410	10	∫	∫	PROPN
ma-55	410	11	t	t	PROPN
ma-55	410	12	0	0	NUM
ma-55	410	13	(	(	PUNCT
ma-55	410	14	∆pu	∆pu	NOUN
ma-55	410	15	1	1	NUM
ma-55	410	16	,	,	PUNCT
ma-55	410	17	wt)ds	wt)ds	X
ma-55	411	1	−	−	PROPN
ma-55	411	2	∫	∫	PROPN
ma-55	411	3	t	t	PROPN
ma-55	411	4	0	0	NUM
ma-55	411	5	(	(	PUNCT
ma-55	411	6	∆pu	∆pu	NOUN
ma-55	411	7	2	2	NUM
ma-55	411	8	,	,	PUNCT
ma-55	411	9	wt)ds	wt)ds	X
ma-55	412	1	+	+	CCONJ
ma-55	412	2	∫	∫	PROPN
ma-55	412	3	t	t	PROPN
ma-55	412	4	0	0	NUM
ma-55	413	1	(	(	PUNCT
ma-55	413	2	∫	∫	PROPN
ma-55	413	3	t	t	PROPN
ma-55	413	4	0	0	NUM
ma-55	413	5	g(t	g(t	PROPN
ma-55	413	6	−	−	PROPN
ma-55	413	7	s)∆w(s)ds	s)∆w(s)ds	PROPN
ma-55	413	8	,	,	PUNCT
ma-55	413	9	wt	wt	NOUN
ma-55	413	10	)	)	PUNCT
ma-55	413	11	ds	ds	PROPN
ma-55	413	12	+	+	NUM
ma-55	413	13	∫	∫	PROPN
ma-55	413	14	t	t	PROPN
ma-55	413	15	0	0	NUM
ma-55	413	16	(	(	PUNCT
ma-55	413	17	∆wt	∆wt	NUM
ma-55	413	18	,	,	PUNCT
ma-55	413	19	wt)ds	wt)ds	X
ma-55	414	1	−	−	PROPN
ma-55	414	2	∫	∫	PROPN
ma-55	414	3	t	t	PROPN
ma-55	414	4	0	0	NUM
ma-55	414	5	(	(	PUNCT
ma-55	414	6	|u1|r−2u1	|u1|r−2u1	X
ma-55	414	7	ln	ln	ADJ
ma-55	414	8	|u1|	|u1|	ADJ
ma-55	414	9	,	,	PUNCT
ma-55	414	10	wt)ds	wt)ds	X
ma-55	414	11	+	+	CCONJ
ma-55	414	12	∫	∫	PROPN
ma-55	414	13	t	t	PROPN
ma-55	414	14	0	0	NUM
ma-55	414	15	(	(	PUNCT
ma-55	414	16	|u2|r−2u2	|u2|r−2u2	VERB
ma-55	414	17	ln	ln	ADJ
ma-55	414	18	|u2|	|u2|	VERB
ma-55	414	19	,	,	PUNCT
ma-55	414	20	wt)ds	wt)ds	X
ma-55	414	21	≥	≥	NOUN
ma-55	414	22	0	0	NUM
ma-55	414	23	,	,	PUNCT
ma-55	414	24	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	414	25	eur	eur	PROPN
ma-55	414	26	.	.	PUNCT
ma-55	415	1	j.	j.	PROPN
ma-55	415	2	math	math	PROPN
ma-55	415	3	.	.	PUNCT
ma-55	416	1	anal	anal	PROPN
ma-55	416	2	.	.	PUNCT
ma-55	417	1	10.28924	10.28924	NUM
ma-55	417	2	/	/	SYM
ma-55	417	3	ada	ada	PROPN
ma-55	417	4	/	/	SYM
ma-55	417	5	ma.2.5	ma.2.5	PROPN
ma-55	417	6	18thus	18thus	ADJ
ma-55	417	7	,	,	PUNCT
ma-55	417	8	we	we	PRON
ma-55	417	9	have	have	VERB
ma-55	417	10	1	1	NUM
ma-55	417	11	2	2	NUM
ma-55	417	12	∫	∫	NOUN
ma-55	417	13	t	t	NOUN
ma-55	417	14	0	0	NUM
ma-55	418	1	d	d	NOUN
ma-55	418	2	dt	dt	X
ma-55	418	3	(	(	PUNCT
ma-55	418	4	|wt(t)|2	|wt(t)|2	PROPN
ma-55	418	5	+	+	CCONJ
ma-55	418	6	|∆w(t)|2	|∆w(t)|2	NOUN
ma-55	418	7	)	)	PUNCT
ma-55	418	8	ds	ds	PROPN
ma-55	418	9	+	+	NUM
ma-55	418	10	∫	∫	PROPN
ma-55	418	11	t	t	NOUN
ma-55	418	12	0	0	NUM
ma-55	418	13	|∇wt(t)|2ds	|∇wt(t)|2ds	VERB
ma-55	418	14	≤	≤	ADJ
ma-55	418	15	∫	∫	PROPN
ma-55	419	1	t	t	PROPN
ma-55	419	2	0	0	NUM
ma-55	420	1	〈	〈	PROPN
ma-55	420	2	∆pu1(t)−	∆pu1(t)−	PROPN
ma-55	420	3	∆pu	∆pu	NOUN
ma-55	420	4	2(t	2(t	NUM
ma-55	420	5	)	)	PUNCT
ma-55	420	6	,	,	PUNCT
ma-55	420	7	wt(t)〉ds	wt(t)〉ds	X
ma-55	420	8	+	+	CCONJ
ma-55	420	9	∫	∫	PROPN
ma-55	420	10	t	t	PROPN
ma-55	420	11	0	0	NUM
ma-55	420	12	∫	∫	PROPN
ma-55	420	13	t	t	PROPN
ma-55	420	14	0	0	NUM
ma-55	420	15	g(t	g(t	PROPN
ma-55	420	16	−	−	PROPN
ma-55	420	17	s)(∇w(s),∇wt(t))dsdσ∫	s)(∇w(s),∇wt(t))dsdσ∫	PROPN
ma-55	420	18	t	t	NOUN
ma-55	420	19	0	0	NUM
ma-55	420	20	(	(	PUNCT
ma-55	420	21	|u1(t)|r−2u1(t	|u1(t)|r−2u1(t	PROPN
ma-55	420	22	)	)	PUNCT
ma-55	420	23	ln	ln	NOUN
ma-55	420	24	|u1(t)|	|u1(t)|	ADP
ma-55	420	25	−	−	PROPN
ma-55	420	26	|u2(t)|r−2u2(t	|u2(t)|r−2u2(t	NOUN
ma-55	420	27	)	)	PUNCT
ma-55	420	28	ln	ln	ADJ
ma-55	420	29	|u2(t)|	|u2(t)|	VERB
ma-55	420	30	,	,	PUNCT
ma-55	420	31	wt(t	wt(t	NUM
ma-55	420	32	)	)	PUNCT
ma-55	420	33	)	)	PUNCT
ma-55	421	1	ds	ds	PROPN
ma-55	421	2	.	.	PROPN
ma-55	421	3	by	by	ADP
ma-55	421	4	lemma	lemma	PROPN
ma-55	421	5	2.2	2.2	NUM
ma-55	421	6	,	,	PUNCT
ma-55	421	7	we	we	PRON
ma-55	421	8	derive	derive	VERB
ma-55	421	9	1	1	NUM
ma-55	421	10	2	2	NUM
ma-55	421	11	∫	∫	NOUN
ma-55	421	12	t	t	NOUN
ma-55	421	13	0	0	NUM
ma-55	422	1	d	d	NOUN
ma-55	422	2	dt	dt	X
ma-55	422	3	{	{	PUNCT
ma-55	422	4	|wt(t)|2	|wt(t)|2	PROPN
ma-55	422	5	+	+	NUM
ma-55	422	6	|∆w(t)|2	|∆w(t)|2	NOUN
ma-55	422	7	−	−	PROPN
ma-55	423	1	(	(	PUNCT
ma-55	423	2	∫	∫	PROPN
ma-55	423	3	t	t	PROPN
ma-55	423	4	0	0	NUM
ma-55	423	5	g(s)ds	g(s)ds	PROPN
ma-55	423	6	)	)	PUNCT
ma-55	423	7	|∇w(t)|2	|∇w(t)|2	PROPN
ma-55	424	1	+	+	CCONJ
ma-55	424	2	(	(	PUNCT
ma-55	424	3	g	g	PROPN
ma-55	424	4	�	�	PROPN
ma-55	424	5	∇w)(t	∇w)(t	PROPN
ma-55	424	6	)	)	PUNCT
ma-55	424	7	}	}	PUNCT
ma-55	424	8	ds	ds	PROPN
ma-55	424	9	+	+	CCONJ
ma-55	424	10	∫	∫	PROPN
ma-55	424	11	t	t	NOUN
ma-55	424	12	0	0	NUM
ma-55	424	13	|∇wt(t)|2ds	|∇wt(t)|2ds	VERB
ma-55	424	14	≤	≤	ADJ
ma-55	424	15	∫	∫	PROPN
ma-55	424	16	t	t	PROPN
ma-55	424	17	0	0	NUM
ma-55	424	18	|〈∆pu1(t)−	|〈∆pu1(t)−	PROPN
ma-55	424	19	∆pu	∆pu	NOUN
ma-55	424	20	2(t	2(t	NUM
ma-55	424	21	)	)	PUNCT
ma-55	424	22	,	,	PUNCT
ma-55	424	23	wt(t)〉|ds	wt(t)〉|ds	PUNCT
ma-55	425	1	+	+	X
ma-55	426	1	∫	∫	PROPN
ma-55	426	2	t	t	PROPN
ma-55	426	3	0	0	NUM
ma-55	426	4	∫	∫	PROPN
ma-55	426	5	ω	ω	PROPN
ma-55	426	6	(	(	PUNCT
ma-55	426	7	|u1(t)|r−2u1(t	|u1(t)|r−2u1(t	PROPN
ma-55	426	8	)	)	PUNCT
ma-55	426	9	ln	ln	NOUN
ma-55	426	10	|u1(t)|	|u1(t)|	ADP
ma-55	426	11	−	−	PROPN
ma-55	426	12	|u2(t)|r−2u2(t	|u2(t)|r−2u2(t	NOUN
ma-55	426	13	)	)	PUNCT
ma-55	426	14	ln	ln	ADJ
ma-55	426	15	|u2(t)|	|u2(t)|	VERB
ma-55	426	16	,	,	PUNCT
ma-55	426	17	wt(t	wt(t	NUM
ma-55	426	18	)	)	PUNCT
ma-55	426	19	)	)	PUNCT
ma-55	426	20	dxds	dxds	NOUN
ma-55	426	21	.	.	PUNCT
ma-55	427	1	(	(	PUNCT
ma-55	427	2	6.1	6.1	NUM
ma-55	427	3	)	)	PUNCT
ma-55	427	4	from	from	ADP
ma-55	427	5	mean	mean	NOUN
ma-55	427	6	value	value	NOUN
ma-55	427	7	theorem	theorem	NOUN
ma-55	427	8	,	,	PUNCT
ma-55	427	9	|〈∆pu1(t)−	|〈∆pu1(t)−	PROPN
ma-55	427	10	∆pu	∆pu	NOUN
ma-55	427	11	2(t	2(t	NUM
ma-55	427	12	)	)	PUNCT
ma-55	427	13	,	,	PUNCT
ma-55	427	14	wt(t)〉|	wt(t)〉|	PROPN
ma-55	427	15	≤	≤	PROPN
ma-55	427	16	c	c	PROPN
ma-55	427	17	(	(	PUNCT
