id	sid	tid	token	lemma	pos
ma-28	1	1	2022	2022	NUM
ma-28	1	2	ada	ada	PROPN
ma-28	1	3	academica	academica	PROPN
ma-28	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-28	1	5	.	.	PUNCT
ma-28	2	1	j.	j.	PROPN
ma-28	2	2	math	math	PROPN
ma-28	2	3	.	.	PUNCT
ma-28	3	1	anal	anal	ADJ
ma-28	3	2	.	.	PUNCT
ma-28	3	3	2	2	NUM
ma-28	3	4	(	(	PUNCT
ma-28	3	5	2022	2022	NUM
ma-28	3	6	)	)	PUNCT
ma-28	3	7	8doi	8doi	NUM
ma-28	3	8	:	:	PUNCT
ma-28	3	9	10.28924	10.28924	NUM
ma-28	3	10	/	/	SYM
ma-28	3	11	ada	ada	NOUN
ma-28	3	12	/	/	SYM
ma-28	3	13	ma.2.8	ma.2.8	PROPN
ma-28	3	14	stability	stability	NOUN
ma-28	3	15	of	of	ADP
ma-28	3	16	positive	positive	ADJ
ma-28	3	17	weak	weak	ADJ
ma-28	3	18	solution	solution	NOUN
ma-28	3	19	for	for	ADP
ma-28	3	20	generalized	generalized	ADJ
ma-28	3	21	weighted	weight	VERB
ma-28	3	22	p	p	PROPN
ma-28	3	23	-	-	PUNCT
ma-28	3	24	fisher	fisher	NOUN
ma-28	3	25	-	-	PUNCT
ma-28	3	26	kolmogoroff	kolmogoroff	NOUN
ma-28	3	27	nonlinear	nonlinear	ADJ
ma-28	3	28	stationary	stationary	ADJ
ma-28	3	29	-	-	PUNCT
ma-28	3	30	state	state	NOUN
ma-28	3	31	problem	problem	NOUN
ma-28	3	32	salah	salah	PROPN
ma-28	3	33	a.	a.	PROPN
ma-28	3	34	khafagy∗	khafagy∗	PROPN
ma-28	3	35	,	,	PUNCT
ma-28	3	36	hassan	hassan	PROPN
ma-28	3	37	m.	m.	PROPN
ma-28	3	38	serag	serag	PROPN
ma-28	3	39	department	department	PROPN
ma-28	3	40	of	of	ADP
ma-28	3	41	mathematics	mathematic	NOUN
ma-28	3	42	,	,	PUNCT
ma-28	3	43	faculty	faculty	NOUN
ma-28	3	44	of	of	ADP
ma-28	3	45	science	science	NOUN
ma-28	3	46	,	,	PUNCT
ma-28	3	47	al	al	PROPN
ma-28	3	48	-	-	PUNCT
ma-28	3	49	azhar	azhar	PROPN
ma-28	3	50	university	university	PROPN
ma-28	3	51	,	,	PUNCT
ma-28	3	52	nasr	nasr	PROPN
ma-28	3	53	city	city	PROPN
ma-28	3	54	(	(	PUNCT
ma-28	3	55	11884	11884	NUM
ma-28	3	56	)	)	PUNCT
ma-28	3	57	,	,	PUNCT
ma-28	3	58	cairo	cairo	PROPN
ma-28	3	59	,	,	PUNCT
ma-28	3	60	egypt	egypt	PROPN
ma-28	3	61	salahabdelnaby.211@azhar.edu.eg	salahabdelnaby.211@azhar.edu.eg	PROPN
ma-28	3	62	,	,	PUNCT
ma-28	3	63	serraghm@yahoo.com	serraghm@yahoo.com	X
ma-28	4	1	∗correspondence	∗correspondence	NOUN
ma-28	4	2	:	:	PUNCT
ma-28	4	3	salahabdelnaby.211@azhar.edu.eg	salahabdelnaby.211@azhar.edu.eg	X
ma-28	4	4	abstract	abstract	NOUN
ma-28	4	5	.	.	PUNCT
ma-28	5	1	in	in	ADP
ma-28	5	2	the	the	DET
ma-28	5	3	present	present	ADJ
ma-28	5	4	paper	paper	NOUN
ma-28	5	5	,	,	PUNCT
ma-28	5	6	we	we	PRON
ma-28	5	7	investigate	investigate	VERB
ma-28	5	8	the	the	DET
ma-28	5	9	stability	stability	NOUN
ma-28	5	10	results	result	NOUN
ma-28	5	11	of	of	ADP
ma-28	5	12	positive	positive	ADJ
ma-28	5	13	weak	weak	ADJ
ma-28	5	14	solution	solution	NOUN
ma-28	5	15	for	for	ADP
ma-28	5	16	thegeneralized	thegeneralize	VERB
ma-28	5	17	fisher	fisher	PROPN
ma-28	5	18	–	–	PUNCT
ma-28	5	19	kolmogoroff	kolmogoroff	PROPN
ma-28	5	20	nonlinear	nonlinear	ADJ
ma-28	5	21	stationary	stationary	ADJ
ma-28	5	22	-	-	PUNCT
ma-28	5	23	state	state	NOUN
ma-28	5	24	problem	problem	NOUN
ma-28	5	25	involving	involving	AUX
ma-28	5	26	weighted	weight	VERB
ma-28	5	27	p	p	ADJ
ma-28	5	28	-	-	PUNCT
ma-28	5	29	laplacianoperator	laplacianoperator	NOUN
ma-28	5	30	−d∆p	−d∆p	PROPN
ma-28	5	31	,	,	PUNCT
ma-28	5	32	pu	pu	PROPN
ma-28	5	33	=	=	SYM
ma-28	5	34	ka(x)u[ν	ka(x)u[ν	PROPN
ma-28	5	35	−	−	NOUN
ma-28	6	1	υu	υu	X
ma-28	6	2	]	]	X
ma-28	6	3	in	in	ADP
ma-28	6	4	ω	ω	PROPN
ma-28	6	5	,	,	PUNCT
ma-28	6	6	bu	bu	ADP
ma-28	6	7	=	=	NOUN
ma-28	6	8	0	0	NUM
ma-28	6	9	on	on	ADP
ma-28	6	10	∂ω	∂ω	PROPN
ma-28	6	11	,	,	PUNCT
ma-28	6	12	where	where	SCONJ
ma-28	6	13	∆p	∆p	PROPN
ma-28	6	14	,	,	PUNCT
ma-28	6	15	p	p	NOUN
ma-28	6	16	with	with	ADP
ma-28	6	17	p	p	PROPN
ma-28	6	18	>	>	X
ma-28	6	19	1	1	NUM
ma-28	6	20	and	and	CCONJ
ma-28	7	1	p	p	NOUN
ma-28	7	2	=	=	NOUN
ma-28	7	3	p	p	X
ma-28	7	4	(	(	PUNCT
ma-28	7	5	x)is	x)is	PROPN
ma-28	7	6	a	a	DET
ma-28	7	7	weight	weight	NOUN
ma-28	7	8	function	function	NOUN
ma-28	7	9	,	,	PUNCT
ma-28	7	10	denotes	denote	VERB
ma-28	7	11	the	the	DET
ma-28	7	12	weighted	weighted	ADJ
ma-28	7	13	p	p	NOUN
ma-28	7	14	-	-	PUNCT
ma-28	7	15	laplacian	laplacian	NOUN
ma-28	7	16	defined	define	VERB
ma-28	7	17	by	by	ADP
ma-28	7	18	∆p	∆p	PROPN
ma-28	7	19	,	,	PUNCT
ma-28	7	20	pu	pu	PROPN
ma-28	7	21	≡	≡	PROPN
ma-28	7	22	div	div	X
ma-28	8	1	[	[	X
ma-28	8	2	p	p	X
ma-28	8	3	(	(	PUNCT
ma-28	8	4	x)|∇u|p−2∇u],the	x)|∇u|p−2∇u],the	DET
ma-28	8	5	continuous	continuous	ADJ
ma-28	8	6	function	function	NOUN
ma-28	8	7	a(x	a(x	NOUN
ma-28	8	8	)	)	PUNCT
ma-28	8	9	:	:	PUNCT
ma-28	9	1	ω	ω	X
ma-28	9	2	→	→	SYM
ma-28	9	3	r	r	NOUN
ma-28	9	4	satisfies	satisfie	NOUN
ma-28	9	5	either	either	CCONJ
ma-28	9	6	a(x	a(x	NOUN
ma-28	9	7	)	)	PUNCT
ma-28	9	8	>	>	X
ma-28	9	9	0	0	NUM
ma-28	9	10	or	or	CCONJ
ma-28	9	11	a(x	a(x	NOUN
ma-28	9	12	)	)	PUNCT
ma-28	9	13	<	<	X
ma-28	9	14	0	0	NUM
ma-28	9	15	for	for	ADP
ma-28	9	16	all	all	DET
ma-28	9	17	x	x	SYM
ma-28	9	18	∈	∈	PROPN
ma-28	9	19	ω	ω	PROPN
ma-28	9	20	,	,	PUNCT
ma-28	9	21	d	d	PROPN
ma-28	9	22	,	,	PUNCT
ma-28	9	23	k	k	NOUN
ma-28	9	24	,	,	PUNCT
ma-28	9	25	νand	νand	NOUN
ma-28	9	26	υ	υ	NOUN
ma-28	9	27	are	be	AUX
ma-28	9	28	positive	positive	ADJ
ma-28	9	29	parameters	parameter	NOUN
ma-28	9	30	and	and	CCONJ
ma-28	9	31	ω	ω	NUM
ma-28	9	32	⊂	⊂	PROPN
ma-28	9	33	rn	rn	PROPN
ma-28	9	34	is	be	AUX
ma-28	9	35	a	a	DET
ma-28	9	36	bounded	bounded	ADJ
ma-28	9	37	domain	domain	NOUN
ma-28	9	38	with	with	ADP
ma-28	9	39	smooth	smooth	ADJ
ma-28	9	40	boundary	boundary	NOUN
ma-28	9	41	bu	bu	INTJ
ma-28	9	42	=	=	SYM
ma-28	9	43	δh(x)u	δh(x)u	X
ma-28	9	44	+	+	CCONJ
ma-28	9	45	(	(	PUNCT
ma-28	9	46	1−	1−	NUM
ma-28	9	47	δ	δ	PROPN
ma-28	9	48	)	)	PUNCT
ma-28	9	49	∂u	∂u	PROPN
ma-28	9	50	∂n	∂n	PROPN
ma-28	9	51	where	where	SCONJ
ma-28	9	52	δ	δ	X
ma-28	9	53	∈	∈	PROPN
ma-28	10	1	[	[	X
ma-28	10	2	0	0	NUM
ma-28	10	3	,	,	PUNCT
ma-28	10	4	1	1	NUM
ma-28	10	5	]	]	PUNCT
ma-28	10	6	,	,	PUNCT
ma-28	10	7	h	h	NOUN
ma-28	10	8	:	:	PUNCT
ma-28	10	9	∂ω→	∂ω→	X
ma-28	10	10	r+	r+	NOUN
ma-28	10	11	with	with	ADP
ma-28	10	12	h	h	NOUN
ma-28	10	13	=	=	NOUN
ma-28	10	14	1	1	NUM
ma-28	10	15	when	when	SCONJ
ma-28	10	16	δ	δ	PROPN
ma-28	10	17	=	=	NOUN
ma-28	10	18	1	1	NUM
ma-28	10	19	.	.	NOUN
ma-28	10	20	1	1	NUM
ma-28	10	21	.	.	X
ma-28	11	1	introduction	introduction	NOUN
ma-28	11	2	:	:	PUNCT
ma-28	11	3	in	in	ADP
ma-28	11	4	this	this	DET
ma-28	11	5	paper	paper	NOUN
ma-28	11	6	we	we	PRON
ma-28	11	7	study	study	VERB
ma-28	11	8	the	the	DET
ma-28	11	9	stability	stability	NOUN
ma-28	11	10	results	result	NOUN
ma-28	11	11	of	of	ADP
ma-28	11	12	positive	positive	ADJ
ma-28	11	13	weak	weak	ADJ
ma-28	11	14	solution	solution	NOUN
ma-28	11	15	for	for	ADP
ma-28	11	16	the	the	DET
ma-28	11	17	generalized	generalize	VERB
ma-28	11	18	weighted	weight	VERB
ma-28	11	19	p	p	PROPN
ma-28	11	20	-	-	PUNCT
ma-28	11	21	fisher	fisher	NOUN
ma-28	11	22	–	–	PUNCT
ma-28	11	23	kolmogoroff	kolmogoroff	PROPN
ma-28	11	24	nonlinear	nonlinear	ADJ
ma-28	11	25	stationary	stationary	ADJ
ma-28	11	26	-	-	PUNCT
ma-28	11	27	state	state	NOUN