ma-55	427	18	|∇u1(t)|p−2	|∇u1(t)|p−2	PROPN
ma-55	427	19	2(p−1	2(p−1	NOUN
ma-55	427	20	)	)	PUNCT
ma-55	427	21	+	+	CCONJ
ma-55	427	22	|∇u2(t)|p−2	|∇u2(t)|p−2	VERB
ma-55	427	23	2(p−1	2(p−1	ADJ
ma-55	427	24	)	)	PUNCT
ma-55	427	25	)	)	PUNCT
ma-55	428	1	|∇w(t)|2(p−1)|∇wt(t)|	|∇w(t)|2(p−1)|∇wt(t)|	ADP
ma-55	428	2	≤	≤	ADJ
ma-55	428	3	c|∆w(t)|2	c|∆w(t)|2	NOUN
ma-55	429	1	+	+	CCONJ
ma-55	429	2	1	1	NUM
ma-55	429	3	4	4	NUM
ma-55	429	4	|∇wt(t)|2	|∇wt(t)|2	NOUN
ma-55	429	5	,	,	PUNCT
ma-55	429	6	(	(	PUNCT
ma-55	429	7	6.2	6.2	NUM
ma-55	429	8	)	)	PUNCT
ma-55	429	9	for	for	ADP
ma-55	429	10	some	some	DET
ma-55	429	11	constant	constant	ADJ
ma-55	429	12	c	c	NOUN
ma-55	429	13	>	>	X
ma-55	429	14	0	0	PROPN
ma-55	429	15	,	,	PUNCT
ma-55	429	16	and∫	and∫	PROPN
ma-55	429	17	t	t	PROPN
ma-55	429	18	0	0	NUM
ma-55	429	19	∫	∫	PROPN
ma-55	430	1	ω	ω	PROPN
ma-55	430	2	(	(	PUNCT
ma-55	430	3	|u1(t)|r−2u1(t	|u1(t)|r−2u1(t	PROPN
ma-55	430	4	)	)	PUNCT
ma-55	430	5	ln	ln	NOUN
ma-55	430	6	|u1(t)|	|u1(t)|	ADP
ma-55	430	7	−	−	PROPN
ma-55	430	8	|u2(t)|r−2u2(t	|u2(t)|r−2u2(t	NOUN
ma-55	430	9	)	)	PUNCT
ma-55	430	10	ln	ln	ADJ
ma-55	430	11	|u2(t)|	|u2(t)|	VERB
ma-55	430	12	,	,	PUNCT
ma-55	430	13	wt(t	wt(t	NUM
ma-55	430	14	)	)	PUNCT
ma-55	430	15	)	)	PUNCT
ma-55	431	1	dxds	dxds	VERB
ma-55	431	2	≤	≤	NUM
ma-55	431	3	∫	∫	PROPN
ma-55	432	1	t	t	PROPN
ma-55	432	2	0	0	NUM
ma-55	432	3	∫	∫	PROPN
ma-55	432	4	ω	ω	NUM
ma-55	432	5	|θu1(t	|θu1(t	PROPN
ma-55	432	6	)	)	PUNCT
ma-55	433	1	+	+	CCONJ
ma-55	433	2	(	(	PUNCT
ma-55	433	3	1−	1−	NUM
ma-55	433	4	θ)u2(t))|r−2|w(t)||wt(t)|dxds	θ)u2(t))|r−2|w(t)||wt(t)|dxds	NOUN
ma-55	433	5	+	+	PROPN
ma-55	433	6	(	(	PUNCT
ma-55	433	7	r	r	NOUN
ma-55	433	8	−	−	NOUN
ma-55	433	9	1	1	NUM
ma-55	433	10	)	)	PUNCT
ma-55	433	11	∫	∫	PROPN
ma-55	433	12	t	t	PROPN
ma-55	433	13	0	0	NUM
ma-55	433	14	∫	∫	PROPN
ma-55	433	15	ω	ω	NUM
ma-55	433	16	|θu1(t	|θu1(t	PROPN
ma-55	433	17	)	)	PUNCT
ma-55	434	1	+	+	CCONJ
ma-55	434	2	(	(	PUNCT
ma-55	434	3	1−	1−	NUM
ma-55	434	4	θ)u2|r−2	θ)u2|r−2	NOUN
ma-55	434	5	ln	ln	ADJ
ma-55	434	6	|θu1(t	|θu1(t	NOUN
ma-55	434	7	)	)	PUNCT
ma-55	435	1	+	+	ADJ
ma-55	435	2	(	(	PUNCT
ma-55	435	3	1−	1−	NUM
ma-55	435	4	θ)u2(t)||w(t)|wt(t)|dxds	θ)u2(t)||w(t)|wt(t)|dxds	PROPN
ma-55	435	5	=	=	PUNCT
ma-55	435	6	i1	i1	PROPN
ma-55	435	7	+	+	CCONJ
ma-55	435	8	i2	i2	PROPN
ma-55	435	9	,	,	PUNCT
ma-55	435	10	0	0	PUNCT
ma-55	435	11	<	<	X
ma-55	435	12	θ	θ	X
ma-55	435	13	<	<	X
ma-55	435	14	1	1	NUM
ma-55	435	15	.	.	PUNCT
ma-55	435	16	(	(	PUNCT
ma-55	435	17	6.3	6.3	NUM
ma-55	435	18	)	)	PUNCT
ma-55	435	19	hence	hence	ADV
ma-55	435	20	,	,	PUNCT
ma-55	435	21	from	from	ADP
ma-55	435	22	the	the	DET
ma-55	435	23	hölder	hölder	NOUN
ma-55	435	24	inequality	inequality	NOUN
ma-55	435	25	and	and	CCONJ
ma-55	435	26	sobolev	sobolev	NOUN
ma-55	435	27	inequality	inequality	NOUN
ma-55	435	28	,	,	PUNCT
ma-55	435	29	we	we	PRON
ma-55	435	30	have∫	have∫	VERB
ma-55	435	31	ω	ω	NUM
ma-55	435	32	|θu1(t	|θu1(t	NUM
ma-55	435	33	)	)	PUNCT
ma-55	436	1	+	+	CCONJ
ma-55	436	2	(	(	PUNCT
ma-55	436	3	1−	1−	NUM
ma-55	436	4	θ)u2(t))|r−2|w(t)||wt(t)|dx	θ)u2(t))|r−2|w(t)||wt(t)|dx	NOUN
ma-55	436	5	≤	≤	NOUN
ma-55	436	6	|θu1(t	|θu1(t	NOUN
ma-55	436	7	)	)	PUNCT
ma-55	437	1	+	+	CCONJ
ma-55	437	2	(	(	PUNCT
ma-55	437	3	1−	1−	NUM
ma-55	437	4	θ)u2(t)|r−2	θ)u2(t)|r−2	NOUN
ma-55	437	5	n(r−2	n(r−2	X
ma-55	437	6	)	)	PUNCT
ma-55	437	7	|w(t)|	|w(t)|	PROPN
ma-55	437	8	2n	2n	NUM
ma-55	437	9	n−2	n−2	PROPN
ma-55	437	10	|wt(t)|	|wt(t)|	PROPN
ma-55	437	11	≤	≤	PROPN
ma-55	437	12	cr−2	cr−2	X
ma-55	437	13	1	1	NUM
ma-55	437	14	c2c3|∆w(t)||∇wt(t)|	c2c3|∆w(t)||∇wt(t)|	NOUN
ma-55	437	15	≤	≤	ADJ
ma-55	437	16	c|∆w(t)|2	c|∆w(t)|2	NOUN
ma-55	438	1	+	+	CCONJ
ma-55	438	2	1	1	NUM
ma-55	438	3	4	4	NUM
ma-55	438	4	|∇wt(t)|2	|∇wt(t)|2	NOUN
ma-55	438	5	,	,	PUNCT
ma-55	438	6	(	(	PUNCT
ma-55	438	7	6.4	6.4	NUM
ma-55	438	8	)	)	PUNCT
ma-55	438	9	where	where	SCONJ
ma-55	438	10	c1	c1	PROPN
ma-55	438	11	,	,	PUNCT
ma-55	438	12	c2	c2	PROPN
ma-55	438	13	and	and	CCONJ
ma-55	438	14	c3	c3	PROPN
ma-55	438	15	are	be	AUX
ma-55	438	16	constants	constant	NOUN
ma-55	438	17	satisfying	satisfy	VERB
ma-55	438	18	|θu1(t	|θu1(t	PROPN
ma-55	438	19	)	)	PUNCT
ma-55	439	1	+	+	CCONJ
ma-55	439	2	(	(	PUNCT
ma-55	439	3	1−	1−	NUM
ma-55	439	4	θ)u2(t)|r−2	θ)u2(t)|r−2	NOUN
ma-55	439	5	n(r−2	n(r−2	X
ma-55	439	6	)	)	PUNCT
ma-55	439	7	|	|	ADV
ma-55	439	8	≤	≤	NUM
ma-55	439	9	c1|θu1(t	c1|θu1(t	X
ma-55	439	10	)	)	PUNCT
ma-55	440	1	+	+	CCONJ
ma-55	440	2	(	(	PUNCT
ma-55	440	3	1−	1−	NUM
ma-55	440	4	θ)u2(t)|	θ)u2(t)|	PROPN
ma-55	440	5	,	,	PUNCT
ma-55	440	6	|w(t)|	|w(t)|	PROPN
ma-55	440	7	2n	2n	NUM
ma-55	440	8	n−2	n−2	PROPN
ma-55	440	9	≤	≤	NOUN
ma-55	440	10	c|w(t)|	c|w(t)|	NUM
ma-55	440	11	≤	≤	NUM
ma-55	440	12	c2|∆w(t)|	c2|∆w(t)|	ADJ
ma-55	440	13	and	and	CCONJ
ma-55	440	14	|w(t)|	|w(t)|	PROPN
ma-55	440	15	≤	≤	PROPN
ma-55	440	16	c3|∇w(t)|	c3|∇w(t)|	NOUN
ma-55	440	17	.	.	PUNCT
ma-55	441	1	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	441	2	eur	eur	PROPN
ma-55	441	3	.	.	PUNCT
ma-55	442	1	j.	j.	PROPN
ma-55	442	2	math	math	PROPN
ma-55	442	3	.	.	PUNCT
ma-55	443	1	anal	anal	PROPN
ma-55	443	2	.	.	PUNCT
ma-55	444	1	10.28924	10.28924	NUM
ma-55	444	2	/	/	SYM
ma-55	444	3	ada	ada	PROPN
ma-55	444	4	/	/	SYM
ma-55	444	5	ma.2.5	ma.2.5	PROPN
ma-55	444	6	19	19	NUM
ma-55	444	7	also	also	ADV
ma-55	444	8	we	we	PRON
ma-55	444	9	used	use	VERB
ma-55	444	10	the	the	DET
ma-55	444	11	condition	condition	NOUN
ma-55	444	12	n(p	n(p	PROPN
ma-55	444	13	−	−	NUM
ma-55	444	14	2	2	NUM
ma-55	444	15	)	)	PUNCT
ma-55	444	16	<	<	X
ma-55	444	17	2n	2n	NUM
ma-55	444	18	n	n	CCONJ
ma-55	444	19	−	−	PROPN
ma-55	444	20	2	2	NUM
ma-55	444	21	.	.	PUNCT
ma-55	445	1	now	now	ADV
ma-55	445	2	,	,	PUNCT
ma-55	445	3	using	use	VERB
ma-55	445	4	the	the	DET
ma-55	445	5	calculation	calculation	NOUN
ma-55	445	6	similar	similar	ADJ
ma-55	445	7	to	to	ADP
ma-55	445	8	(	(	PUNCT
ma-55	445	9	5.24	5.24	NUM
ma-55	445	10	)	)	PUNCT
ma-55	445	11	,	,	PUNCT
ma-55	445	12	it	it	PRON
ma-55	445	13	follows	follow	VERB
ma-55	445	14	that∫	that∫	PROPN
ma-55	445	15	ω	ω	NUM
ma-55	445	16	|θu1(t	|θu1(t	PROPN
ma-55	445	17	)	)	PUNCT
ma-55	446	1	+	+	CCONJ
ma-55	446	2	(	(	PUNCT
ma-55	446	3	1−	1−	NUM
ma-55	446	4	θ)u2|r−2	θ)u2|r−2	NOUN
ma-55	446	5	ln	ln	ADJ
ma-55	446	6	|θu1(t	|θu1(t	NOUN
ma-55	446	7	)	)	PUNCT
ma-55	447	1	+	+	CCONJ
ma-55	447	2	(	(	PUNCT
ma-55	447	3	1−	1−	NUM
ma-55	447	4	θ)u2(t)|ndx	θ)u2(t)|ndx	NOUN
ma-55	447	5	≤	≤	NUM
ma-55	447	6	(	(	PUNCT
ma-55	447	7	e(r	e(r	CCONJ
ma-55	447	8	−	−	PROPN
ma-55	448	1	2)−n)|ω|+	2)−n)|ω|+	NUM
ma-55	448	2	(	(	PUNCT
ma-55	448	3	e(r	e(r	PROPN
ma-55	448	4	−	−	NUM
ma-55	448	5	2))−n|θu1(t	2))−n|θu1(t	NUM
ma-55	448	6	)	)	PUNCT
ma-55	449	1	+	+	CCONJ
ma-55	449	2	(	(	PUNCT
ma-55	449	3	1−	1−	NUM
ma-55	449	4	θ)u2(t)|n(r−2	θ)u2(t)|n(r−2	PROPN
ma-55	449	5	)	)	PUNCT
ma-55	449	6	n(r−2	n(r−2	NOUN
ma-55	449	7	)	)	PUNCT
ma-55	449	8	≤	≤	NOUN
ma-55	449	9	(	(	PUNCT
ma-55	449	10	e(r	e(r	CCONJ
ma-55	449	11	−	−	PROPN
ma-55	450	1	2)−n)|ω|+	2)−n)|ω|+	NUM
ma-55	450	2	(	(	PUNCT
ma-55	450	3	e(r	e(r	PROPN
ma-55	450	4	−	−	PROPN
ma-55	450	5	2))−1c4|θu1(t	2))−1c4|θu1(t	NUM
ma-55	450	6	)	)	PUNCT
ma-55	450	7	+	+	CCONJ
ma-55	450	8	(	(	PUNCT
ma-55	450	9	1−	1−	NUM
ma-55	450	10	θ)u2(t)|n(r−2	θ)u2(t)|n(r−2	NOUN
ma-55	450	11	)	)	PUNCT
ma-55	450	12	≤	≤	NOUN
ma-55	450	13	c.	c.	NOUN
ma-55	450	14	(	(	PUNCT
ma-55	450	15	6.5	6.5	NUM
ma-55	450	16	)	)	PUNCT
ma-55	450	17	inserting	insert	VERB
ma-55	450	18	(	(	PUNCT
ma-55	450	19	6.5	6.5	NUM
ma-55	450	20	)	)	PUNCT
ma-55	450	21	into	into	ADP
ma-55	450	22	i2	i2	PROPN
ma-55	450	23	,	,	PUNCT
ma-55	450	24	we	we	PRON
ma-55	450	25	have	have	VERB
ma-55	450	26	i2	i2	NOUN
ma-55	450	27	=	=	PUNCT
ma-55	450	28	(	(	PUNCT
ma-55	450	29	r	r	NOUN
ma-55	450	30	−	−	PROPN
ma-55	450	31	1	1	NUM
ma-55	450	32	)	)	PUNCT
ma-55	450	33	∫	∫	PROPN
ma-55	451	1	t	t	PROPN
ma-55	451	2	0	0	NUM
ma-55	451	3	∫	∫	PROPN
ma-55	451	4	ω	ω	NUM
ma-55	451	5	|θu1(t	|θu1(t	PROPN
ma-55	451	6	)	)	PUNCT
ma-55	452	1	+	+	CCONJ
ma-55	452	2	(	(	PUNCT
ma-55	452	3	1−	1−	NUM
ma-55	452	4	θ)u2|r−2	θ)u2|r−2	NOUN
ma-55	452	5	ln	ln	ADJ
ma-55	452	6	|θu1(t	|θu1(t	NOUN
ma-55	452	7	)	)	PUNCT
ma-55	453	1	+	+	ADJ
ma-55	453	2	(	(	PUNCT
ma-55	453	3	1−	1−	NUM
ma-55	453	4	θ)u2(t)||w(t)|wt(t)|dxds	θ)u2(t)||w(t)|wt(t)|dxds	PROPN
ma-55	453	5	≤	≤	NOUN
ma-55	453	6	(	(	PUNCT
ma-55	453	7	r	r	NOUN
ma-55	453	8	−	−	PROPN
ma-55	453	9	1	1	NUM
ma-55	453	10	)	)	PUNCT
ma-55	453	11	∫	∫	PROPN
ma-55	454	1	t	t	PROPN
ma-55	454	2	0	0	NUM
ma-55	455	1	(	(	PUNCT
ma-55	455	2	∫	∫	PROPN
ma-55	455	3	ω	ω	PROPN
ma-55	455	4	||θu1(t	||θu1(t	NUM
ma-55	455	5	)	)	PUNCT
ma-55	456	1	+	+	CCONJ
ma-55	456	2	(	(	PUNCT
ma-55	456	3	1−	1−	NUM
ma-55	456	4	θ)u2|r−2	θ)u2|r−2	NOUN
ma-55	456	5	ln	ln	ADJ
ma-55	456	6	|θu1(t	|θu1(t	NOUN
ma-55	456	7	)	)	PUNCT
ma-55	457	1	+	+	CCONJ
ma-55	457	2	θu2(t)||ndx	θu2(t)||ndx	X
ma-55	457	3	)	)	PUNCT
ma-55	457	4	1	1	NUM
ma-55	458	1	n	n	NUM
ma-55	458	2	×|wt(t)||w(t)|	×|wt(t)||w(t)|	NUM
ma-55	458	3	2n	2n	NUM
ma-55	458	4	n−2	n−2	PROPN
ma-55	458	5	ds	d	VERB
ma-55	458	6	≤	≤	ADJ
ma-55	458	7	c|∆w(t)|2	c|∆w(t)|2	NOUN
ma-55	459	1	+	+	CCONJ
ma-55	459	2	1	1	NUM
ma-55	459	3	4	4	NUM
ma-55	459	4	|∇wt(t)|2	|∇wt(t)|2	NOUN
ma-55	459	5	.	.	PUNCT
ma-55	460	1	(	(	PUNCT
ma-55	460	2	6.6	6.6	NUM
ma-55	460	3	)	)	PUNCT
ma-55	460	4	by	by	ADP
ma-55	460	5	(	(	PUNCT
ma-55	460	6	6.1	6.1	NUM
ma-55	460	7	)	)	PUNCT
ma-55	460	8	,	,	PUNCT
ma-55	460	9	(	(	PUNCT
ma-55	460	10	6.2	6.2	NUM
ma-55	460	11	)	)	PUNCT
ma-55	460	12	,	,	PUNCT
ma-55	460	13	(	(	PUNCT
ma-55	460	14	6.4	6.4	NUM
ma-55	460	15	)	)	PUNCT
ma-55	460	16	and	and	CCONJ
ma-55	460	17	(	(	PUNCT
ma-55	460	18	6.6	6.6	NUM
ma-55	460	19	)	)	PUNCT
ma-55	460	20	we	we	PRON
ma-55	460	21	get∫	get∫	VERB
ma-55	461	1	t	t	NOUN
ma-55	461	2	0	0	PUNCT
ma-55	462	1	d	d	NOUN
ma-55	462	2	dt	dt	X
ma-55	462	3	{	{	PUNCT
ma-55	462	4	|wt(t)|2	|wt(t)|2	PROPN
ma-55	462	5	+	+	NUM
ma-55	462	6	|∆w(t)|2	|∆w(t)|2	NOUN
ma-55	462	7	−	−	PROPN
ma-55	462	8	(	(	PUNCT
ma-55	462	9	∫	∫	PROPN
ma-55	462	10	t	t	PROPN
ma-55	462	11	0	0	NUM
ma-55	462	12	g(s)ds	g(s)ds	PROPN
ma-55	462	13	)	)	PUNCT
ma-55	462	14	|∇w(t)|2	|∇w(t)|2	PROPN
ma-55	462	15	+	+	CCONJ
ma-55	462	16	(	(	PUNCT
ma-55	462	17	g	g	PROPN
ma-55	462	18	�	�	PROPN
ma-55	462	19	∇w)(t	∇w)(t	PROPN
ma-55	462	20	)	)	PUNCT
ma-55	462	21	}	}	PUNCT
ma-55	462	22	ds	ds	PROPN
ma-55	462	23	+	+	CCONJ
ma-55	462	24	∫	∫	PROPN
ma-55	462	25	t	t	NOUN
ma-55	462	26	0	0	NUM
ma-55	462	27	|∇wt(t)|2ds	|∇wt(t)|2ds	NOUN
ma-55	462	28	≤	≤	NUM
ma-55	462	29	c	c	PROPN
ma-55	462	30	∫	∫	PROPN
ma-55	462	31	t	t	PROPN
ma-55	462	32	0	0	NUM
ma-55	462	33	(	(	PUNCT
ma-55	462	34	|∆w(t)|2	|∆w(t)|2	PROPN
ma-55	462	35	+	+	CCONJ
ma-55	462	36	|∇wt(t)|2)ds	|∇wt(t)|2)ds	PROPN
ma-55	462	37	.	.	PUNCT
ma-55	463	1	(	(	PUNCT
ma-55	463	2	6.7	6.7	NUM
ma-55	463	3	)	)	PUNCT
ma-55	463	4	putting	putting	NOUN
ma-55	463	5	,	,	PUNCT
ma-55	463	6	φ(t	φ(t	PROPN
ma-55	463	7	)	)	PUNCT
ma-55	463	8	=	=	SYM
ma-55	464	1	|wt(t)|2	|wt(t)|2	NOUN
ma-55	464	2	+	+	CCONJ
ma-55	464	3	|∆w(t)|2	|∆w(t)|2	NOUN
ma-55	464	4	−	−	PROPN
ma-55	464	5	(	(	PUNCT
ma-55	464	6	∫	∫	PROPN
ma-55	464	7	t	t	PROPN
ma-55	464	8	0	0	NUM
ma-55	464	9	g(s)ds	g(s)ds	PROPN
ma-55	464	10	)	)	PUNCT
ma-55	464	11	|∇w(t)|2	|∇w(t)|2	PROPN
ma-55	464	12	+	+	CCONJ
ma-55	464	13	(	(	PUNCT
ma-55	464	14	g	g	PROPN
ma-55	464	15	�	�	PROPN
ma-55	464	16	∇w)(t	∇w)(t	PROPN
ma-55	464	17	)	)	PUNCT
ma-55	464	18	and	and	CCONJ
ma-55	464	19	using	use	VERB
ma-55	464	20	(	(	PUNCT
ma-55	464	21	h3	h3	NOUN