ma-28	11	28	problem	problem	NOUN
ma-28	11	29	−d∆p	−d∆p	PROPN
ma-28	11	30	,	,	PUNCT
ma-28	11	31	pu	pu	PROPN
ma-28	11	32	=	=	PUNCT
ma-28	11	33	ka(x)f	ka(x)f	PROPN
ma-28	11	34	(	(	PUNCT
ma-28	11	35	u	u	NOUN
ma-28	11	36	)	)	PUNCT
ma-28	11	37	=	=	SYM
ma-28	11	38	ka(x)u[ν	ka(x)u[ν	PROPN
ma-28	12	1	−	−	NOUN
ma-28	13	1	υu	υu	X
ma-28	13	2	]	]	X
ma-28	13	3	in	in	ADP
ma-28	13	4	ω	ω	PROPN
ma-28	13	5	,	,	PUNCT
ma-28	13	6	bu	bu	ADP
ma-28	13	7	=	=	NOUN
ma-28	13	8	0	0	NUM
ma-28	13	9	on	on	ADP
ma-28	13	10	∂ω	∂ω	PROPN
ma-28	13	11	,	,	PUNCT
ma-28	13	12	}	}	PUNCT
ma-28	13	13	(	(	PUNCT
ma-28	13	14	1.1	1.1	NUM
ma-28	13	15	)	)	PUNCT
ma-28	13	16	where	where	SCONJ
ma-28	13	17	∆p	∆p	PROPN
ma-28	13	18	,	,	PUNCT
ma-28	13	19	p	p	NOUN
ma-28	13	20	with	with	ADP
ma-28	13	21	p	p	PROPN
ma-28	13	22	>	>	X
ma-28	13	23	1	1	NUM
ma-28	13	24	and	and	CCONJ
ma-28	13	25	p	p	NOUN
ma-28	13	26	=	=	NOUN
ma-28	13	27	p	p	X
ma-28	13	28	(	(	PUNCT
ma-28	13	29	x	x	X
ma-28	13	30	)	)	PUNCT
ma-28	13	31	is	be	AUX
ma-28	13	32	a	a	DET
ma-28	13	33	weight	weight	NOUN
ma-28	13	34	function	function	NOUN
ma-28	13	35	,	,	PUNCT
ma-28	13	36	denotes	denote	VERB
ma-28	13	37	the	the	DET
ma-28	13	38	weighted	weighted	ADJ
ma-28	13	39	p	p	X
ma-28	13	40	-	-	PUNCT
ma-28	13	41	laplaciandefined	laplaciandefined	ADJ
ma-28	13	42	by	by	ADP
ma-28	13	43	∆p	∆p	PROPN
ma-28	13	44	,	,	PUNCT
ma-28	13	45	pu	pu	PROPN
ma-28	13	46	≡	≡	PROPN
ma-28	13	47	div	div	X
ma-28	14	1	[	[	X
ma-28	14	2	p	p	X
ma-28	14	3	(	(	PUNCT
ma-28	14	4	x)|∇u|p−2∇u	x)|∇u|p−2∇u	X
ma-28	14	5	]	]	PUNCT
ma-28	14	6	(	(	PUNCT
ma-28	14	7	see	see	VERB
ma-28	14	8	for	for	ADP
ma-28	14	9	details	detail	NOUN
ma-28	14	10	[	[	X
ma-28	14	11	6	6	NUM
ma-28	14	12	]	]	NUM
ma-28	14	13	)	)	PUNCT
ma-28	14	14	,	,	PUNCT
ma-28	14	15	the	the	DET
ma-28	14	16	continuous	continuous	ADJ
ma-28	14	17	function	function	NOUN
ma-28	14	18	a(x	a(x	NOUN
ma-28	14	19	)	)	PUNCT
ma-28	14	20	:	:	PUNCT
ma-28	14	21	ω→	ω→	X
ma-28	14	22	rsatisfies	rsatisfie	NOUN
ma-28	14	23	either	either	CCONJ
ma-28	14	24	a(x	a(x	NOUN
ma-28	14	25	)	)	PUNCT
ma-28	14	26	>	>	X
ma-28	14	27	0	0	NUM
ma-28	14	28	or	or	CCONJ
ma-28	14	29	a(x	a(x	NOUN
ma-28	14	30	)	)	PUNCT
ma-28	14	31	<	<	X
ma-28	14	32	0	0	NUM
ma-28	14	33	for	for	ADP
ma-28	14	34	all	all	DET
ma-28	14	35	x	x	SYM
ma-28	14	36	∈	∈	PROPN
ma-28	14	37	ω	ω	PROPN
ma-28	14	38	,	,	PUNCT
ma-28	14	39	d	d	PROPN
ma-28	14	40	,	,	PUNCT
ma-28	14	41	k	k	NOUN
ma-28	14	42	,	,	PUNCT
ma-28	14	43	ν	ν	NOUN
ma-28	14	44	,	,	PUNCT
ma-28	14	45	ν	ν	NOUN
ma-28	14	46	and	and	CCONJ
ma-28	14	47	υ	υ	NOUN
ma-28	14	48	are	be	AUX
ma-28	14	49	positive	positive	ADJ
ma-28	14	50	parameter	parameter	NOUN
ma-28	14	51	and	and	CCONJ
ma-28	14	52	ω	ω	NUM
ma-28	14	53	⊂	⊂	PROPN
ma-28	14	54	rn	rn	PROPN
ma-28	14	55	is	be	AUX
ma-28	14	56	a	a	DET
ma-28	14	57	bounded	bounded	ADJ
ma-28	14	58	domain	domain	NOUN
ma-28	14	59	with	with	ADP
ma-28	14	60	smooth	smooth	ADJ
ma-28	14	61	boundary	boundary	NOUN
ma-28	14	62	bu	bu	INTJ
ma-28	14	63	=	=	SYM
ma-28	14	64	δh(x)u	δh(x)u	X
ma-28	14	65	+	+	CCONJ
ma-28	14	66	(	(	PUNCT
ma-28	14	67	1	1	NUM
ma-28	14	68	−	−	PROPN
ma-28	14	69	δ)∂u∂n	δ)∂u∂n	NOUN
ma-28	14	70	where	where	SCONJ
ma-28	14	71	δ	δ	PROPN
ma-28	14	72	∈	∈	PROPN
ma-28	15	1	[	[	X
ma-28	15	2	0	0	NUM
ma-28	15	3	,	,	PUNCT
ma-28	15	4	1	1	NUM
ma-28	15	5	]	]	PUNCT
ma-28	15	6	,	,	PUNCT
ma-28	15	7	h	h	NOUN
ma-28	15	8	:	:	PUNCT
ma-28	15	9	∂ω	∂ω	ADJ
ma-28	15	10	→	→	NOUN
ma-28	15	11	r+	r+	NOUN
ma-28	15	12	with	with	ADP
ma-28	15	13	h	h	NOUN
ma-28	15	14	=	=	NOUN
ma-28	15	15	1	1	NUM
ma-28	15	16	when	when	SCONJ
ma-28	15	17	δ	δ	PROPN
ma-28	15	18	=	=	SYM
ma-28	15	19	1	1	X
ma-28	15	20	.	.	PUNCT
ma-28	15	21	system	system	NOUN
ma-28	15	22	(	(	PUNCT
ma-28	15	23	1.1	1.1	NUM
ma-28	15	24	)	)	PUNCT
ma-28	15	25	is	be	AUX
ma-28	15	26	the	the	DET
ma-28	15	27	generalized	generalize	VERB
ma-28	15	28	weighted	weight	VERB
ma-28	15	29	p	p	PROPN
ma-28	15	30	-	-	PUNCT
ma-28	15	31	fisher	fisher	NOUN
ma-28	15	32	–	–	PUNCT
ma-28	15	33	kolmogoroff	kolmogoroff	PROPN
ma-28	15	34	nonlinear	nonlinear	ADJ
ma-28	15	35	stationary	stationary	ADJ
ma-28	15	36	-	-	PUNCT
ma-28	15	37	state	state	NOUN
ma-28	15	38	problem	problem	NOUN
ma-28	16	1	[	[	X
ma-28	16	2	21	21	NUM
ma-28	16	3	]	]	PUNCT
ma-28	16	4	,	,	PUNCT
ma-28	16	5	where	where	SCONJ
ma-28	16	6	d	d	NOUN
ma-28	16	7	is	be	AUX
ma-28	16	8	the	the	DET
ma-28	16	9	diffusion	diffusion	NOUN
ma-28	16	10	coefficient	coefficient	NOUN
ma-28	16	11	,	,	PUNCT
ma-28	16	12	k	k	PROPN
ma-28	16	13	is	be	AUX
ma-28	16	14	theis	theis	NOUN
ma-28	16	15	the	the	DET
ma-28	16	16	linear	linear	PROPN
ma-28	16	17	reproduction	reproduction	NOUN
ma-28	16	18	rate	rate	NOUN
ma-28	16	19	and	and	CCONJ
ma-28	16	20	u	u	NOUN
ma-28	16	21	is	be	AUX
ma-28	16	22	the	the	DET
ma-28	16	23	population	population	NOUN
ma-28	16	24	density	density	NOUN
ma-28	16	25	.	.	PUNCT
ma-28	17	1	situations	situation	NOUN
ma-28	17	2	where	where	SCONJ
ma-28	17	3	d	d	NOUN
ma-28	17	4	is	be	AUX
ma-28	17	5	space	space	NOUN
ma-28	17	6	-	-	PUNCT
ma-28	17	7	dependent	dependent	ADJ
ma-28	17	8	are	be	AUX
ma-28	17	9	arising	arise	VERB
ma-28	17	10	in	in	ADP
ma-28	17	11	more	more	ADJ
ma-28	17	12	and	and	CCONJ
ma-28	17	13	more	more	ADJ
ma-28	17	14	modelling	modelling	ADJ
ma-28	17	15	situations	situation	NOUN
ma-28	17	16	of	of	ADP
ma-28	17	17	biomedical	biomedical	ADJ
ma-28	17	18	importance	importance	NOUN
ma-28	17	19	from	from	ADP
ma-28	17	20	diffusionof	diffusionof	NOUN
ma-28	17	21	genetically	genetically	ADV
ma-28	17	22	engineered	engineer	VERB
ma-28	17	23	organisms	organism	NOUN
ma-28	17	24	in	in	ADP
ma-28	17	25	heterogeneous	heterogeneous	ADJ
ma-28	17	26	environments	environment	NOUN
ma-28	17	27	to	to	ADP
ma-28	17	28	the	the	DET
ma-28	17	29	effect	effect	NOUN
ma-28	17	30	of	of	ADP
ma-28	17	31	white	white	ADJ
ma-28	17	32	and	and	CCONJ
ma-28	17	33	grey	grey	PROPN
ma-28	17	34	received	receive	VERB
ma-28	17	35	:	:	PUNCT
ma-28	17	36	12	12	NUM
ma-28	17	37	sep	sep	NOUN
ma-28	17	38	2021	2021	NUM
ma-28	17	39	.	.	PUNCT
ma-28	18	1	key	key	ADJ
ma-28	18	2	words	word	NOUN
ma-28	18	3	and	and	CCONJ
ma-28	18	4	phrases	phrase	NOUN
ma-28	18	5	.	.	PUNCT
ma-28	19	1	stability	stability	NOUN
ma-28	19	2	;	;	PUNCT
ma-28	19	3	weak	weak	ADJ
ma-28	19	4	solution	solution	NOUN
ma-28	19	5	;	;	PUNCT
ma-28	19	6	p	p	X
ma-28	19	7	-	-	PUNCT
ma-28	19	8	laplacian.1	laplacian.1	PROPN
ma-28	19	9	https://adac.ee	https://adac.ee	PROPN
ma-28	19	10	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	19	11	eur	eur	PROPN
ma-28	19	12	.	.	PUNCT
ma-28	20	1	j.	j.	PROPN
ma-28	20	2	math	math	PROPN
ma-28	20	3	.	.	PUNCT
ma-28	21	1	anal	anal	PROPN
ma-28	21	2	.	.	PUNCT
ma-28	22	1	10.28924	10.28924	NUM
ma-28	22	2	/	/	SYM
ma-28	22	3	ada	ada	PROPN
ma-28	22	4	/	/	SYM
ma-28	22	5	ma.2.8	ma.2.8	PROPN
ma-28	22	6	2matter	2matter	NUM
ma-28	22	7	in	in	ADP
ma-28	22	8	the	the	DET
ma-28	22	9	growth	growth	NOUN
ma-28	22	10	and	and	CCONJ
ma-28	22	11	spread	spread	VERB
ma-28	22	12	of	of	ADP