ma-55	464	22	)	)	PUNCT
ma-55	464	23	,	,	PUNCT
ma-55	464	24	we	we	PRON
ma-55	464	25	have	have	VERB
ma-55	464	26	|∆w(t)|2	|∆w(t)|2	NOUN
ma-55	464	27	−	−	PROPN
ma-55	465	1	(	(	PUNCT
ma-55	465	2	∫	∫	PROPN
ma-55	465	3	t	t	PROPN
ma-55	465	4	0	0	NUM
ma-55	465	5	g(s)ds	g(s)ds	PROPN
ma-55	465	6	)	)	PUNCT
ma-55	465	7	|∇w(t)|2	|∇w(t)|2	PROPN
ma-55	465	8	≥	≥	PROPN
ma-55	465	9	i|∆w(t)|2	i|∆w(t)|2	PROPN
ma-55	465	10	≥	≥	PROPN
ma-55	465	11	0	0	NUM
ma-55	465	12	.	.	PUNCT
ma-55	466	1	as	as	ADP
ma-55	466	2	(	(	PUNCT
ma-55	466	3	g	g	PROPN
ma-55	466	4	�	�	PROPN
ma-55	466	5	∇w)(t	∇w)(t	PROPN
ma-55	466	6	)	)	PUNCT
ma-55	466	7	≥	≥	NOUN
ma-55	466	8	0	0	NUM
ma-55	466	9	,	,	PUNCT
ma-55	466	10	we	we	PRON
ma-55	466	11	have	have	VERB
ma-55	466	12	from	from	ADP
ma-55	466	13	(	(	PUNCT
ma-55	466	14	6.7	6.7	NUM
ma-55	466	15	)	)	PUNCT
ma-55	467	1	that	that	PRON
ma-55	467	2	∫	∫	PROPN
ma-55	468	1	t	t	NOUN
ma-55	468	2	0	0	NUM
ma-55	468	3	d	d	PRON
ma-55	468	4	dt	dt	X
ma-55	468	5	φ(t	φ(t	PROPN
ma-55	468	6	)	)	PUNCT
ma-55	468	7	≤	≤	NOUN
ma-55	468	8	cφ(t	cφ(t	PUNCT
ma-55	468	9	)	)	PUNCT
ma-55	468	10	and	and	CCONJ
ma-55	468	11	because	because	SCONJ
ma-55	468	12	φ(0	φ(0	ADJ
ma-55	468	13	)	)	PUNCT
ma-55	468	14	=	=	SYM
ma-55	468	15	0	0	NUM
ma-55	468	16	,	,	PUNCT
ma-55	468	17	followsfrom	followsfrom	ADP
ma-55	468	18	the	the	DET
ma-55	468	19	gronwall	gronwall	ADJ
ma-55	468	20	lemma	lemma	PROPN
ma-55	468	21	that	that	SCONJ
ma-55	468	22	|wt(t)|2	|wt(t)|2	NOUN
ma-55	468	23	+	+	CCONJ
ma-55	468	24	i|∆w(t)|2	i|∆w(t)|2	PROPN
ma-55	468	25	≤	≤	PROPN
ma-55	468	26	φ(t	φ(t	PROPN
ma-55	468	27	)	)	PUNCT
ma-55	468	28	≤	≤	NOUN
ma-55	468	29	0	0	NUM
ma-55	468	30	,	,	PUNCT
ma-55	468	31	which	which	PRON
ma-55	468	32	proves	prove	VERB
ma-55	468	33	that	that	PRON
ma-55	468	34	w	w	NOUN
ma-55	468	35	=	=	NOUN
ma-55	468	36	0	0	NUM
ma-55	468	37	in	in	ADP
ma-55	468	38	h1	h1	NOUN
ma-55	468	39	0(ω	0(ω	NUM
ma-55	468	40	)	)	PUNCT
ma-55	468	41	∩h2(ω	∩h2(ω	ADJ
ma-55	468	42	)	)	PUNCT
ma-55	468	43	.	.	PUNCT
ma-55	469	1	references	reference	NOUN
ma-55	469	2	[	[	X
ma-55	469	3	1	1	NUM
ma-55	469	4	]	]	PUNCT
ma-55	469	5	l.	l.	PROPN
ma-55	469	6	an	an	PROPN
ma-55	469	7	,	,	PUNCT
ma-55	469	8	a.	a.	PROPN
ma-55	469	9	pierce	pierce	PROPN
ma-55	469	10	,	,	PUNCT
ma-55	469	11	the	the	DET
ma-55	469	12	effect	effect	NOUN
ma-55	469	13	of	of	ADP
ma-55	469	14	microstructure	microstructure	NOUN
ma-55	469	15	on	on	ADP
ma-55	469	16	elastic	elastic	ADJ
ma-55	469	17	-	-	PUNCT
ma-55	469	18	plastic	plastic	NOUN
ma-55	469	19	models	model	NOUN
ma-55	469	20	,	,	PUNCT
ma-55	469	21	siam	siam	PROPN
ma-55	469	22	j.	j.	PROPN
ma-55	469	23	appl	appl	PROPN
ma-55	469	24	.	.	PROPN
ma-55	469	25	math	math	PROPN
ma-55	469	26	.	.	PUNCT
ma-55	470	1	54(3	54(3	NUM
ma-55	470	2	)	)	PUNCT
ma-55	470	3	(	(	PUNCT
ma-55	470	4	1994	1994	NUM
ma-55	470	5	)	)	PUNCT
ma-55	470	6	708	708	NUM
ma-55	470	7	-	-	SYM
ma-55	470	8	730	730	NUM
ma-55	470	9	.	.	PUNCT
ma-55	471	1	https://doi.org/10.1137/s0036139992238498[2	https://doi.org/10.1137/s0036139992238498[2	PROPN
ma-55	471	2	]	]	PUNCT
ma-55	472	1	l.	l.	PROPN
ma-55	472	2	an	an	PRON
ma-55	472	3	,	,	PUNCT
ma-55	472	4	a.	a.	PROPN
ma-55	472	5	pierce	pierce	PROPN
ma-55	472	6	,	,	PUNCT
ma-55	472	7	a	a	DET
ma-55	472	8	weakly	weakly	ADJ
ma-55	472	9	nonlinear	nonlinear	ADJ
ma-55	472	10	analysis	analysis	NOUN
ma-55	472	11	of	of	ADP
ma-55	472	12	elastoplastic	elastoplastic	ADJ
ma-55	472	13	-	-	PUNCT
ma-55	472	14	microstructure	microstructure	NOUN
ma-55	472	15	models	model	NOUN
ma-55	472	16	,	,	PUNCT
ma-55	472	17	siam	siam	PROPN
ma-55	472	18	j.	j.	PROPN
ma-55	472	19	appl	appl	PROPN
ma-55	472	20	.	.	PROPN
ma-55	472	21	math	math	PROPN
ma-55	472	22	.	.	PUNCT
ma-55	473	1	55(1)(1995	55(1)(1995	NUM
ma-55	473	2	)	)	PUNCT
ma-55	473	3	136	136	NUM
ma-55	473	4	-	-	SYM
ma-55	473	5	155	155	NUM
ma-55	473	6	.	.	PUNCT
ma-55	474	1	https://doi.org/10.1137/s0036139993255327[3	https://doi.org/10.1137/s0036139993255327[3	PROPN
ma-55	474	2	]	]	PUNCT
ma-55	474	3	a.	a.	PROPN
ma-55	474	4	andrade	andrade	PROPN
ma-55	474	5	,	,	PUNCT
ma-55	474	6	m.	m.	NOUN
ma-55	474	7	a.	a.	PROPN
ma-55	474	8	jorge	jorge	PROPN
ma-55	474	9	silva	silva	PROPN
ma-55	474	10	,	,	PUNCT
ma-55	474	11	t.	t.	PROPN
ma-55	474	12	f.	f.	PROPN
ma-55	474	13	ma	ma	PROPN
ma-55	474	14	,	,	PUNCT
ma-55	474	15	exponential	exponential	ADJ
ma-55	474	16	stability	stability	NOUN
ma-55	474	17	for	for	ADP
ma-55	474	18	a	a	DET
ma-55	474	19	plate	plate	NOUN
ma-55	474	20	equation	equation	NOUN
ma-55	474	21	with	with	ADP
ma-55	474	22	p	p	NOUN
ma-55	474	23	-	-	PUNCT
ma-55	474	24	laplacian	laplacian	NOUN
ma-55	474	25	and	and	CCONJ
ma-55	474	26	memoryterms	memoryterm	NOUN
ma-55	474	27	,	,	PUNCT
ma-55	474	28	math	math	NOUN
ma-55	474	29	.	.	PUNCT
ma-55	475	1	meth	meth	NOUN
ma-55	475	2	.	.	PUNCT
ma-55	476	1	appl	appl	PROPN
ma-55	476	2	.	.	PUNCT
ma-55	477	1	sci	sci	PROPN
ma-55	477	2	.	.	PUNCT
ma-55	477	3	35(4	35(4	NUM
ma-55	477	4	)	)	PUNCT
ma-55	477	5	(	(	PUNCT
ma-55	477	6	2012	2012	NUM
ma-55	477	7	)	)	PUNCT
ma-55	477	8	417	417	NUM
ma-55	477	9	-	-	SYM
ma-55	477	10	426	426	NUM
ma-55	477	11	.	.	PUNCT
ma-55	477	12	https://doi.org/10.1002/mma.1552	https://doi.org/10.1002/mma.1552	VERB
ma-55	477	13	https://doi.org/10.28924/ada/ma.2.5	https://doi.org/10.28924/ada/ma.2.5	PROPN
ma-55	477	14	https://doi.org/10.1137/s0036139992238498	https://doi.org/10.1137/s0036139992238498	PRON
ma-55	477	15	https://doi.org/10.1137/s0036139993255327	https://doi.org/10.1137/s0036139993255327	NUM
ma-55	477	16	https://doi.org/10.1002/mma.1552	https://doi.org/10.1002/mma.1552	NOUN
ma-55	477	17	eur	eur	NOUN
ma-55	477	18	.	.	PUNCT
ma-55	478	1	j.	j.	PROPN
ma-55	478	2	math	math	PROPN
ma-55	478	3	.	.	PUNCT
ma-55	479	1	anal	anal	PROPN
ma-55	479	2	.	.	PUNCT
ma-55	480	1	10.28924	10.28924	NUM
ma-55	480	2	/	/	SYM
ma-55	480	3	ada	ada	PROPN
ma-55	480	4	/	/	SYM
ma-55	480	5	ma.2.5	ma.2.5	PROPN
ma-55	480	6	20	20	NUM
ma-55	481	1	[	[	SYM
ma-55	481	2	4	4	NUM
ma-55	481	3	]	]	PUNCT
ma-55	481	4	a.	a.	NOUN
ma-55	481	5	ambrosetti	ambrosetti	PROPN
ma-55	481	6	,	,	PUNCT
ma-55	482	1	p.	p.	PROPN
ma-55	482	2	h.	h.	PROPN
ma-55	482	3	rabinowitz	rabinowitz	PROPN
ma-55	482	4	,	,	PUNCT
ma-55	482	5	dual	dual	ADJ
ma-55	482	6	variational	variational	ADJ
ma-55	482	7	methods	method	NOUN
ma-55	482	8	in	in	ADP
ma-55	482	9	critical	critical	ADJ
ma-55	482	10	point	point	NOUN
ma-55	482	11	theory	theory	NOUN
ma-55	482	12	and	and	CCONJ
ma-55	482	13	applications	application	NOUN
ma-55	482	14	,	,	PUNCT
ma-55	482	15	j.	j.	PROPN
ma-55	482	16	functionalanalysis	functionalanalysis	PROPN
ma-55	482	17	14(4	14(4	NUM
ma-55	482	18	)	)	PUNCT
ma-55	482	19	(	(	PUNCT
ma-55	482	20	1973	1973	NUM
ma-55	482	21	)	)	PUNCT
ma-55	482	22	349	349	NUM
ma-55	482	23	-	-	SYM
ma-55	482	24	381	381	NUM
ma-55	482	25	.	.	PUNCT
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ma-55	483	3	g.	g.	PROPN
ma-55	483	4	m.	m.	PROPN
ma-55	483	5	araújo	araújo	PROPN
ma-55	483	6	,	,	PUNCT
ma-55	483	7	s.	s.	PROPN
ma-55	483	8	b.	b.	PROPN
ma-55	483	9	menezes	menezes	PROPN
ma-55	483	10	,	,	PUNCT
ma-55	483	11	on	on	ADP
ma-55	483	12	a	a	DET
ma-55	483	13	variational	variational	ADJ
ma-55	483	14	inequality	inequality	NOUN
ma-55	483	15	for	for	ADP
ma-55	483	16	the	the	DET
ma-55	483	17	navier	navier	NOUN
ma-55	483	18	-	-	PUNCT
ma-55	483	19	stokes	stoke	NOUN
ma-55	483	20	operator	operator	NOUN
ma-55	483	21	with	with	ADP
ma-55	483	22	variable	variable	ADJ
ma-55	483	23	viscosity	viscosity	NOUN
ma-55	483	24	,	,	PUNCT
ma-55	483	25	commun	commun	PROPN
ma-55	483	26	.	.	PUNCT
ma-55	484	1	pur	pur	PROPN
ma-55	484	2	.	.	PUNCT
ma-55	485	1	appl	appl	PROPN
ma-55	485	2	.	.	PUNCT
ma-55	486	1	anal	anal	PROPN
ma-55	486	2	.	.	PUNCT
ma-55	487	1	1(3	1(3	NUM
ma-55	487	2	)	)	PUNCT
ma-55	487	3	(	(	PUNCT
ma-55	487	4	2006	2006	NUM
ma-55	487	5	)	)	PUNCT
ma-55	487	6	583	583	NUM
ma-55	487	7	-	-	SYM
ma-55	487	8	596	596	NUM
ma-55	487	9	.	.	PUNCT
ma-55	488	1	https://doi.org/10.3934/cpaa.2006.5.583[6	https://doi.org/10.3934/cpaa.2006.5.583[6	PROPN
ma-55	488	2	]	]	X
ma-55	488	3	g.	g.	PROPN
ma-55	488	4	m.	m.	PROPN
ma-55	488	5	araújo	araújo	PROPN
ma-55	488	6	,	,	PUNCT
ma-55	488	7	s.	s.	PROPN
ma-55	488	8	b.	b.	PROPN
ma-55	488	9	menezes	menezes	PROPN
ma-55	488	10	,	,	PUNCT
ma-55	488	11	a.	a.	NOUN
ma-55	488	12	o.	o.	NOUN
ma-55	488	13	marinho	marinho	PROPN
ma-55	488	14	,	,	PUNCT
ma-55	488	15	on	on	ADP
ma-55	488	16	a	a	DET
ma-55	488	17	variational	variational	ADJ
ma-55	488	18	inequality	inequality	NOUN
ma-55	488	19	for	for	ADP
ma-55	488	20	the	the	DET
ma-55	488	21	equation	equation	NOUN
ma-55	488	22	of	of	ADP
ma-55	488	23	motion	motion	NOUN
ma-55	488	24	of	of	ADP
ma-55	488	25	oldroyd	oldroyd	VERB
ma-55	488	26	fluid	fluid	NOUN
ma-55	488	27	,	,	PUNCT
ma-55	488	28	electron	electron	NOUN
ma-55	488	29	j.	j.	PROPN
ma-55	488	30	differential	differential	PROPN
ma-55	488	31	equations	equation	NOUN
ma-55	488	32	69	69	NUM
ma-55	488	33	(	(	PUNCT
ma-55	488	34	2009	2009	NUM
ma-55	488	35	)	)	PUNCT
ma-55	488	36	1	1	NUM
ma-55	488	37	-	-	SYM
ma-55	488	38	16	16	NUM
ma-55	488	39	.	.	PUNCT
ma-55	489	1	http://ejde.math.txstate.edu[7	http://ejde.math.txstate.edu[7	PROPN
ma-55	489	2	]	]	X
ma-55	489	3	g.	g.	PROPN
ma-55	489	4	m.	m.	PROPN
ma-55	489	5	araújo	araújo	PROPN
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ma-55	489	7	m.	m.	PROPN
ma-55	489	8	a.	a.	PROPN
ma-55	489	9	f.	f.	PROPN
ma-55	489	10	araújo	araújo	PROPN
ma-55	489	11	,	,	PUNCT
ma-55	489	12	d.	d.	PROPN
ma-55	489	13	c.	c.	PROPN
ma-55	489	14	pereira	pereira	PROPN
ma-55	489	15	,	,	PUNCT
ma-55	489	16	on	on	ADP
ma-55	489	17	a	a	DET
ma-55	489	18	variational	variational	ADJ
ma-55	489	19	inequality	inequality	NOUN
ma-55	489	20	for	for	ADP
ma-55	489	21	a	a	DET
ma-55	489	22	plate	plate	NOUN
ma-55	489	23	equation	equation	NOUN
ma-55	489	24	with	with	ADP
ma-55	489	25	p	p	NOUN
ma-55	489	26	-	-	PUNCT
ma-55	489	27	laplacian	laplacian	ADJ
ma-55	489	28	andmemory	andmemory	ADJ
ma-55	489	29	terms	term	NOUN
ma-55	489	30	,	,	PUNCT
ma-55	489	31	appl	appl	PROPN
ma-55	489	32	.	.	PROPN
ma-55	490	1	anal	anal	PROPN
ma-55	490	2	.	.	PUNCT
ma-55	491	1	1	1	NUM
ma-55	491	2	(	(	PUNCT
ma-55	491	3	2020	2020	NUM
ma-55	491	4	)	)	PUNCT
ma-55	491	5	1	1	NUM
ma-55	491	6	-	-	SYM
ma-55	491	7	14	14	NUM
ma-55	491	8	.	.	PUNCT
ma-55	492	1	https://doi.org/10.1080/00036811.2020.1766028[8	https://doi.org/10.1080/00036811.2020.1766028[8	NOUN
ma-55	492	2	]	]	X
ma-55	492	3	m.	m.	NOUN
ma-55	492	4	bokalo	bokalo	PROPN
ma-55	492	5	,	,	PUNCT
ma-55	492	6	o.	o.	PROPN
ma-55	492	7	sus	sus	PROPN
ma-55	492	8	,	,	PUNCT
ma-55	492	9	evolutionary	evolutionary	ADJ
ma-55	492	10	variational	variational	ADJ
ma-55	492	11	inequalities	inequality	NOUN
ma-55	492	12	with	with	ADP
ma-55	492	13	volterra	volterra	PROPN
ma-55	492	14	type	type	PROPN
ma-55	492	15	operators	operator	NOUN
ma-55	492	16	,	,	PUNCT
ma-55	492	17	mathematics	mathematic	NOUN
ma-55	492	18	and	and	CCONJ
ma-55	492	19	statistics7(5	statistics7(5	NOUN
ma-55	492	20	)	)	PUNCT
ma-55	492	21	(	(	PUNCT
ma-55	492	22	2019	2019	NUM
ma-55	492	23	)	)	PUNCT
ma-55	492	24	182	182	NUM
ma-55	492	25	-	-	SYM
ma-55	492	26	190	190	NUM
ma-55	492	27	.	.	PUNCT
ma-55	493	1	https://doi.org/10.13189/ms.2019.070504[9	https://doi.org/10.13189/ms.2019.070504[9	NOUN
ma-55	493	2	]	]	PUNCT
ma-55	493	3	a.	a.	NOUN
ma-55	493	4	bensoussan	bensoussan	PROPN
ma-55	493	5	,	,	PUNCT
ma-55	493	6	j.	j.	PROPN
ma-55	493	7	l.	l.	PROPN
ma-55	493	8	lions	lions	PROPN
ma-55	493	9	,	,	PUNCT