ma-28	22	13	brain	brain	NOUN
ma-28	22	14	tumours	tumour	NOUN
ma-28	22	15	.	.	PUNCT
ma-28	23	1	problem	problem	NOUN
ma-28	23	2	(	(	PUNCT
ma-28	23	3	1.1	1.1	NUM
ma-28	23	4	)	)	PUNCT
ma-28	23	5	arises	arise	VERB
ma-28	23	6	from	from	ADP
ma-28	23	7	the	the	DET
ma-28	23	8	population	population	NOUN
ma-28	23	9	biologyof	biologyof	VERB
ma-28	23	10	one	one	NUM
ma-28	23	11	species.systems	species.system	NOUN
ma-28	23	12	of	of	ADP
ma-28	23	13	type	type	NOUN
ma-28	23	14	(	(	PUNCT
ma-28	23	15	1.1	1.1	NUM
ma-28	23	16	)	)	PUNCT
ma-28	23	17	have	have	AUX
ma-28	23	18	received	receive	VERB
ma-28	23	19	considerable	considerable	ADJ
ma-28	23	20	attention	attention	NOUN
ma-28	23	21	in	in	ADP
ma-28	23	22	the	the	DET
ma-28	23	23	last	last	ADJ
ma-28	23	24	decade	decade	NOUN
ma-28	23	25	(	(	PUNCT
ma-28	23	26	see	see	VERB
ma-28	23	27	,	,	PUNCT
ma-28	23	28	e.g.	e.g.	ADV
ma-28	23	29	,	,	PUNCT
ma-28	23	30	[	[	X
ma-28	23	31	18,19,24]and	18,19,24]and	NOUN
ma-28	23	32	the	the	DET
ma-28	23	33	references	reference	NOUN
ma-28	23	34	therein	therein	ADV
ma-28	23	35	)	)	PUNCT
ma-28	23	36	.	.	PUNCT
ma-28	24	1	it	it	PRON
ma-28	24	2	has	have	AUX
ma-28	24	3	been	be	AUX
ma-28	24	4	shown	show	VERB
ma-28	24	5	that	that	SCONJ
ma-28	24	6	for	for	ADP
ma-28	24	7	some	some	DET
ma-28	24	8	certian	certian	ADJ
ma-28	24	9	values	value	NOUN
ma-28	24	10	of	of	ADP
ma-28	24	11	ν	ν	PROPN
ma-28	24	12	,	,	PUNCT
ma-28	24	13	υ	υ	NOUN
ma-28	24	14	,	,	PUNCT
ma-28	24	15	system	system	NOUN
ma-28	24	16	(	(	PUNCT
ma-28	24	17	1.1)has	1.1)has	NUM
ma-28	24	18	a	a	DET
ma-28	24	19	rich	rich	ADJ
ma-28	24	20	mathematical	mathematical	ADJ
ma-28	24	21	structure	structure	NOUN
ma-28	24	22	.	.	PUNCT
ma-28	25	1	in	in	ADP
ma-28	25	2	[	[	X
ma-28	25	3	8	8	NUM
ma-28	25	4	,	,	PUNCT
ma-28	25	5	23	23	NUM
ma-28	25	6	]	]	PUNCT
ma-28	25	7	the	the	DET
ma-28	25	8	system	system	NOUN
ma-28	25	9	(	(	PUNCT
ma-28	25	10	1.1	1.1	NUM
ma-28	25	11	)	)	PUNCT
ma-28	25	12	is	be	AUX
ma-28	25	13	considered	consider	VERB
ma-28	25	14	under	under	ADP
ma-28	25	15	the	the	DET
ma-28	25	16	hypothesis	hypothesis	NOUN
ma-28	25	17	p	p	NOUN
ma-28	25	18	(	(	PUNCT
ma-28	25	19	x	x	NOUN
ma-28	25	20	)	)	PUNCT
ma-28	25	21	=	=	SYM
ma-28	25	22	(	(	PUNCT
ma-28	25	23	k	k	NOUN
ma-28	25	24	/	/	SYM
ma-28	25	25	d	d	NOUN
ma-28	25	26	)	)	PUNCT
ma-28	26	1	=	=	SYM
ma-28	26	2	1	1	NUM
ma-28	26	3	,	,	PUNCT
ma-28	26	4	p	p	NOUN
ma-28	26	5	=	=	SYM
ma-28	26	6	2	2	NUM
ma-28	26	7	and	and	CCONJ
ma-28	26	8	f	f	PROPN
ma-28	26	9	(	(	PUNCT
ma-28	26	10	u	u	NOUN
ma-28	26	11	)	)	PUNCT
ma-28	26	12	=	=	PUNCT
ma-28	26	13	u.this	u.this	NOUN
ma-28	26	14	corresponds	correspond	VERB
ma-28	26	15	to	to	ADP
ma-28	26	16	the	the	DET
ma-28	26	17	emden	emden	ADJ
ma-28	26	18	-	-	PUNCT
ma-28	26	19	fowler	fowler	PROPN
ma-28	26	20	stationary	stationary	PROPN
ma-28	26	21	-	-	PUNCT
ma-28	26	22	stateproblem	stateproblem	NOUN
ma-28	26	23	of	of	ADP
ma-28	26	24	polytropic	polytropic	ADJ
ma-28	26	25	index	index	NOUN
ma-28	26	26	of	of	ADP
ma-28	26	27	order	order	NOUN
ma-28	26	28	one	one	NUM
ma-28	26	29	.	.	PUNCT
ma-28	27	1	while	while	SCONJ
ma-28	27	2	in	in	ADP
ma-28	27	3	[	[	PUNCT
ma-28	27	4	9	9	NUM
ma-28	27	5	,	,	PUNCT
ma-28	27	6	19	19	NUM
ma-28	27	7	]	]	PUNCT
ma-28	27	8	,	,	PUNCT
ma-28	27	9	system	system	NOUN
ma-28	27	10	(	(	PUNCT
ma-28	27	11	1.1	1.1	NUM
ma-28	27	12	)	)	PUNCT
ma-28	27	13	is	be	AUX
ma-28	27	14	considered	consider	VERB
ma-28	27	15	under	under	ADP
ma-28	27	16	thehypothesis	thehypothesis	NOUN
ma-28	27	17	p	p	X
ma-28	27	18	(	(	PUNCT
ma-28	27	19	x	x	NOUN
ma-28	27	20	)	)	PUNCT
ma-28	27	21	=	=	SYM
ma-28	27	22	(	(	PUNCT
ma-28	27	23	k	k	NOUN
ma-28	27	24	/	/	SYM
ma-28	27	25	d	d	NOUN
ma-28	27	26	)	)	PUNCT
ma-28	28	1	=	=	SYM
ma-28	28	2	1	1	NUM
ma-28	28	3	,	,	PUNCT
ma-28	28	4	p	p	NOUN
ma-28	28	5	=	=	SYM
ma-28	28	6	2	2	NUM
ma-28	28	7	and	and	CCONJ
ma-28	28	8	f	f	PROPN
ma-28	28	9	(	(	PUNCT
ma-28	28	10	u	u	NOUN
ma-28	28	11	)	)	PUNCT
ma-28	29	1	=	=	SYM
ma-28	29	2	u	u	NOUN
ma-28	29	3	−	−	PROPN
ma-28	29	4	u2,where	u2,where	ADV
ma-28	29	5	u	u	NOUN
ma-28	29	6	is	be	AUX
ma-28	29	7	the	the	DET
ma-28	29	8	population	population	NOUN
ma-28	29	9	denistyof	denistyof	NOUN
ma-28	29	10	degree	degree	NOUN
ma-28	29	11	two.this	two.this	X
ma-28	29	12	corresponds	correspond	VERB
ma-28	29	13	to	to	ADP
ma-28	29	14	the	the	DET
ma-28	29	15	logestic	logestic	ADJ
ma-28	29	16	nonlinear	nonlinear	ADJ
ma-28	29	17	stationary	stationary	ADJ
ma-28	29	18	-	-	PUNCT
ma-28	29	19	state	state	NOUN
ma-28	29	20	problem	problem	NOUN
ma-28	29	21	.	.	PUNCT
ma-28	30	1	due	due	ADP
ma-28	30	2	to	to	ADP
ma-28	30	3	theappearance	theappearance	NOUN
ma-28	30	4	of	of	ADP
ma-28	30	5	weighted	weight	VERB
ma-28	30	6	p	p	PROPN
ma-28	30	7	-	-	PUNCT
ma-28	30	8	laplacian	laplacian	ADJ
ma-28	30	9	operator	operator	NOUN
ma-28	30	10	in	in	ADP
ma-28	30	11	(	(	PUNCT
ma-28	30	12	1.1	1.1	NUM
ma-28	30	13	)	)	PUNCT
ma-28	30	14	and	and	CCONJ
ma-28	30	15	the	the	DET
ma-28	30	16	particular	particular	ADJ
ma-28	30	17	cases	case	NOUN
ma-28	30	18	;	;	PUNCT
ma-28	30	19	the	the	DET
ma-28	30	20	extensions	extension	NOUN
ma-28	30	21	arechallenging	arechallenge	VERB
ma-28	30	22	and	and	CCONJ
ma-28	30	23	nontrivial.many	nontrivial.many	PRON
ma-28	30	24	authors	author	NOUN
ma-28	30	25	are	be	AUX
ma-28	30	26	interested	interested	ADJ
ma-28	30	27	in	in	ADP
ma-28	30	28	the	the	DET
ma-28	30	29	study	study	NOUN
ma-28	30	30	of	of	ADP
ma-28	30	31	stability	stability	NOUN
ma-28	30	32	and	and	CCONJ
ma-28	30	33	instability	instability	NOUN
ma-28	30	34	of	of	ADP
ma-28	30	35	nonnegative	nonnegative	ADJ
ma-28	30	36	solutions	solution	NOUN
ma-28	30	37	oflinear	oflinear	NOUN
ma-28	31	1	[	[	X
ma-28	31	2	2	2	NUM
ma-28	31	3	]	]	PUNCT
ma-28	31	4	,	,	PUNCT
ma-28	31	5	semilinear	semilinear	NOUN
ma-28	31	6	(	(	PUNCT
ma-28	31	7	see	see	VERB
ma-28	31	8	[	[	X
ma-28	31	9	10,26	10,26	NUM
ma-28	31	10	]	]	PUNCT
ma-28	31	11	)	)	PUNCT
ma-28	31	12	,	,	PUNCT
ma-28	31	13	semiposiotne	semiposiotne	NOUN
ma-28	31	14	(	(	PUNCT
ma-28	31	15	see	see	VERB
ma-28	31	16	[	[	X
ma-28	31	17	3,25	3,25	NUM
ma-28	31	18	]	]	X
ma-28	31	19	)	)	PUNCT
ma-28	31	20	,	,	PUNCT
ma-28	31	21	nonlinear	nonlinear	NOUN
ma-28	31	22	(	(	PUNCT
ma-28	31	23	see	see	VERB
ma-28	31	24	[	[	X
ma-28	31	25	1,16	1,16	NOUN
ma-28	31	26	]	]	X
ma-28	31	27	)	)	PUNCT
ma-28	31	28	and	and	CCONJ
ma-28	31	29	singular	singular	NOUN
ma-28	31	30	(	(	PUNCT
ma-28	31	31	see[17	see[17	PROPN
ma-28	31	32	]	]	PUNCT
ma-28	31	33	)	)	PUNCT
ma-28	31	34	systems	system	NOUN
ma-28	31	35	,	,	PUNCT
ma-28	31	36	due	due	ADP
ma-28	31	37	to	to	ADP
ma-28	31	38	the	the	DET
ma-28	31	39	great	great	ADJ
ma-28	31	40	number	number	NOUN
ma-28	31	41	of	of	ADP
ma-28	31	42	applications	application	NOUN
ma-28	31	43	in	in	ADP
ma-28	31	44	reaction	reaction	NOUN
ma-28	31	45	-	-	PUNCT
ma-28	31	46	diffusion	diffusion	NOUN
ma-28	31	47	problems	problem	NOUN
ma-28	31	48	,	,	PUNCT
ma-28	31	49	in	in	ADP
ma-28	31	50	autocatalyticreaction	autocatalyticreaction	NOUN