ma-55	493	10	contrôle	contrôle	X
ma-55	493	11	impulsionnel	impulsionnel	NOUN
ma-55	493	12	et	et	PROPN
ma-55	493	13	inèquations	inèquation	NOUN
ma-55	493	14	quasi	quasi	NOUN
ma-55	493	15	variationnelles	variationnelle	NOUN
ma-55	493	16	,	,	PUNCT
ma-55	493	17	math	math	NOUN
ma-55	493	18	.	.	PUNCT
ma-55	494	1	models	model	NOUN
ma-55	494	2	methodsinform	methodsinform	VERB
ma-55	494	3	.	.	PUNCT
ma-55	495	1	sci	sci	PROPN
ma-55	495	2	.	.	PROPN
ma-55	495	3	11	11	NUM
ma-55	495	4	,	,	PUNCT
ma-55	495	5	gauthier	gauthier	NOUN
ma-55	495	6	-	-	PUNCT
ma-55	495	7	villars	villar	NOUN
ma-55	495	8	,	,	PUNCT
ma-55	495	9	paris	paris	PROPN
ma-55	495	10	,	,	PUNCT
ma-55	495	11	1982.[10	1982.[10	NUM
ma-55	495	12	]	]	PUNCT
ma-55	495	13	m.	m.	NOUN
ma-55	495	14	m.	m.	PROPN
ma-55	495	15	cavalcanti	cavalcanti	PROPN
ma-55	495	16	,	,	PUNCT
ma-55	495	17	v.	v.	ADP
ma-55	495	18	n.	n.	PROPN
ma-55	495	19	domigos	domigos	PROPN
ma-55	495	20	cavalcanti	cavalcanti	PROPN
ma-55	495	21	,	,	PUNCT
ma-55	495	22	t.	t.	PROPN
ma-55	495	23	f.	f.	PROPN
ma-55	495	24	ma	ma	PROPN
ma-55	495	25	,	,	PUNCT
ma-55	495	26	exponential	exponential	ADJ
ma-55	495	27	decay	decay	NOUN
ma-55	495	28	of	of	ADP
ma-55	495	29	the	the	DET
ma-55	495	30	viscoelastic	viscoelastic	PROPN
ma-55	495	31	euler	euler	PROPN
ma-55	495	32	-	-	PUNCT
ma-55	495	33	bernoulli	bernoulli	NOUN
ma-55	495	34	withnonlocal	withnonlocal	ADJ
ma-55	495	35	dissipation	dissipation	NOUN
ma-55	495	36	in	in	ADP
ma-55	495	37	general	general	ADJ
ma-55	495	38	domains	domain	NOUN
ma-55	495	39	,	,	PUNCT
ma-55	495	40	differ	differ	VERB
ma-55	495	41	.	.	PUNCT
ma-55	496	1	integral	integral	PROPN
ma-55	496	2	.	.	PUNCT
ma-55	497	1	equ	equ	PROPN
ma-55	497	2	.	.	PUNCT
ma-55	497	3	17(5	17(5	NUM
ma-55	497	4	-	-	SYM
ma-55	497	5	6	6	NUM
ma-55	497	6	)	)	PUNCT
ma-55	497	7	(	(	PUNCT
ma-55	497	8	2004	2004	NUM
ma-55	497	9	)	)	PUNCT
ma-55	497	10	495	495	NUM
ma-55	497	11	-	-	SYM
ma-55	497	12	510.[11	510.[11	NUM
ma-55	497	13	]	]	PUNCT
ma-55	497	14	m.	m.	NOUN
ma-55	497	15	m.	m.	PROPN
ma-55	497	16	cavalcanti	cavalcanti	PROPN
ma-55	497	17	,	,	PUNCT
ma-55	497	18	h.	h.	PROPN
ma-55	497	19	p.	p.	PROPN
ma-55	497	20	oquendo	oquendo	PROPN
ma-55	497	21	,	,	PUNCT
ma-55	497	22	frictional	frictional	ADJ
ma-55	497	23	versus	versus	ADP
ma-55	497	24	viscoelastic	viscoelastic	NOUN
ma-55	497	25	damping	damp	VERB
ma-55	497	26	in	in	ADP
ma-55	497	27	a	a	DET
ma-55	497	28	semi	semi	ADJ
ma-55	497	29	linear	linear	PROPN
ma-55	497	30	wave	wave	NOUN
ma-55	497	31	equation	equation	NOUN
ma-55	497	32	,	,	PUNCT
ma-55	497	33	siam	siam	ADJ
ma-55	497	34	j.control	j.control	NOUN
ma-55	497	35	.	.	PUNCT
ma-55	498	1	optim	optim	PROPN
ma-55	498	2	.	.	PUNCT
ma-55	499	1	14(4	14(4	NUM
ma-55	499	2	)	)	PUNCT
ma-55	499	3	(	(	PUNCT
ma-55	499	4	2003	2003	NUM
ma-55	499	5	)	)	PUNCT
ma-55	499	6	1310	1310	NUM
ma-55	499	7	-	-	SYM
ma-55	499	8	1324	1324	NUM
ma-55	499	9	.	.	PUNCT
ma-55	500	1	https://doi.org/10.1137/s0363012902408010[12	https://doi.org/10.1137/s0363012902408010[12	PROPN
ma-55	500	2	]	]	X
ma-55	500	3	i.	i.	PROPN
ma-55	500	4	chueshov	chueshov	PROPN
ma-55	500	5	,	,	PUNCT
ma-55	500	6	i.	i.	PROPN
ma-55	500	7	lasiecka	lasiecka	PROPN
ma-55	500	8	,	,	PUNCT
ma-55	500	9	existence	existence	NOUN
ma-55	500	10	and	and	CCONJ
ma-55	500	11	uniqueness	uniqueness	NOUN
ma-55	500	12	of	of	ADP
ma-55	500	13	weak	weak	ADJ
ma-55	500	14	solutions	solution	NOUN
ma-55	500	15	and	and	CCONJ
ma-55	500	16	attractors	attractor	NOUN
ma-55	500	17	global	global	ADJ
ma-55	500	18	for	for	ADP
ma-55	500	19	a	a	DET
ma-55	500	20	class	class	NOUN
ma-55	500	21	of	of	ADP
ma-55	500	22	nonlinear	nonlinear	ADJ
ma-55	500	23	2dkirchhoff	2dkirchhoff	NUM
ma-55	500	24	-	-	PUNCT
ma-55	500	25	boussinesq	boussinesq	NOUN
ma-55	500	26	models	model	NOUN
ma-55	500	27	,	,	PUNCT
ma-55	500	28	discret	discret	ADJ
ma-55	500	29	.	.	PUNCT
ma-55	501	1	contin	contin	NOUN
ma-55	501	2	.	.	PUNCT
ma-55	502	1	dyn	dyn	PROPN
ma-55	502	2	.	.	PUNCT
ma-55	503	1	s.	s.	PROPN
ma-55	503	2	15(3	15(3	PROPN
ma-55	503	3	)	)	PUNCT
ma-55	503	4	(	(	PUNCT
ma-55	503	5	2006	2006	NUM
ma-55	503	6	)	)	PUNCT
ma-55	503	7	777	777	NUM
ma-55	503	8	-	-	SYM
ma-55	503	9	809	809	NUM
ma-55	503	10	.	.	PUNCT
ma-55	504	1	https://doi.org/10.3934/dcds	https://doi.org/10.3934/dcd	NOUN
ma-55	504	2	.	.	PUNCT
ma-55	505	1	2006.15.777[13	2006.15.777[13	NUM
ma-55	505	2	]	]	X
ma-55	505	3	c.	c.	PROPN
ma-55	505	4	m.	m.	PROPN
ma-55	505	5	dafermos	dafermos	PROPN
ma-55	505	6	,	,	PUNCT
ma-55	505	7	asymptotic	asymptotic	ADJ
ma-55	505	8	stability	stability	NOUN
ma-55	505	9	in	in	ADP
ma-55	505	10	viscoelasticity	viscoelasticity	NOUN
ma-55	505	11	,	,	PUNCT
ma-55	505	12	arch	arch	NOUN
ma-55	505	13	.	.	PUNCT
ma-55	506	1	ration	ration	NOUN
ma-55	506	2	.	.	PUNCT
ma-55	507	1	mech	mech	PROPN
ma-55	507	2	.	.	PUNCT
ma-55	508	1	anal	anal	PROPN
ma-55	508	2	.	.	PUNCT
ma-55	509	1	37	37	NUM
ma-55	509	2	(	(	PUNCT
ma-55	509	3	1970	1970	NUM
ma-55	509	4	)	)	PUNCT
ma-55	509	5	297	297	NUM
ma-55	509	6	-	-	SYM
ma-55	509	7	308	308	NUM
ma-55	509	8	.	.	PUNCT
ma-55	509	9	https	https	NOUN
ma-55	509	10	:	:	PUNCT
ma-55	509	11	//doi.org/10.1007	//doi.org/10.1007	PROPN
ma-55	509	12	/	/	SYM
ma-55	509	13	bf00251609[14	bf00251609[14	PROPN
ma-55	509	14	]	]	PUNCT
ma-55	510	1	p.	p.	PROPN
ma-55	510	2	hartman	hartman	PROPN
ma-55	510	3	,	,	PUNCT
ma-55	510	4	g.	g.	PROPN
ma-55	510	5	stampacchia	stampacchia	PROPN
ma-55	510	6	,	,	PUNCT
ma-55	510	7	on	on	ADP
ma-55	510	8	some	some	DET
ma-55	510	9	nonlinear	nonlinear	ADJ
ma-55	510	10	elliptic	elliptic	ADJ
ma-55	510	11	differential	differential	ADJ
ma-55	510	12	functional	functional	ADJ
ma-55	510	13	equations	equation	NOUN
ma-55	510	14	,	,	PUNCT
ma-55	510	15	acta	acta	PROPN
ma-55	510	16	math	math	PROPN
ma-55	510	17	.	.	PUNCT
ma-55	511	1	115	115	NUM
ma-55	511	2	(	(	PUNCT
ma-55	511	3	1966)271	1966)271	PROPN
ma-55	511	4	-	-	PUNCT
ma-55	511	5	310	310	NUM
ma-55	511	6	.	.	PUNCT
ma-55	512	1	https://doi.org/10.1007/bf02392210[15	https://doi.org/10.1007/bf02392210[15	NOUN
ma-55	512	2	]	]	X
ma-55	512	3	n.	n.	PROPN
ma-55	512	4	kikuchi	kikuchi	PROPN
ma-55	512	5	,	,	PUNCT
ma-55	512	6	j.	j.	PROPN
ma-55	512	7	t.	t.	PROPN
ma-55	512	8	oden	oden	PROPN
ma-55	512	9	,	,	PUNCT
ma-55	512	10	contacts	contact	NOUN
ma-55	512	11	problems	problem	NOUN
ma-55	512	12	in	in	ADP
ma-55	512	13	elasticity	elasticity	NOUN
ma-55	512	14	:	:	PUNCT
ma-55	512	15	a	a	DET
ma-55	512	16	study	study	NOUN
ma-55	512	17	of	of	ADP
ma-55	512	18	variational	variational	ADJ
ma-55	512	19	inequalities	inequality	NOUN
ma-55	512	20	and	and	CCONJ
ma-55	512	21	finite	finite	ADJ
ma-55	512	22	elementmethods	elementmethod	NOUN
ma-55	512	23	,	,	PUNCT
ma-55	512	24	siam	siam	PROPN
ma-55	512	25	,	,	PUNCT
ma-55	512	26	philadelphia	philadelphia	PROPN
ma-55	512	27	,	,	PUNCT
ma-55	512	28	1988.[16	1988.[16	PROPN
ma-55	512	29	]	]	X
ma-55	512	30	j.	j.	PROPN
ma-55	512	31	l.	l.	PROPN
ma-55	512	32	lions	lions	PROPN
ma-55	512	33	,	,	PUNCT
ma-55	512	34	quelques	quelques	PROPN
ma-55	512	35	méthodes	méthode	NOUN
ma-55	512	36	de	de	PROPN
ma-55	512	37	resolution	resolution	PROPN
ma-55	512	38	des	des	PROPN
ma-55	512	39	problémes	problémes	PROPN
ma-55	512	40	aux	aux	PROPN
ma-55	512	41	limites	limites	PROPN
ma-55	512	42	non	non	PROPN
ma-55	512	43	linéaires	linéaires	PROPN
ma-55	512	44	,	,	PUNCT
ma-55	512	45	dunod	dunod	PROPN
ma-55	512	46	,	,	PUNCT
ma-55	512	47	paris	paris	PROPN
ma-55	512	48	,	,	PUNCT
ma-55	512	49	1969.[17	1969.[17	NUM
ma-55	512	50	]	]	PUNCT
ma-55	512	51	l.	l.	PROPN
ma-55	512	52	e.	e.	PROPN
ma-55	512	53	payne	payne	PROPN
ma-55	512	54	,	,	PUNCT
ma-55	512	55	d.	d.	PROPN
ma-55	512	56	h.	h.	PROPN
ma-55	512	57	sattinger	sattinger	PROPN
ma-55	512	58	,	,	PUNCT
ma-55	512	59	saddle	saddle	NOUN
ma-55	512	60	points	point	NOUN
ma-55	512	61	and	and	CCONJ
ma-55	512	62	instability	instability	NOUN
ma-55	512	63	of	of	ADP
ma-55	512	64	nonlinear	nonlinear	ADJ
ma-55	512	65	hyperbolic	hyperbolic	ADJ
ma-55	512	66	equations	equation	NOUN
ma-55	512	67	,	,	PUNCT
ma-55	512	68	israel	israel	PROPN
ma-55	512	69	j.	j.	PROPN
ma-55	512	70	math	math	PROPN
ma-55	512	71	.	.	PUNCT
ma-55	513	1	22(1975	22(1975	NUM
ma-55	513	2	)	)	PUNCT
ma-55	513	3	273	273	NUM
ma-55	513	4	-	-	SYM
ma-55	513	5	303	303	NUM
ma-55	513	6	.	.	PUNCT
ma-55	514	1	https://doi.org/10.1007/bf02761595[18	https://doi.org/10.1007/bf02761595[18	NOUN
ma-55	514	2	]	]	X
ma-55	514	3	c.	c.	PROPN
ma-55	514	4	a.	a.	PROPN
ma-55	514	5	raposo	raposo	PROPN
ma-55	514	6	,	,	PUNCT
ma-55	514	7	d.	d.	PROPN
ma-55	514	8	c.	c.	PROPN
ma-55	514	9	pereira	pereira	PROPN
ma-55	514	10	,	,	PUNCT
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ma-55	514	22	-	-	PUNCT
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ma-55	514	35	equations	equation	NOUN
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ma-55	514	37	(	(	PUNCT
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ma-55	514	39	)	)	PUNCT
ma-55	514	40	1	1	NUM
ma-55	514	41	-	-	SYM
ma-55	514	42	14	14	NUM
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ma-55	515	7	a.	a.	PROPN
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ma-55	515	12	pereira	pereira	PROPN
ma-55	515	13	,	,	PUNCT
ma-55	515	14	c.	c.	PROPN
ma-55	515	15	h.	h.	PROPN
ma-55	515	16	maranhão	maranhão	PROPN
ma-55	515	17	,	,	PUNCT
ma-55	515	18	unilateral	unilateral	ADJ
ma-55	515	19	problem	problem	NOUN
ma-55	515	20	for	for	ADP
ma-55	515	21	a	a	DET
ma-55	515	22	nonlinear	nonlinear	ADJ
ma-55	515	23	wave	wave	NOUN
ma-55	515	24	equation	equation	NOUN
ma-55	515	25	with	with	ADP
ma-55	515	26	p	p	NOUN
ma-55	515	27	-	-	PUNCT
ma-55	515	28	laplacianoperator	laplacianoperator	NOUN
ma-55	515	29	,	,	PUNCT
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ma-55	518	1	11(1	11(1	NUM
ma-55	518	2	)	)	PUNCT
ma-55	518	3	(	(	PUNCT
ma-55	518	4	2021	2021	NUM
ma-55	518	5	)	)	PUNCT
ma-55	518	6	546	546	NUM
ma-55	518	7	-	-	SYM
ma-55	518	8	555	555	NUM
ma-55	518	9	.	.	PUNCT
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ma-55	519	5	,	,	PUNCT
ma-55	519	6	global	global	ADJ
ma-55	519	7	existence	existence	NOUN
ma-55	519	8	and	and	CCONJ
ma-55	519	9	asymptotic	asymptotic	ADJ
ma-55	519	10	behavior	behavior	NOUN
ma-55	519	11	of	of	ADP
ma-55	519	12	solutions	solution	NOUN
ma-55	519	13	for	for	ADP
ma-55	519	14	a	a	DET
ma-55	519	15	class	class	NOUN
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ma-55	519	17	nonlinear	nonlinear	ADJ
ma-55	519	18	degenerate	degenerate	ADJ
ma-55	519	19	wave	wave	NOUN
ma-55	519	20	equations	equation	NOUN
ma-55	519	21	,	,	PUNCT
ma-55	519	22	differ	differ	VERB
ma-55	519	23	.	.	PUNCT
ma-55	520	1	equ	equ	PROPN
ma-55	520	2	.	.	PUNCT
ma-55	520	3	nonlinear	nonlinear	PROPN
ma-55	520	4	mech	mech	PROPN
ma-55	520	5	.	.	PUNCT
ma-55	521	1	2007	2007	NUM
ma-55	521	2	(	(	PUNCT
ma-55	521	3	2007	2007	NUM
ma-55	521	4	)	)	PUNCT
ma-55	521	5	1	1	NUM
ma-55	521	6	-	-	SYM
ma-55	521	7	9	9	NUM
ma-55	521	8	.	.	PUNCT
ma-55	522	1	https://doi.org/10.1155/2007/19685[21	https://doi.org/10.1155/2007/19685[21	PROPN
ma-55	522	2	]	]	PUNCT
ma-55	522	3	m.	m.	PROPN
ma-55	522	4	willem	willem	PROPN
ma-55	522	5	,	,	PUNCT
ma-55	522	6	minimax	minimax	NOUN
ma-55	522	7	theorems	theorem	NOUN
ma-55	522	8	.	.	PUNCT
ma-55	523	1	progress	progress	NOUN
ma-55	523	2	in	in	ADP
ma-55	523	3	nonlinear	nonlinear	ADJ
ma-55	523	4	differential	differential	ADJ
ma-55	523	5	equations	equation	NOUN
ma-55	523	6	and	and	CCONJ
ma-55	523	7	their	their	PRON
ma-55	523	8	applications	application	NOUN