ma-28	31	51	,	,	PUNCT
ma-28	31	52	in	in	ADP
ma-28	31	53	temperature	temperature	NOUN
ma-28	31	54	on	on	ADP
ma-28	31	55	plasma	plasma	NOUN
ma-28	31	56	,	,	PUNCT
ma-28	31	57	population	population	NOUN
ma-28	31	58	dynamics	dynamic	NOUN
ma-28	31	59	,	,	PUNCT
ma-28	31	60	etc	etc	X
ma-28	31	61	.	.	X
ma-28	31	62	;	;	PUNCT
ma-28	31	63	see	see	VERB
ma-28	31	64	[	[	X
ma-28	31	65	4	4	NUM
ma-28	31	66	,	,	PUNCT
ma-28	31	67	23	23	NUM
ma-28	31	68	]	]	PUNCT
ma-28	31	69	and	and	CCONJ
ma-28	31	70	references	reference	NOUN
ma-28	31	71	therein.also	therein.also	ADV
ma-28	31	72	,	,	PUNCT
ma-28	31	73	in	in	ADP
ma-28	31	74	the	the	DET
ma-28	31	75	recent	recent	ADJ
ma-28	31	76	past	past	NOUN
ma-28	31	77	,	,	PUNCT
ma-28	31	78	many	many	ADJ
ma-28	31	79	authors	author	NOUN
ma-28	31	80	devoted	devote	VERB
ma-28	31	81	their	their	PRON
ma-28	31	82	attention	attention	NOUN
ma-28	31	83	to	to	PART
ma-28	31	84	study	study	VERB
ma-28	31	85	the	the	DET
ma-28	31	86	weighted	weighted	ADJ
ma-28	31	87	p	p	ADJ
ma-28	31	88	-	-	PUNCT
ma-28	31	89	laplaciannonlinear	laplaciannonlinear	NOUN
ma-28	31	90	systems	system	NOUN
ma-28	31	91	(	(	PUNCT
ma-28	31	92	see	see	VERB
ma-28	31	93	[	[	X
ma-28	31	94	11,12,14,15]).tertikas	11,12,14,15]).tertikas	NUM
ma-28	31	95	in	in	ADP
ma-28	31	96	[	[	X
ma-28	31	97	25	25	NUM
ma-28	31	98	]	]	PUNCT
ma-28	31	99	have	have	AUX
ma-28	31	100	been	be	AUX
ma-28	31	101	proved	prove	VERB
ma-28	31	102	the	the	DET
ma-28	31	103	stability	stability	NOUN
ma-28	31	104	and	and	CCONJ
ma-28	31	105	instability	instability	NOUN
ma-28	31	106	results	result	NOUN
ma-28	31	107	of	of	ADP
ma-28	31	108	positive	positive	ADJ
ma-28	31	109	solutions	solution	NOUN
ma-28	31	110	for	for	ADP
ma-28	31	111	thesemilinear	thesemilinear	ADJ
ma-28	31	112	system	system	NOUN
ma-28	31	113	−∆u	−∆u	PRON
ma-28	32	1	=	=	PUNCT
ma-28	32	2	λf	λf	X
ma-28	32	3	(	(	PUNCT
ma-28	32	4	u	u	NOUN
ma-28	32	5	)	)	PUNCT
ma-28	32	6	in	in	ADP
ma-28	32	7	ω	ω	PROPN
ma-28	32	8	,	,	PUNCT
ma-28	32	9	bu	bu	ADP
ma-28	32	10	=	=	NOUN
ma-28	32	11	0	0	NUM
ma-28	32	12	on	on	ADP
ma-28	32	13	∂ω	∂ω	PROPN
ma-28	32	14	,	,	PUNCT
ma-28	32	15	under	under	ADP
ma-28	32	16	various	various	ADJ
ma-28	32	17	choices	choice	NOUN
ma-28	32	18	of	of	ADP
ma-28	32	19	the	the	DET
ma-28	32	20	function	function	NOUN
ma-28	32	21	f	f	PROPN
ma-28	32	22	.	.	PUNCT
ma-28	33	1	in	in	ADP
ma-28	33	2	[	[	X
ma-28	33	3	3	3	NUM
ma-28	33	4	]	]	PUNCT
ma-28	33	5	,	,	PUNCT
ma-28	33	6	the	the	DET
ma-28	33	7	authors	author	NOUN
ma-28	33	8	have	have	AUX
ma-28	33	9	been	be	AUX
ma-28	33	10	studied	study	VERB
ma-28	33	11	the	the	DET
ma-28	33	12	uniqueness	uniqueness	ADJ
ma-28	33	13	andstability	andstability	NOUN
ma-28	33	14	of	of	ADP
ma-28	33	15	nonnegative	nonnegative	ADJ
ma-28	33	16	solutions	solution	NOUN
ma-28	33	17	for	for	ADP
ma-28	33	18	classes	class	NOUN
ma-28	33	19	of	of	ADP
ma-28	33	20	nonlinear	nonlinear	ADJ
ma-28	33	21	elliptic	elliptic	ADJ
ma-28	33	22	dirichlet	dirichlet	PROPN
ma-28	33	23	problems	problem	NOUN
ma-28	33	24	in	in	ADP
ma-28	33	25	a	a	DET
ma-28	33	26	ball	ball	NOUN
ma-28	33	27	,	,	PUNCT
ma-28	33	28	whenthe	whenthe	ADJ
ma-28	33	29	nonlinearity	nonlinearity	NOUN
ma-28	33	30	is	be	AUX
ma-28	33	31	monotone	monotone	ADJ
ma-28	33	32	,	,	PUNCT
ma-28	33	33	negative	negative	ADJ
ma-28	33	34	at	at	ADP
ma-28	33	35	the	the	DET
ma-28	33	36	origin	origin	NOUN
ma-28	33	37	,	,	PUNCT
ma-28	33	38	and	and	CCONJ
ma-28	33	39	either	either	CCONJ
ma-28	33	40	concave	concave	NOUN
ma-28	33	41	or	or	CCONJ
ma-28	33	42	convex	convex	NOUN
ma-28	33	43	.	.	PUNCT
ma-28	34	1	in	in	ADP
ma-28	34	2	the	the	DET
ma-28	34	3	case	case	NOUN
ma-28	34	4	p	p	X
ma-28	34	5	(	(	PUNCT
ma-28	34	6	x	x	NOUN
ma-28	34	7	)	)	PUNCT
ma-28	34	8	=	=	SYM
ma-28	34	9	a(x	a(x	PROPN
ma-28	34	10	)	)	PUNCT
ma-28	34	11	=	=	SYM
ma-28	34	12	1	1	NUM
ma-28	34	13	,	,	PUNCT
ma-28	34	14	p	p	NOUN
ma-28	34	15	=	=	SYM
ma-28	34	16	2	2	NUM
ma-28	34	17	and	and	CCONJ
ma-28	34	18	a	a	DET
ma-28	34	19	function	function	NOUN
ma-28	34	20	λf	λf	INTJ
ma-28	34	21	(	(	PUNCT
ma-28	34	22	u	u	NOUN
ma-28	34	23	)	)	PUNCT
ma-28	34	24	instead	instead	ADV
ma-28	34	25	of	of	ADP
ma-28	34	26	λuα	λuα	NOUN
ma-28	35	1	+	+	NOUN
ma-28	35	2	uβ	uβ	PROPN
ma-28	35	3	,	,	PUNCT
ma-28	35	4	system	system	NOUN
ma-28	35	5	(	(	PUNCT
ma-28	35	6	1.1	1.1	NUM
ma-28	35	7	)	)	PUNCT
ma-28	35	8	have	have	AUX
ma-28	35	9	been	be	AUX
ma-28	35	10	studiedby	studiedby	ADJ
ma-28	35	11	several	several	ADJ
ma-28	35	12	authors	author	NOUN
ma-28	35	13	(	(	PUNCT
ma-28	35	14	see	see	VERB
ma-28	35	15	[	[	X
ma-28	35	16	5	5	NUM
ma-28	35	17	,	,	PUNCT
ma-28	35	18	7	7	NUM
ma-28	35	19	,	,	PUNCT
ma-28	35	20	20]).khafagy	20]).khafagy	NUM
ma-28	35	21	in	in	ADP
ma-28	35	22	[	[	PUNCT
ma-28	35	23	13	13	NUM
ma-28	35	24	]	]	PUNCT
ma-28	35	25	have	have	AUX
ma-28	35	26	been	be	AUX
ma-28	35	27	studied	study	VERB
ma-28	35	28	the	the	DET
ma-28	35	29	stability	stability	NOUN
ma-28	35	30	and	and	CCONJ
ma-28	35	31	instability	instability	NOUN
ma-28	35	32	of	of	ADP
ma-28	35	33	positive	positive	ADJ
ma-28	35	34	weak	weak	ADJ
ma-28	35	35	solution	solution	NOUN
ma-28	35	36	for	for	ADP
ma-28	35	37	thenonlinear	thenonlinear	ADJ
ma-28	35	38	system	system	NOUN
ma-28	35	39	−∆p	−∆p	NOUN
ma-28	35	40	,	,	PUNCT
ma-28	35	41	pu	pu	PROPN
ma-28	35	42	+	+	CCONJ
ma-28	36	1	a(x)|u|p−2u	a(x)|u|p−2u	X
ma-28	36	2	=	=	PUNCT
ma-28	36	3	λb(x)uα	λb(x)uα	X
ma-28	36	4	in	in	ADP
ma-28	36	5	ω	ω	PROPN
ma-28	36	6	,	,	PUNCT
ma-28	36	7	bu	bu	ADP
ma-28	36	8	=	=	NOUN
ma-28	36	9	0	0	NUM
ma-28	36	10	on	on	ADP
ma-28	36	11	∂ω	∂ω	PROPN
ma-28	36	12	.	.	PUNCT
ma-28	36	13	}	}	PUNCT
ma-28	36	14	(	(	PUNCT
ma-28	36	15	1.2	1.2	NUM
ma-28	36	16	)	)	PUNCT
ma-28	36	17	where	where	SCONJ
ma-28	36	18	0	0	NUM
ma-28	36	19	<	<	X
ma-28	36	20	α	α	X
ma-28	36	21	<	<	X
ma-28	36	22	p	p	X
ma-28	36	23	−	−	PROPN
ma-28	36	24	1	1	NUM
ma-28	36	25	.	.	PUNCT
ma-28	37	1	he	he	PRON
ma-28	37	2	proved	prove	VERB
ma-28	37	3	that	that	SCONJ
ma-28	37	4	if	if	SCONJ
ma-28	37	5	0	0	NUM
ma-28	37	6	<	<	X
ma-28	37	7	α	α	X
ma-28	37	8	<	<	X
ma-28	37	9	p	p	X
ma-28	37	10	−	−	PROPN
ma-28	37	11	1	1	NUM
ma-28	37	12	and	and	CCONJ
ma-28	37	13	b(x	b(x	NOUN
ma-28	37	14	)	)	PUNCT
ma-28	37	15	>	>	X
ma-28	38	1	0	0	PUNCT
ma-28	38	2	(	(	PUNCT
ma-28	38	3	<	<	NOUN
ma-28	38	4	0	0	NUM
ma-28	38	5	)	)	PUNCT
ma-28	38	6	for	for	ADP
ma-28	38	7	all	all	DET
ma-28	38	8	x	x	SYM
ma-28	38	9	∈	∈	PROPN
ma-28	38	10	ω	ω	PROPN
ma-28	38	11	,	,	PUNCT
ma-28	38	12	thenevery	thenevery	ADP
ma-28	38	13	positive	positive	ADJ
ma-28	38	14	weak	weak	ADJ
ma-28	38	15	solution	solution	NOUN
ma-28	38	16	u	u	NOUN
ma-28	38	17	of	of	ADP
ma-28	38	18	(	(	PUNCT
ma-28	38	19	1.2	1.2	NUM
ma-28	38	20	)	)	PUNCT
ma-28	38	21	is	be	AUX
ma-28	38	22	linearly	linearly	ADV
ma-28	38	23	stable	stable	ADJ
ma-28	38	24	(	(	PUNCT
ma-28	38	25	unstable	unstable	ADJ
ma-28	38	26	)	)	PUNCT
ma-28	38	27	respectively	respectively	ADV
ma-28	38	28	.	.	PUNCT
ma-28	39	1	definition	definition	NOUN
ma-28	39	2	1.1	1.1	NUM
ma-28	39	3	.	.	PUNCT
ma-28	40	1	we	we	PRON
ma-28	40	2	recall	recall	VERB
ma-28	40	3	that	that	SCONJ
ma-28	40	4	,	,	PUNCT
ma-28	40	5	if	if	SCONJ
ma-28	40	6	u	u	PRON
ma-28	40	7	be	be	VERB
ma-28	40	8	any	any	DET
ma-28	40	9	positive	positive	ADJ
ma-28	40	10	weak	weak	ADJ
ma-28	40	11	solution	solution	NOUN
ma-28	40	12	of	of	ADP
ma-28	40	13	(	(	PUNCT
ma-28	40	14	1.1	1.1	NUM
ma-28	40	15	)	)	PUNCT
ma-28	40	16	,	,	PUNCT
ma-28	40	17	then	then	ADV
ma-28	40	18	the	the	DET
ma-28	40	19	linearizedequation	linearizedequation	NOUN
ma-28	40	20	of	of	ADP
ma-28	40	21	(	(	PUNCT
ma-28	40	22	1.1	1.1	NUM
ma-28	40	23	)	)	PUNCT
ma-28	40	24	about	about	ADP
ma-28	40	25	u	u	NOUN
ma-28	40	26	is	be	AUX
ma-28	40	27	given	give	VERB
ma-28	40	28	by	by	ADP
ma-28	40	29	−(p	−(p	NOUN
ma-28	40	30	−	−	PROPN
ma-28	40	31	1)div	1)div	NUM
ma-28	40	32	[	[	X
ma-28	40	33	p	p	X
ma-28	40	34	(	(	PUNCT
ma-28	40	35	x)|∇u|p−2∇φ]−	x)|∇u|p−2∇φ]−	X
ma-28	40	36	(	(	PUNCT
ma-28	40	37	k	k	X
ma-28	40	38	/	/	SYM
ma-28	40	39	d)a(x)[ν	d)a(x)[ν	NOUN
ma-28	40	40	−	−	PROPN
ma-28	40	41	2υu]φ	2υu]φ	NOUN
ma-28	41	1	=	=	SYM
ma-28	41	2	µφ	µφ	PROPN
ma-28	41	3	,	,	PUNCT
ma-28	41	4	x	x	SYM
ma-28	41	5	∈	∈	PROPN
ma-28	41	6	ω	ω	PROPN
ma-28	41	7	,	,	PUNCT
ma-28	41	8	bφ	bφ	NOUN
ma-28	41	9	=	=	PUNCT
ma-28	41	10	0	0	NUM
ma-28	41	11	,	,	PUNCT
ma-28	41	12	x	x	X
ma-28	41	13	∈	∈	PROPN
ma-28	41	14	∂ω	∂ω	PROPN
ma-28	41	15	,	,	PUNCT
ma-28	41	16	}	}	PUNCT
ma-28	41	17	(	(	PUNCT
ma-28	41	18	1.3	1.3	NUM
ma-28	41	19	)	)	PUNCT
ma-28	41	20	where	where	SCONJ
ma-28	41	21	µ	µ	NOUN
ma-28	41	22	is	be	AUX
ma-28	41	23	the	the	DET
ma-28	41	24	eigenvalue	eigenvalue	NOUN
ma-28	41	25	corresponding	corresponding	NOUN
ma-28	41	26	to	to	ADP
ma-28	41	27	the	the	DET
ma-28	41	28	eigenfunction	eigenfunction	NOUN
ma-28	41	29	φ	φ	PROPN
ma-28	41	30	.	.	PROPN
ma-28	41	31	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	41	32	eur	eur	PROPN
ma-28	41	33	.	.	PUNCT
ma-28	42	1	j.	j.	PROPN
ma-28	42	2	math	math	PROPN
ma-28	42	3	.	.	PUNCT
ma-28	43	1	anal	anal	PROPN
ma-28	43	2	.	.	PUNCT
ma-28	44	1	10.28924	10.28924	NUM
ma-28	44	2	/	/	SYM
ma-28	44	3	ada	ada	PROPN
ma-28	44	4	/	/	SYM
ma-28	44	5	ma.2.8	ma.2.8	PROPN
ma-28	44	6	3	3	NUM
ma-28	44	7	definition	definition	NOUN
ma-28	44	8	1.2	1.2	NUM
ma-28	44	9	.	.	PUNCT
ma-28	45	1	[	[	X
ma-28	45	2	3	3	X
ma-28	45	3	]	]	X
ma-28	45	4	a	a	DET
ma-28	45	5	solution	solution	NOUN
ma-28	45	6	u	u	NOUN
ma-28	45	7	of	of	ADP
ma-28	45	8	(	(	PUNCT
ma-28	45	9	1.1	1.1	NUM
ma-28	45	10	)	)	PUNCT
ma-28	45	11	is	be	AUX
ma-28	45	12	called	call	VERB
ma-28	45	13	stable	stable	ADJ
ma-28	45	14	solution	solution	NOUN
ma-28	45	15	if	if	SCONJ
ma-28	45	16	all	all	PRON
ma-28	45	17	eigenvalues	eigenvalue	VERB
ma-28	45	18	of	of	ADP
ma-28	45	19	(	(	PUNCT
ma-28	45	20	1.3	1.3	NUM
ma-28	45	21	)	)	PUNCT
ma-28	45	22	arestrictly	arestrictly	ADV
ma-28	45	23	positive	positive	ADJ
ma-28	45	24	,	,	PUNCT
ma-28	45	25	which	which	PRON
ma-28	45	26	can	can	AUX
ma-28	45	27	be	be	AUX
ma-28	45	28	implied	imply	VERB
ma-28	45	29	if	if	SCONJ
ma-28	45	30	the	the	DET
ma-28	45	31	principal	principal	NOUN
ma-28	45	32	eigenvalue	eigenvalue	PROPN
ma-28	45	33	µ1	µ1	PROPN
ma-28	45	34	>	>	X
ma-28	45	35	0	0	PROPN
ma-28	45	36	.	.	PUNCT
ma-28	46	1	otherwise	otherwise	ADV
ma-28	46	2	u	u	NOUN
ma-28	46	3	unstable	unstable	ADJ
ma-28	46	4	.	.	PUNCT
ma-28	47	1	2	2	X
ma-28	47	2	.	.	X
ma-28	47	3	main	main	ADJ
ma-28	47	4	results	result	NOUN
ma-28	47	5	the	the	DET
ma-28	47	6	main	main	ADJ
ma-28	47	7	goal	goal	NOUN
ma-28	47	8	of	of	ADP
ma-28	47	9	this	this	DET
ma-28	47	10	section	section	NOUN
ma-28	47	11	is	be	AUX
ma-28	47	12	to	to	PART
ma-28	47	13	prove	prove	VERB
ma-28	47	14	the	the	DET
ma-28	47	15	stability	stability	NOUN
ma-28	47	16	and	and	CCONJ
ma-28	47	17	instability	instability	NOUN
ma-28	47	18	of	of	ADP
ma-28	47	19	the	the	DET
ma-28	47	20	positive	positive	ADJ
ma-28	47	21	weak	weak	ADJ
ma-28	47	22	solution	solution	NOUN
ma-28	47	23	u	u	NOUN
ma-28	47	24	of	of	ADP
ma-28	47	25	(	(	PUNCT
ma-28	47	26	1.1	1.1	NUM
ma-28	47	27	)	)	PUNCT
ma-28	47	28	.	.	PUNCT
ma-28	48	1	our	our	PRON
ma-28	48	2	main	main	ADJ
ma-28	48	3	results	result	NOUN
ma-28	48	4	are	be	AUX
ma-28	48	5	formulate	formulate	ADJ
ma-28	48	6	in	in	ADP
ma-28	48	7	the	the	DET
ma-28	48	8	following	follow	VERB
ma-28	48	9	theorems	theorem	NOUN
ma-28	48	10	.	.	PUNCT
ma-28	49	1	theorem	theorem	VERB
ma-28	49	2	2.1	2.1	NUM
ma-28	49	3	.	.	PUNCT
ma-28	50	1	if	if	SCONJ
ma-28	50	2	α+	α+	ADP
ma-28	50	3	1	1	NUM
ma-28	50	4	<	<	X
ma-28	50	5	p	p	X
ma-28	50	6	<	<	X
ma-28	50	7	β	β	X
ma-28	50	8	+	+	CCONJ
ma-28	50	9	1	1	NUM
ma-28	50	10	and	and	CCONJ
ma-28	50	11	a(x	a(x	PROPN
ma-28	50	12	)	)	PUNCT
ma-28	50	13	>	>	X
ma-28	50	14	0	0	PUNCT
ma-28	51	1	for	for	ADP
ma-28	51	2	all	all	DET
ma-28	51	3	x	x	SYM
ma-28	51	4	∈	∈	PROPN
ma-28	51	5	ω	ω	NOUN
ma-28	51	6	,	,	PUNCT
ma-28	51	7	then	then	ADV
ma-28	51	8	every	every	DET
ma-28	51	9	positive	positive	ADJ
ma-28	51	10	weak	weak	ADJ
ma-28	51	11	solution	solution	NOUN
ma-28	51	12	of	of	ADP
ma-28	51	13	(	(	PUNCT
ma-28	51	14	1.1	1.1	NUM
ma-28	51	15	)	)	PUNCT
ma-28	51	16	is	be	AUX
ma-28	51	17	linearly	linearly	ADV
ma-28	51	18	stable	stable	ADJ
ma-28	51	19	.	.	PUNCT
ma-28	52	1	proof	proof	NOUN
ma-28	52	2	.	.	PUNCT
ma-28	53	1	let	let	VERB
ma-28	53	2	u0	u0	ADJ
ma-28	53	3	be	be	AUX
ma-28	53	4	any	any	DET
ma-28	53	5	positive	positive	ADJ
ma-28	53	6	weak	weak	ADJ
ma-28	53	7	solution	solution	NOUN
ma-28	53	8	of	of	ADP
ma-28	53	9	(	(	PUNCT
ma-28	53	10	1.1	1.1	NUM
ma-28	53	11	)	)	PUNCT
ma-28	53	12	,	,	PUNCT
ma-28	53	13	then	then	ADV
ma-28	53	14	the	the	DET
ma-28	53	15	linearized	linearize	VERB
ma-28	53	16	equation	equation	NOUN
ma-28	53	17	bout	bout	NOUN
ma-28	53	18	u0	u0	NOUN
ma-28	53	19	is	be	AUX
ma-28	53	20	−(p	−(p	ADJ
ma-28	53	21	−	−	NOUN
ma-28	53	22	1)div	1)div	NUM
ma-28	54	1	[	[	X
ma-28	54	2	p	p	X
ma-28	54	3	(	(	PUNCT
ma-28	54	4	x)|∇u0|p−2∇φ]−	x)|∇u0|p−2∇φ]−	PROPN
ma-28	54	5	(	(	PUNCT
ma-28	54	6	k	k	X
ma-28	54	7	/	/	SYM
ma-28	54	8	d)a(x)[ν	d)a(x)[ν	PROPN
ma-28	54	9	−	−	PROPN
ma-28	54	10	2υu0]φ	2υu0]φ	NUM
ma-28	54	11	=	=	SYM
ma-28	54	12	µφ	µφ	PROPN
ma-28	54	13	,	,	PUNCT
ma-28	54	14	x	x	PUNCT
ma-28	54	15	∈	∈	PROPN
ma-28	54	16	ω	ω	NUM
ma-28	54	17	bφ	bφ	NOUN
ma-28	54	18	=	=	SYM
ma-28	54	19	0	0	NUM
ma-28	54	20	,	,	PUNCT
ma-28	54	21	x	x	SYM
ma-28	54	22	∈	∈	PROPN
ma-28	54	23	∂ω	∂ω	PROPN
ma-28	54	24	.	.	PUNCT
ma-28	54	25	}	}	PUNCT
ma-28	54	26	(	(	PUNCT
ma-28	54	27	2.1	2.1	NUM
ma-28	54	28	)	)	PUNCT
ma-28	54	29	let	let	VERB
ma-28	54	30	µ1	µ1	PROPN
ma-28	54	31	be	be	AUX
ma-28	54	32	the	the	DET
ma-28	54	33	first	first	ADJ
ma-28	54	34	eigenvalue	eigenvalue	NOUN
ma-28	54	35	of	of	ADP
ma-28	54	36	(	(	PUNCT
ma-28	54	37	2.1	2.1	NUM
ma-28	54	38	)	)	PUNCT
ma-28	54	39	and	and	CCONJ
ma-28	54	40	let	let	VERB
ma-28	54	41	ψ(x	ψ(x	PROPN
ma-28	54	42	)	)	PUNCT
ma-28	54	43	≥	≥	X
ma-28	54	44	0	0	NUM
ma-28	54	45	be	be	AUX
ma-28	54	46	the	the	DET