ma-55	523	9	24	24	NUM
ma-55	523	10	,	,	PUNCT
ma-55	523	11	birkhouserboston	birkhouserboston	PROPN
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ma-55	523	15	,	,	PUNCT
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ma-55	523	17	,	,	PUNCT
ma-55	523	18	1996.[22	1996.[22	PROPN
ma-55	523	19	]	]	X
ma-55	523	20	y.	y.	PROPN
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ma-55	523	22	,	,	PUNCT
ma-55	523	23	longtime	longtime	ADJ
ma-55	523	24	behavior	behavior	NOUN
ma-55	523	25	for	for	ADP
ma-55	523	26	a	a	DET
ma-55	523	27	nonlinear	nonlinear	ADJ
ma-55	523	28	wave	wave	NOUN
ma-55	523	29	equation	equation	NOUN
ma-55	523	30	arising	arise	VERB
ma-55	523	31	in	in	ADP
ma-55	523	32	elastoplastic	elastoplastic	ADJ
ma-55	523	33	flow	flow	NOUN
ma-55	523	34	,	,	PUNCT
ma-55	523	35	math	math	NOUN
ma-55	523	36	.	.	PUNCT
ma-55	524	1	meth	meth	NOUN
ma-55	524	2	.	.	PUNCT
ma-55	525	1	appl	appl	PROPN
ma-55	525	2	.	.	PUNCT
ma-55	526	1	sci.32(9	sci.32(9	NOUN
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ma-55	526	3	(	(	PUNCT
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ma-55	526	5	)	)	PUNCT
ma-55	526	6	1082	1082	NUM
ma-55	526	7	-	-	SYM
ma-55	526	8	1104	1104	NUM
ma-55	526	9	.	.	PUNCT
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ma-55	527	3	y.	y.	PROPN
ma-55	527	4	zhijian	zhijian	PROPN
ma-55	527	5	,	,	PUNCT
ma-55	527	6	global	global	ADJ
ma-55	527	7	attractores	attractore	NOUN
ma-55	527	8	and	and	CCONJ
ma-55	527	9	their	their	PRON
ma-55	527	10	hausdorff	hausdorff	NOUN
ma-55	527	11	dimensions	dimension	NOUN
ma-55	527	12	for	for	ADP
ma-55	527	13	a	a	DET
ma-55	527	14	class	class	NOUN
ma-55	527	15	of	of	ADP
ma-55	527	16	kirchhoff	kirchhoff	NOUN
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ma-55	527	19	j.	j.	PROPN
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ma-55	529	2	)	)	PUNCT
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ma-55	530	3	y.	y.	PROPN
ma-55	530	4	zhijian	zhijian	PROPN
ma-55	530	5	,	,	PUNCT
ma-55	530	6	j.	j.	PROPN
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ma-55	530	8	,	,	PUNCT
ma-55	530	9	global	global	ADJ
ma-55	530	10	attractor	attractor	NOUN
ma-55	530	11	for	for	ADP
ma-55	530	12	a	a	DET
ma-55	530	13	class	class	NOUN
ma-55	530	14	of	of	ADP
ma-55	530	15	kirchhoff	kirchhoff	NOUN
ma-55	530	16	models	model	NOUN
ma-55	530	17	,	,	PUNCT
ma-55	530	18	j.	j.	PROPN
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ma-55	530	20	.	.	PUNCT
ma-55	531	1	phys	phy	NOUN
ma-55	531	2	.	.	PUNCT
ma-55	532	1	50(3	50(3	X
ma-55	532	2	)	)	PUNCT
ma-55	532	3	(	(	PUNCT
ma-55	532	4	2009	2009	NUM
ma-55	532	5	)	)	PUNCT
ma-55	532	6	032701	032701	NUM
ma-55	532	7	.	.	PUNCT
ma-55	533	1	https	https	NOUN
ma-55	533	2	:	:	PUNCT
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ma-55	535	1	https://doi.org/10.3934/cpaa.2006.5.583	https://doi.org/10.3934/cpaa.2006.5.583	PUNCT
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ma-55	535	4	https://doi.org/10.13189/ms.2019.070504	https://doi.org/10.13189/ms.2019.070504	PROPN
ma-55	535	5	https://doi.org/10.1137/s0363012902408010	https://doi.org/10.1137/s0363012902408010	NUM
ma-55	535	6	https://doi.org/10.3934/dcds.2006.15.777	https://doi.org/10.3934/dcds.2006.15.777	ADJ
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ma-55	535	10	https://doi.org/10.1007/bf02392210	https://doi.org/10.1007/bf02392210	NOUN
ma-55	536	1	https://doi.org/10.1007/bf02761595	https://doi.org/10.1007/bf02761595	NOUN
ma-55	536	2	https://ejde.math.txstate.edu/volumes/2015/137/abstr.html	https://ejde.math.txstate.edu/volumes/2015/137/abstr.html	X
ma-55	536	3	https://ejde.math.txstate.edu/volumes/2015/137/abstr.html	https://ejde.math.txstate.edu/volumes/2015/137/abstr.html	PROPN
ma-55	537	1	https://doi.org/10.11948/20200147	https://doi.org/10.11948/20200147	PRON
ma-55	537	2	https://doi.org/10.1155/2007/19685	https://doi.org/10.1155/2007/19685	PROPN
ma-55	537	3	https://doi.org/10.1002/mma.1080	https://doi.org/10.1002/mma.1080	PROPN
ma-55	537	4	https://doi.org/10.1063/1.3303633	https://doi.org/10.1063/1.3303633	PROPN
ma-55	537	5	https://doi.org/10.1063/1.3085951	https://doi.org/10.1063/1.3085951	PROPN
ma-55	537	6	https://doi.org/10.1063/1.3085951	https://doi.org/10.1063/1.3085951	PROPN
ma-55	537	7	1	1	NUM
ma-55	537	8	.	.	PUNCT
ma-55	537	9	introduction	introduction	NOUN
ma-55	537	10	2	2	NUM
ma-55	537	11	.	.	PUNCT
ma-55	537	12	preliminaries	preliminary	NOUN
ma-55	537	13	3	3	NUM
ma-55	537	14	.	.	PUNCT
ma-55	537	15	potential	potential	ADJ
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ma-55	537	17	4	4	NUM
ma-55	537	18	.	.	PUNCT
ma-55	537	19	existence	existence	NOUN
ma-55	537	20	of	of	ADP
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ma-55	537	22	solutions	solution	NOUN
ma-55	537	23	5	5	NUM
ma-55	537	24	.	.	PUNCT
ma-55	537	25	penalization	penalization	NOUN
ma-55	537	26	method	method	NOUN
ma-55	537	27	5.1	5.1	NUM
ma-55	537	28	.	.	PUNCT
ma-55	537	29	approximate	approximate	ADJ
ma-55	537	30	problem	problem	NOUN
ma-55	537	31	5.2	5.2	NUM
ma-55	537	32	.	.	PUNCT
ma-55	538	1	first	first	ADV
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ma-55	538	3	5.3	5.3	NUM
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ma-55	538	6	estimate	estimate	NOUN
ma-55	538	7	5.4	5.4	NUM
ma-55	538	8	.	.	PUNCT
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ma-55	538	10	estimate	estimate	NOUN
ma-55	538	11	5.5	5.5	NUM
ma-55	538	12	.	.	PUNCT
ma-55	539	1	strong	strong	ADJ
ma-55	539	2	solution	solution	NOUN
ma-55	539	3	6	6	NUM
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ma-55	540	1	uniqueness	uniqueness	NOUN
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