ma-28	54	47	corresponding	correspond	VERB
ma-28	54	48	eigenfunction.multiplying	eigenfunction.multiplying	PROPN
ma-28	54	49	(	(	PUNCT
ma-28	54	50	1.1	1.1	NUM
ma-28	54	51	)	)	PUNCT
ma-28	54	52	by	by	ADP
ma-28	54	53	ψ	ψ	X
ma-28	54	54	and	and	CCONJ
ma-28	54	55	integrating	integrate	VERB
ma-28	54	56	over	over	ADP
ma-28	54	57	ω	ω	PROPN
ma-28	54	58	,	,	PUNCT
ma-28	54	59	we	we	PRON
ma-28	54	60	have	have	VERB
ma-28	54	61	−	−	PROPN
ma-28	54	62	∫	∫	PROPN
ma-28	54	63	ω	ω	X
ma-28	54	64	ψdiv	ψdiv	X
ma-28	55	1	[	[	X
ma-28	55	2	p	p	X
ma-28	55	3	(	(	PUNCT
ma-28	55	4	x)|∇u0|p−2∇u0]dx	x)|∇u0|p−2∇u0]dx	PROPN
ma-28	55	5	=	=	SYM
ma-28	55	6	(	(	PUNCT
ma-28	55	7	k	k	NOUN
ma-28	55	8	/	/	SYM
ma-28	55	9	d	d	NOUN
ma-28	55	10	)	)	PUNCT
ma-28	55	11	∫	∫	PROPN
ma-28	55	12	ω	ω	NUM
ma-28	55	13	a(x)[νu0	a(x)[νu0	ADJ
ma-28	55	14	−	−	PROPN
ma-28	55	15	υu2	υu2	NOUN
ma-28	55	16	0	0	NUM
ma-28	55	17	]	]	X
ma-28	55	18	ψdx	ψdx	NOUN
ma-28	55	19	.	.	PUNCT
ma-28	56	1	(	(	PUNCT
ma-28	56	2	2.2	2.2	NUM
ma-28	56	3	)	)	PUNCT
ma-28	56	4	the	the	DET
ma-28	56	5	first	first	ADJ
ma-28	56	6	term	term	NOUN
ma-28	56	7	of	of	ADP
ma-28	56	8	the	the	DET
ma-28	56	9	l.h.s	l.h.s	NOUN
ma-28	56	10	.	.	PUNCT
ma-28	57	1	of	of	ADP
ma-28	57	2	(	(	PUNCT
ma-28	57	3	2.2	2.2	NUM
ma-28	57	4	)	)	PUNCT
ma-28	57	5	may	may	AUX
ma-28	57	6	be	be	AUX
ma-28	57	7	written	write	VERB
ma-28	57	8	in	in	ADP
ma-28	57	9	the	the	DET
ma-28	57	10	form∫	form∫	PROPN
ma-28	57	11	ω	ω	X
ma-28	57	12	ψdiv	ψdiv	X
ma-28	58	1	[	[	X
ma-28	58	2	p	p	X
ma-28	58	3	(	(	PUNCT
ma-28	58	4	x)|∇u0|p−2∇u0]dx	x)|∇u0|p−2∇u0]dx	PROPN
ma-28	58	5	=	=	SYM
ma-28	58	6	∫	∫	PROPN
ma-28	58	7	ω	ω	PROPN
ma-28	58	8	ψ∇u0∇[p	ψ∇u0∇[p	NUM
ma-28	58	9	(	(	PUNCT
ma-28	58	10	x)|∇u0|p−2]dx	x)|∇u0|p−2]dx	PROPN
ma-28	58	11	+	+	CCONJ
ma-28	58	12	∫	∫	PROPN
ma-28	58	13	ω	ω	NUM
ma-28	58	14	ψ[p	ψ[p	NOUN
ma-28	58	15	(	(	PUNCT
ma-28	58	16	x)|∇u0|p−2]div(∇u0)dx	x)|∇u0|p−2]div(∇u0)dx	PROPN
ma-28	58	17	.	.	PUNCT
ma-28	59	1	applying	apply	VERB
ma-28	59	2	green	green	PROPN
ma-28	59	3	’s	’s	PART
ma-28	59	4	first	first	ADJ
ma-28	59	5	identity	identity	NOUN
ma-28	59	6	,	,	PUNCT
ma-28	59	7	we	we	PRON
ma-28	59	8	have∫	have∫	VERB
ma-28	59	9	ω	ω	X
ma-28	59	10	ψdiv	ψdiv	PROPN
ma-28	60	1	[	[	X
ma-28	60	2	p	p	X
ma-28	60	3	(	(	PUNCT
ma-28	60	4	x)|∇u0|p−2∇u0]dx	x)|∇u0|p−2∇u0]dx	PROPN
ma-28	60	5	=	=	SYM
ma-28	60	6	∫	∫	PROPN
ma-28	60	7	ω	ω	PROPN
ma-28	60	8	ψ∇u0∇[p	ψ∇u0∇[p	NUM
ma-28	60	9	(	(	PUNCT
ma-28	60	10	x)|∇u0|p−2]dx	x)|∇u0|p−2]dx	PROPN
ma-28	60	11	−	−	PROPN
ma-28	60	12	∫	∫	PROPN
ma-28	60	13	ω	ω	NUM
ma-28	60	14	∇[ψ(p	∇[ψ(p	PROPN
ma-28	60	15	(	(	PUNCT
ma-28	60	16	x)|∇u0|p−2)∇u0dx	x)|∇u0|p−2)∇u0dx	X
ma-28	60	17	+	+	NUM
ma-28	60	18	∫	∫	PROPN
ma-28	60	19	∂ω	∂ω	ADJ
ma-28	60	20	ψ[p	ψ[p	NOUN
ma-28	60	21	(	(	PUNCT
ma-28	60	22	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	60	23	]	]	PUNCT
ma-28	60	24	∂u0	∂u0	NOUN
ma-28	60	25	∂n	∂n	PROPN
ma-28	60	26	ds	ds	NOUN
ma-28	60	27	,	,	PUNCT
ma-28	60	28	=	=	PUNCT
ma-28	60	29	−	−	PROPN
ma-28	60	30	∫	∫	PROPN
ma-28	60	31	ω	ω	NUM
ma-28	60	32	∇ψ[p	∇ψ[p	NOUN
ma-28	60	33	(	(	PUNCT
ma-28	60	34	x)|∇u0|p−2]∇u0dx	x)|∇u0|p−2]∇u0dx	PROPN
ma-28	60	35	+	+	CCONJ
ma-28	60	36	∫	∫	PROPN
ma-28	60	37	∂ω	∂ω	ADJ
ma-28	60	38	ψ[p	ψ[p	NOUN
ma-28	60	39	(	(	PUNCT
ma-28	60	40	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	60	41	]	]	PUNCT
ma-28	60	42	∂u0	∂u0	NOUN
ma-28	60	43	∂n	∂n	PROPN
ma-28	60	44	ds	ds	NOUN
ma-28	60	45	.	.	PUNCT
ma-28	61	1	(	(	PUNCT
ma-28	61	2	2.3	2.3	NUM
ma-28	61	3	)	)	PUNCT
ma-28	61	4	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	61	5	eur	eur	PROPN
ma-28	61	6	.	.	PUNCT
ma-28	62	1	j.	j.	PROPN
ma-28	62	2	math	math	PROPN
ma-28	62	3	.	.	PUNCT
ma-28	63	1	anal	anal	PROPN
ma-28	63	2	.	.	PUNCT
ma-28	64	1	10.28924	10.28924	NUM
ma-28	64	2	/	/	SYM
ma-28	64	3	ada	ada	PROPN
ma-28	64	4	/	/	SYM
ma-28	64	5	ma.2.8	ma.2.8	PROPN
ma-28	64	6	4from	4from	NUM
ma-28	64	7	(	(	PUNCT
ma-28	64	8	2.3	2.3	NUM
ma-28	64	9	)	)	PUNCT
ma-28	64	10	in	in	ADP
ma-28	64	11	(	(	PUNCT
ma-28	64	12	2.2	2.2	NUM
ma-28	64	13	)	)	PUNCT
ma-28	64	14	,	,	PUNCT
ma-28	64	15	we	we	PRON
ma-28	64	16	have	have	VERB
ma-28	64	17	(	(	PUNCT
ma-28	65	1	k	k	X
ma-28	65	2	/	/	SYM
ma-28	65	3	d	d	NOUN
ma-28	65	4	)	)	PUNCT
ma-28	65	5	∫	∫	PROPN
ma-28	65	6	ω	ω	NUM
ma-28	65	7	a(x)[νu0	a(x)[νu0	ADJ
ma-28	65	8	−	−	PROPN
ma-28	65	9	υu2	υu2	NOUN
ma-28	65	10	0	0	NUM
ma-28	65	11	]	]	PUNCT
ma-28	65	12	ψdx	ψdx	NOUN
ma-28	65	13	=	=	SYM
ma-28	65	14	∫	∫	PROPN
ma-28	65	15	ω	ω	NUM
ma-28	65	16	∇ψ[p	∇ψ[p	NOUN
ma-28	65	17	(	(	PUNCT
ma-28	65	18	x)|∇u0|p−2]∇u0dx	x)|∇u0|p−2]∇u0dx	PROPN
ma-28	65	19	−	−	PROPN
ma-28	65	20	∫	∫	PROPN
ma-28	66	1	∂ω	∂ω	ADJ
ma-28	66	2	ψ[p	ψ[p	NOUN
ma-28	66	3	(	(	PUNCT
ma-28	66	4	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	66	5	]	]	PUNCT
ma-28	66	6	∂u0	∂u0	NOUN
ma-28	66	7	∂n	∂n	PROPN
ma-28	66	8	ds	ds	PROPN
ma-28	67	1	+	+	CCONJ
ma-28	67	2	∫	∫	PROPN
ma-28	67	3	ω	ω	X
ma-28	67	4	a(x)ψ|u0|p−2u0]dx	a(x)ψ|u0|p−2u0]dx	PROPN
ma-28	67	5	.	.	PUNCT
ma-28	68	1	(	(	PUNCT
ma-28	68	2	2.4	2.4	NUM
ma-28	68	3	)	)	PUNCT
ma-28	68	4	also	also	ADV
ma-28	68	5	,	,	PUNCT
ma-28	68	6	multiplying	multiply	VERB
ma-28	68	7	(	(	PUNCT
ma-28	68	8	2.1	2.1	NUM
ma-28	68	9	)	)	PUNCT
ma-28	68	10	by	by	ADP
ma-28	68	11	(	(	PUNCT
ma-28	68	12	−u0	−u0	NOUN
ma-28	68	13	)	)	PUNCT
ma-28	68	14	and	and	CCONJ
ma-28	68	15	integrating	integrate	VERB
ma-28	68	16	over	over	ADP
ma-28	68	17	ω	ω	PROPN
ma-28	68	18	,	,	PUNCT
ma-28	68	19	we	we	PRON
ma-28	68	20	have	have	VERB
ma-28	68	21	−µ1	−µ1	PROPN
ma-28	68	22	∫	∫	PROPN
ma-28	68	23	ω	ω	PROPN
ma-28	68	24	u0ψdx	u0ψdx	PROPN
ma-28	69	1	=	=	PUNCT
ma-28	69	2	(	(	PUNCT
ma-28	69	3	p	p	NOUN
ma-28	69	4	−	−	PROPN
ma-28	69	5	1	1	NUM
ma-28	69	6	)	)	PUNCT
ma-28	69	7	∫	∫	PROPN
ma-28	70	1	ω	ω	NUM
ma-28	70	2	u0div	u0div	PROPN
ma-28	71	1	[	[	X
ma-28	71	2	p	p	X
ma-28	71	3	(	(	PUNCT
ma-28	71	4	x)|∇u0|p−2∇ψ]dx	x)|∇u0|p−2∇ψ]dx	INTJ
ma-28	71	5	−(p	−(p	NOUN
ma-28	71	6	−	−	NOUN
ma-28	71	7	1	1	NUM
ma-28	71	8	)	)	PUNCT
ma-28	71	9	∫	∫	PROPN
ma-28	72	1	ω	ω	X
ma-28	72	2	u0a(x)|u0|p−2ψ	u0a(x)|u0|p−2ψ	PROPN
ma-28	72	3	+	+	PROPN
ma-28	72	4	λ	λ	PROPN
ma-28	72	5	∫	∫	PROPN
ma-28	72	6	ω	ω	PROPN
ma-28	72	7	a(x)[ν	a(x)[ν	NOUN
ma-28	72	8	−	−	PROPN
ma-28	73	1	2υu0]ψdx	2υu0]ψdx	NUM
ma-28	73	2	.	.	PUNCT
ma-28	74	1	(	(	PUNCT
ma-28	74	2	2.5	2.5	NUM
ma-28	74	3	)	)	PUNCT
ma-28	74	4	the	the	DET
ma-28	74	5	first	first	ADJ
ma-28	74	6	term	term	NOUN
ma-28	74	7	of	of	ADP
ma-28	74	8	the	the	DET
ma-28	74	9	l.h.s	l.h.s	NOUN
ma-28	74	10	.	.	PUNCT
ma-28	75	1	of	of	ADP
ma-28	75	2	(	(	PUNCT
ma-28	75	3	2.5	2.5	NUM
ma-28	75	4	)	)	PUNCT
ma-28	75	5	may	may	AUX
ma-28	75	6	be	be	AUX
ma-28	75	7	written	write	VERB
ma-28	75	8	in	in	ADP
ma-28	75	9	the	the	DET
ma-28	75	10	form∫	form∫	PROPN
ma-28	75	11	ω	ω	NUM
ma-28	75	12	u0div	u0div	PROPN
ma-28	76	1	[	[	X
ma-28	76	2	p	p	X
ma-28	76	3	(	(	PUNCT
ma-28	76	4	x)|∇u0|p−2∇ψ]dx	x)|∇u0|p−2∇ψ]dx	NUM
ma-28	76	5	=	=	SYM
ma-28	76	6	∫	∫	PROPN
ma-28	76	7	ω	ω	NUM
ma-28	76	8	u0[p	u0[p	PROPN
ma-28	76	9	(	(	PUNCT
ma-28	76	10	x)|∇u0|p−2]∇	x)|∇u0|p−2]∇	PROPN
ma-28	76	11	·	·	PUNCT
ma-28	77	1	∇ψdx	∇ψdx	PROPN
ma-28	78	1	+	+	NUM
ma-28	78	2	∫	∫	PROPN
ma-28	78	3	ω	ω	NUM
ma-28	78	4	u0∇ψ∇[p	u0∇ψ∇[p	PROPN
ma-28	78	5	(	(	PUNCT
ma-28	78	6	x)|∇u0|p−2]dx	x)|∇u0|p−2]dx	AUX
ma-28	78	7	.	.	PUNCT
ma-28	79	1	using	use	VERB
ma-28	79	2	green	green	PROPN
ma-28	79	3	’s	’s	PART
ma-28	79	4	first	first	ADJ
ma-28	79	5	identity	identity	NOUN
ma-28	79	6	,	,	PUNCT
ma-28	79	7	one	one	NUM
ma-28	79	8	have∫	have∫	NOUN
ma-28	79	9	ω	ω	NOUN
ma-28	79	10	u0div	u0div	PROPN
ma-28	80	1	[	[	X
ma-28	80	2	p	p	X
ma-28	80	3	(	(	PUNCT
ma-28	80	4	x)|∇u0|p−2∇ψ]dx	x)|∇u0|p−2∇ψ]dx	NUM
ma-28	80	5	=	=	SYM
ma-28	80	6	−	−	PROPN
ma-28	80	7	∫	∫	PROPN
ma-28	80	8	ω	ω	PROPN
ma-28	80	9	∇[u0p	∇[u0p	PROPN
ma-28	80	10	(	(	PUNCT
ma-28	80	11	x)|∇u0|p−2]∇ψ	x)|∇u0|p−2]∇ψ	PROPN
ma-28	80	12	+	+	CCONJ
ma-28	80	13	∫	∫	PROPN
ma-28	80	14	ω	ω	NUM
ma-28	80	15	u0∇[p	u0∇[p	ADJ
ma-28	80	16	(	(	PUNCT
ma-28	80	17	x)|∇u0|p−2]∇ψdx	x)|∇u0|p−2]∇ψdx	PROPN
ma-28	80	18	+	+	NUM
ma-28	80	19	∫	∫	PROPN
ma-28	80	20	∂ω	∂ω	ADJ
ma-28	80	21	u0[p	u0[p	NOUN
ma-28	80	22	(	(	PUNCT
ma-28	80	23	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	80	24	]	]	PUNCT
ma-28	80	25	∂ψ	∂ψ	PROPN
ma-28	81	1	∂n	∂n	PROPN
ma-28	81	2	ds	ds	PROPN
ma-28	81	3	,	,	PUNCT
ma-28	81	4	=	=	PUNCT
ma-28	81	5	−	−	PROPN
ma-28	82	1	∫	∫	PROPN
ma-28	82	2	ω	ω	PROPN
ma-28	83	1	[	[	X
ma-28	83	2	p	p	X
ma-28	83	3	(	(	PUNCT
ma-28	83	4	x)|∇u0|p−2]∇u0∇ψ	x)|∇u0|p−2]∇u0∇ψ	PROPN
ma-28	83	5	+	+	CCONJ
ma-28	83	6	∫	∫	PROPN
ma-28	83	7	∂ω	∂ω	ADJ
ma-28	83	8	u0[p	u0[p	NOUN
ma-28	83	9	(	(	PUNCT
ma-28	83	10	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	83	11	]	]	PUNCT
ma-28	83	12	∂ψ	∂ψ	PROPN
ma-28	83	13	∂n	∂n	PROPN
ma-28	83	14	ds	ds	PROPN
ma-28	83	15	.	.	PUNCT
ma-28	84	1	(	(	PUNCT
ma-28	84	2	2.6	2.6	NUM
ma-28	84	3	)	)	PUNCT
ma-28	84	4	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	84	5	eur	eur	PROPN
ma-28	84	6	.	.	PUNCT
ma-28	85	1	j.	j.	PROPN
ma-28	85	2	math	math	PROPN
ma-28	85	3	.	.	PUNCT
ma-28	86	1	anal	anal	PROPN
ma-28	86	2	.	.	PUNCT
ma-28	87	1	10.28924	10.28924	NUM
ma-28	87	2	/	/	SYM
ma-28	87	3	ada	ada	PROPN
ma-28	87	4	/	/	SYM
ma-28	87	5	ma.2.8	ma.2.8	PROPN
ma-28	87	6	5from	5from	NUM
ma-28	87	7	(	(	PUNCT
ma-28	87	8	2.6	2.6	NUM
ma-28	87	9	)	)	PUNCT
ma-28	87	10	in	in	ADP
ma-28	87	11	(	(	PUNCT
ma-28	87	12	2.5	2.5	NUM
ma-28	87	13	)	)	PUNCT
ma-28	87	14	we	we	PRON
ma-28	87	15	have	have	VERB
ma-28	87	16	−µ1	−µ1	PROPN
ma-28	87	17	∫	∫	PROPN
ma-28	87	18	ω	ω	PROPN
ma-28	87	19	u0ψdx	u0ψdx	PROPN
ma-28	88	1	=	=	PUNCT
ma-28	88	2	(	(	PUNCT
ma-28	88	3	p	p	NOUN
ma-28	88	4	−	−	PROPN
ma-28	88	5	1	1	NUM
ma-28	88	6	)	)	PUNCT
ma-28	88	7	[	[	PUNCT
ma-28	88	8	∫	∫	PROPN
ma-28	88	9	∂ω	∂ω	ADJ
ma-28	88	10	u0[p	u0[p	NOUN
ma-28	88	11	(	(	PUNCT
ma-28	88	12	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	88	13	]	]	PUNCT
ma-28	88	14	∂ψ	∂ψ	PROPN
ma-28	89	1	∂n	∂n	PROPN
ma-28	89	2	ds	ds	PRON
ma-28	89	3	−	−	PROPN
ma-28	90	1	∫	∫	PROPN
ma-28	91	1	ω	ω	PROPN
ma-28	92	1	[	[	X
ma-28	92	2	p	p	X
ma-28	92	3	(	(	PUNCT
ma-28	92	4	x)|∇u0|p−2]∇u0∇ψ	x)|∇u0|p−2]∇u0∇ψ	PROPN
ma-28	92	5	]	]	X
ma-28	93	1	+	+	PROPN
ma-28	93	2	(	(	PUNCT
ma-28	93	3	k	k	ADJ
ma-28	93	4	/	/	SYM
ma-28	93	5	d	d	NOUN
ma-28	93	6	)	)	PUNCT
ma-28	93	7	∫	∫	PROPN
ma-28	93	8	ω	ω	NUM
ma-28	93	9	a(x)[νu0	a(x)[νu0	ADJ
ma-28	93	10	−	−	PROPN
ma-28	93	11	2υu2	2υu2	NUM
ma-28	93	12	0	0	NUM
ma-28	93	13	]	]	X
ma-28	93	14	ψdx	ψdx	NOUN
ma-28	93	15	.	.	PUNCT
ma-28	94	1	(	(	PUNCT
ma-28	94	2	2.7	2.7	NUM
ma-28	94	3	)	)	PUNCT
ma-28	94	4	multiplying	multiplying	NOUN
ma-28	94	5	(	(	PUNCT
ma-28	94	6	2.4	2.4	NUM
ma-28	94	7	)	)	PUNCT
ma-28	94	8	by	by	ADP
ma-28	94	9	(	(	PUNCT
ma-28	94	10	p	p	NOUN
ma-28	94	11	−	−	PROPN
ma-28	94	12	1	1	NUM
ma-28	94	13	)	)	PUNCT
ma-28	94	14	and	and	CCONJ
ma-28	94	15	adding	add	VERB
ma-28	94	16	with	with	ADP
ma-28	94	17	(	(	PUNCT
ma-28	94	18	2.7	2.7	NUM
ma-28	94	19	)	)	PUNCT
ma-28	94	20	,	,	PUNCT
ma-28	94	21	we	we	PRON
ma-28	94	22	have	have	VERB
ma-28	94	23	−µ1	−µ1	PROPN
ma-28	94	24	∫	∫	PROPN
ma-28	94	25	ω	ω	PROPN
ma-28	94	26	u0ψdx	u0ψdx	PROPN
ma-28	95	1	=	=	PUNCT
ma-28	95	2	(	(	PUNCT
ma-28	95	3	p	p	NOUN
ma-28	95	4	−	−	PROPN
ma-28	95	5	1	1	NUM
ma-28	95	6	)	)	PUNCT
ma-28	95	7	[	[	PUNCT
ma-28	95	8	∫	∫	PROPN
ma-28	95	9	∂ω	∂ω	ADJ
ma-28	95	10	u0[p	u0[p	NOUN
ma-28	95	11	(	(	PUNCT
ma-28	95	12	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	95	13	]	]	PUNCT
ma-28	95	14	∂ψ	∂ψ	PROPN
ma-28	96	1	∂n	∂n	PROPN
ma-28	96	2	ds	ds	PRON
ma-28	96	3	−	−	NUM
ma-28	96	4	∫	∫	PROPN
ma-28	97	1	∂ω	∂ω	ADJ
ma-28	97	2	ψ[p	ψ[p	NOUN
ma-28	97	3	(	(	PUNCT
ma-28	97	4	x)|∇u0|p−2	x)|∇u0|p−2	X
ma-28	97	5	]	]	PUNCT
ma-28	97	6	∂u0	∂u0	NOUN
ma-28	97	7	∂n	∂n	PROPN
ma-28	97	8	ds	ds	NOUN
ma-28	97	9	]	]	X
ma-28	97	10	+	+	PROPN
ma-28	97	11	(	(	PUNCT
ma-28	97	12	k	k	ADJ
ma-28	97	13	/	/	SYM
ma-28	97	14	d	d	NOUN
ma-28	97	15	)	)	PUNCT
ma-28	97	16	∫	∫	PROPN
ma-28	97	17	ω	ω	NUM
ma-28	97	18	a(x)[νu0	a(x)[νu0	ADJ
ma-28	97	19	−	−	PROPN
ma-28	97	20	2υu2	2υu2	NUM
ma-28	97	21	0	0	NUM
ma-28	97	22	]	]	PUNCT
ma-28	97	23	ψdx	ψdx	ADV
ma-28	97	24	−(p	−(p	ADJ
ma-28	97	25	−	−	PROPN
ma-28	97	26	1)(k	1)(k	NUM
ma-28	97	27	/	/	SYM
ma-28	97	28	d	d	NOUN
ma-28	97	29	)	)	PUNCT
ma-28	97	30	∫	∫	PROPN
ma-28	98	1	ω	ω	NUM
ma-28	98	2	a(x)[νu0	a(x)[νu0	ADJ
ma-28	98	3	−	−	PROPN
ma-28	98	4	υu2	υu2	NOUN
ma-28	98	5	0	0	NUM
ma-28	98	6	]	]	X
ma-28	98	7	ψdx	ψdx	NOUN
ma-28	98	8	.	.	PUNCT
ma-28	99	1	hence	hence	ADV
ma-28	99	2	−µ1	−µ1	ADJ
ma-28	99	3	∫	∫	PROPN
ma-28	99	4	ω	ω	NUM
ma-28	99	5	u0ψdx	u0ψdx	PROPN
ma-28	100	1	=	=	PUNCT
ma-28	100	2	(	(	PUNCT
ma-28	100	3	p	p	NOUN
ma-28	100	4	−	−	PROPN
ma-28	100	5	1	1	NUM
ma-28	100	6	)	)	PUNCT
ma-28	100	7	∫	∫	NOUN
ma-28	101	1	∂ω	∂ω	PROPN
ma-28	102	1	[	[	X
ma-28	102	2	p	p	X
ma-28	102	3	(	(	PUNCT
ma-28	102	4	x)|∇u0|p−2][u0	x)|∇u0|p−2][u0	AUX
ma-28	102	5	∂ψ	∂ψ	VERB
ma-28	102	6	∂n	∂n	PROPN
ma-28	102	7	−	−	PROPN
ma-28	103	1	ψ	ψ	ADP
ma-28	103	2	∂u0	∂u0	NOUN
ma-28	103	3	∂n	∂n	PROPN
ma-28	103	4	]	]	PUNCT
ma-28	103	5	ds	ds	PRON
ma-28	103	6	+	+	ADJ
ma-28	103	7	(	(	PUNCT
ma-28	103	8	k	k	ADJ
ma-28	103	9	/	/	SYM
ma-28	103	10	d	d	NOUN
ma-28	103	11	)	)	PUNCT
ma-28	103	12	∫	∫	PROPN
ma-28	103	13	ω	ω	NUM
ma-28	103	14	a(x)νu0[1−	a(x)νu0[1−	PROPN
ma-28	103	15	(	(	PUNCT
ma-28	103	16	p	p	NOUN
ma-28	103	17	−	−	PROPN
ma-28	103	18	1)]ψdx	1)]ψdx	PROPN
ma-28	104	1	+	+	ADJ
ma-28	104	2	(	(	PUNCT
ma-28	104	3	k	k	ADJ
ma-28	104	4	/	/	SYM
ma-28	104	5	d	d	NOUN
ma-28	104	6	)	)	PUNCT
ma-28	104	7	∫	∫	PROPN
ma-28	105	1	ω	ω	PROPN
ma-28	105	2	a(x)υu2	a(x)υu2	NOUN
ma-28	105	3	0	0	PUNCT
ma-28	106	1	[	[	X
ma-28	106	2	(	(	PUNCT
ma-28	106	3	p	p	X
ma-28	106	4	−	−	PROPN
ma-28	106	5	1)−	1)−	PROPN
ma-28	106	6	2]ψdx	2]ψdx	PROPN
ma-28	106	7	.	.	PUNCT
ma-28	107	1	(	(	PUNCT
ma-28	107	2	2.8	2.8	NUM
ma-28	107	3	)	)	PUNCT
ma-28	107	4	now	now	ADV
ma-28	107	5	,	,	PUNCT
ma-28	107	6	when	when	SCONJ
ma-28	107	7	δ	δ	PROPN
ma-28	107	8	=	=	SYM
ma-28	107	9	1	1	NUM
ma-28	107	10	,	,	PUNCT
ma-28	107	11	we	we	PRON
ma-28	107	12	have	have	VERB
ma-28	107	13	bu0	bu0	NOUN
ma-28	107	14	=	=	SYM
ma-28	107	15	u0	u0	ADJ
ma-28	107	16	=	=	NOUN
ma-28	107	17	0	0	NUM
ma-28	107	18	for	for	ADP
ma-28	107	19	s	s	NOUN
ma-28	107	20	∈	∈	PROPN
ma-28	107	21	∂ω	∂ω	PROPN
ma-28	107	22	and	and	CCONJ
ma-28	107	23	also	also	ADV
ma-28	107	24	we	we	PRON
ma-28	107	25	have	have	VERB
ma-28	107	26	ψ	ψ	X
ma-28	107	27	=	=	SYM
ma-28	107	28	0	0	NUM
ma-28	107	29	for	for	ADP
ma-28	107	30	s	s	PROPN
ma-28	107	31	∈	∈	PROPN
ma-28	107	32	∂ω	∂ω	PROPN
ma-28	107	33	.	.	PUNCT
ma-28	108	1	then∫	then∫	NOUN
ma-28	108	2	∂ω	∂ω	PROPN
ma-28	109	1	[	[	X
ma-28	109	2	p	p	X
ma-28	109	3	(	(	PUNCT
ma-28	109	4	x)|∇u0|p−2][u0	x)|∇u0|p−2][u0	AUX
ma-28	109	5	∂ψ	∂ψ	VERB
ma-28	109	6	∂n	∂n	PROPN
ma-28	109	7	−	−	PROPN
ma-28	110	1	ψ	ψ	ADP
ma-28	110	2	∂u0	∂u0	NOUN
ma-28	110	3	∂n	∂n	PROPN
ma-28	110	4	]	]	PUNCT
ma-28	110	5	ds	ds	X
ma-28	110	6	=	=	ADJ
ma-28	110	7	0	0	PROPN
ma-28	110	8	.	.	PUNCT
ma-28	110	9	(	(	PUNCT
ma-28	110	10	2.9	2.9	NUM
ma-28	110	11	)	)	PUNCT
ma-28	110	12	also	also	ADV
ma-28	110	13	,	,	PUNCT
ma-28	110	14	when	when	SCONJ
ma-28	110	15	δ	δ	PROPN
ma-28	110	16	6=	6=	PROPN
ma-28	110	17	1	1	NUM
ma-28	110	18	,	,	PUNCT
ma-28	110	19	we	we	PRON
ma-28	110	20	have	have	VERB
ma-28	110	21	∂u0	∂u0	NOUN
ma-28	110	22	∂n	∂n	PROPN
ma-28	110	23	=	=	PUNCT
ma-28	110	24	−	−	PROPN
ma-28	111	1	δhu0	δhu0	NOUN
ma-28	111	2	1−	1−	NUM
ma-28	111	3	δ	δ	PROPN
ma-28	111	4	and	and	CCONJ
ma-28	111	5	∂ψ	∂ψ	VERB
ma-28	111	6	∂n	∂n	PROPN
ma-28	111	7	=	=	PUNCT
ma-28	111	8	−	−	PROPN
ma-28	111	9	δhψ	δhψ	PROPN
ma-28	111	10	1−	1−	NUM
ma-28	111	11	δ	δ	PROPN
ma-28	111	12	,	,	PUNCT
ma-28	111	13	which	which	PRON
ma-28	111	14	implies	imply	VERB
ma-28	111	15	again	again	ADV
ma-28	111	16	the	the	DET
ma-28	111	17	result	result	NOUN
ma-28	111	18	given	give	VERB
ma-28	111	19	by	by	ADP
ma-28	111	20	(	(	PUNCT
ma-28	111	21	2.9).hence	2.9).hence	NUM
ma-28	111	22	−	−	NOUN
ma-28	111	23	µ1	µ1	PROPN
ma-28	111	24	∫	∫	PROPN
ma-28	111	25	ω	ω	NUM
ma-28	111	26	u0ψdx	u0ψdx	PROPN
ma-28	112	1	=	=	PRON
ma-28	112	2	(	(	PUNCT
ma-28	112	3	k	k	NOUN
ma-28	112	4	/	/	SYM
ma-28	112	5	d	d	NOUN
ma-28	112	6	)	)	PUNCT
ma-28	112	7	∫	∫	PROPN
ma-28	113	1	ω	ω	NUM
ma-28	113	2	a(x)[νu0[2−	a(x)[νu0[2−	PROPN
ma-28	113	3	p	p	X
ma-28	113	4	]	]	X
ma-28	113	5	+	+	NUM
ma-28	113	6	υu2	υu2	NOUN
ma-28	113	7	0	0	PUNCT
ma-28	114	1	[	[	X
ma-28	114	2	p	p	X
ma-28	114	3	−	−	PROPN
ma-28	114	4	3]]ψdx	3]]ψdx	NUM
ma-28	114	5	.	.	PUNCT
ma-28	115	1	(	(	PUNCT
ma-28	115	2	2.10	2.10	NUM
ma-28	115	3	)	)	PUNCT
ma-28	115	4	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	115	5	eur	eur	PROPN
ma-28	115	6	.	.	PUNCT
ma-28	116	1	j.	j.	PROPN
ma-28	116	2	math	math	PROPN
ma-28	116	3	.	.	PUNCT
ma-28	117	1	anal	anal	PROPN
ma-28	117	2	.	.	PUNCT
ma-28	118	1	10.28924	10.28924	NUM
ma-28	118	2	/	/	SYM
ma-28	118	3	ada	ada	PROPN
ma-28	118	4	/	/	SYM
ma-28	118	5	ma.2.8	ma.2.8	PROPN
ma-28	118	6	6since	6since	NUM
ma-28	118	7	2	2	NUM
ma-28	118	8	<	<	X
ma-28	118	9	p	p	X
ma-28	118	10	<	<	X
ma-28	118	11	3	3	NUM
ma-28	118	12	and	and	CCONJ
ma-28	118	13	a(x	a(x	PROPN
ma-28	118	14	)	)	PUNCT
ma-28	118	15	>	>	X
ma-28	118	16	0	0	PUNCT
ma-28	118	17	for	for	ADP
ma-28	118	18	all	all	PRON
ma-28	118	19	x	x	NOUN
ma-28	118	20	,	,	PUNCT
ma-28	118	21	then	then	ADV
ma-28	118	22	(	(	PUNCT
ma-28	118	23	2.10	2.10	NUM
ma-28	118	24	)	)	PUNCT
ma-28	118	25	becomes	become	VERB
ma-28	118	26	−	−	PROPN
ma-28	118	27	µ1	µ1	PROPN
ma-28	118	28	∫	∫	PROPN
ma-28	118	29	ω	ω	PROPN
ma-28	118	30	u0ψdx	u0ψdx	PROPN
ma-28	118	31	<	<	X
ma-28	118	32	0	0	NUM
ma-28	118	33	,	,	PUNCT
ma-28	118	34	(	(	PUNCT
ma-28	118	35	2.11	2.11	NUM
ma-28	118	36	)	)	PUNCT
ma-28	118	37	so	so	ADV
ma-28	118	38	µ1	µ1	PROPN
ma-28	118	39	>	>	X
ma-28	118	40	0	0	PUNCT
ma-28	119	1	and	and	CCONJ
ma-28	119	2	the	the	DET
ma-28	119	3	result	result	NOUN
ma-28	119	4	follows	follow	VERB
ma-28	119	5	.	.	PUNCT
ma-28	120	1	�	�	PROPN
ma-28	120	2	theorem	theorem	VERB
ma-28	120	3	2.2	2.2	NUM
ma-28	120	4	.	.	PUNCT
ma-28	121	1	if	if	SCONJ
ma-28	121	2	2	2	NUM
ma-28	121	3	<	<	X
ma-28	121	4	p	p	X
ma-28	121	5	<	<	X
ma-28	121	6	3	3	NUM
ma-28	121	7	and	and	CCONJ
ma-28	121	8	a(x	a(x	NOUN
ma-28	121	9	)	)	PUNCT
ma-28	121	10	<	<	X
ma-28	121	11	0	0	NUM
ma-28	121	12	for	for	ADP
ma-28	121	13	all	all	DET
ma-28	121	14	x	x	SYM
ma-28	121	15	∈	∈	PROPN
ma-28	121	16	ω	ω	NOUN
ma-28	121	17	,	,	PUNCT
ma-28	121	18	then	then	ADV
ma-28	121	19	every	every	DET
ma-28	121	20	positive	positive	ADJ
ma-28	121	21	weak	weak	ADJ
ma-28	121	22	solution	solution	NOUN
ma-28	121	23	of	of	ADP
ma-28	121	24	(	(	PUNCT
ma-28	121	25	1.1	1.1	NUM
ma-28	121	26	)	)	PUNCT
ma-28	121	27	is	be	AUX
ma-28	121	28	unstable	unstable	ADJ
ma-28	121	29	.	.	PUNCT
ma-28	122	1	proof	proof	NOUN
ma-28	122	2	.	.	PUNCT
ma-28	123	1	as	as	ADP
ma-28	123	2	in	in	ADP
ma-28	123	3	the	the	DET
ma-28	123	4	proof	proof	NOUN
ma-28	123	5	of	of	ADP
ma-28	123	6	theorem	theorem	NOUN
ma-28	123	7	1	1	NUM
ma-28	123	8	.	.	NUM
ma-28	123	9	,	,	PUNCT
ma-28	123	10	we	we	PRON
ma-28	123	11	have	have	VERB
ma-28	123	12	−	−	PROPN
ma-28	124	1	µ1	µ1	PROPN
ma-28	124	2	∫	∫	PROPN
ma-28	124	3	ω	ω	PROPN
ma-28	124	4	u0ψdx	u0ψdx	PROPN
ma-28	124	5	>	>	X
ma-28	124	6	0	0	NUM
ma-28	124	7	,	,	PUNCT
ma-28	124	8	(	(	PUNCT
ma-28	124	9	2.12	2.12	NUM
ma-28	124	10	)	)	PUNCT
ma-28	124	11	so	so	ADV
ma-28	124	12	µ1	µ1	PROPN
ma-28	124	13	<	<	X
ma-28	124	14	0	0	NUM
ma-28	125	1	and	and	CCONJ
ma-28	125	2	the	the	DET
ma-28	125	3	result	result	NOUN
ma-28	125	4	follows	follow	VERB
ma-28	125	5	.	.	PUNCT
ma-28	126	1	�	�	PROPN
ma-28	126	2	3	3	NUM
ma-28	126	3	.	.	PUNCT
ma-28	126	4	applications	application	NOUN
ma-28	126	5	and	and	CCONJ
ma-28	126	6	related	relate	VERB
ma-28	126	7	results	result	NOUN
ma-28	126	8	here	here	ADV
ma-28	126	9	we	we	PRON
ma-28	126	10	introduce	introduce	VERB
ma-28	126	11	some	some	DET
ma-28	126	12	examples	example	NOUN
ma-28	126	13	to	to	PART
ma-28	126	14	demonstrate	demonstrate	VERB
ma-28	126	15	the	the	DET
ma-28	126	16	effectiveness	effectiveness	NOUN
ma-28	126	17	of	of	ADP
ma-28	126	18	our	our	PRON
ma-28	126	19	results	result	NOUN
ma-28	126	20	.	.	PUNCT
ma-28	127	1	example	example	NOUN
ma-28	127	2	3.1	3.1	NUM
ma-28	127	3	.	.	PUNCT
ma-28	128	1	consider	consider	VERB
ma-28	128	2	the	the	DET
ma-28	128	3	emden	emden	ADJ
ma-28	128	4	-	-	PUNCT
ma-28	128	5	fowler	fowler	ADJ
ma-28	128	6	steady	steady	ADJ
ma-28	128	7	-	-	PUNCT
ma-28	128	8	state	state	NOUN
ma-28	128	9	problem	problem	NOUN
ma-28	128	10	of	of	ADP
ma-28	128	11	polytropic	polytropic	ADJ
ma-28	128	12	index	index	NOUN
ma-28	128	13	of	of	ADP
ma-28	128	14	order	order	NOUN
ma-28	128	15	one	one	NUM
ma-28	129	1	[	[	X
ma-28	129	2	8	8	NUM
ma-28	129	3	]	]	PUNCT
ma-28	129	4	,	,	PUNCT
ma-28	129	5	−∆u	−∆u	NOUN
ma-28	129	6	=	=	SYM
ma-28	130	1	λa(x)u	λa(x)u	PROPN
ma-28	130	2	in	in	ADP
ma-28	130	3	ω	ω	PROPN
ma-28	130	4	,	,	PUNCT
ma-28	130	5	bu	bu	ADP
ma-28	130	6	=	=	NOUN
ma-28	130	7	0	0	NUM
ma-28	130	8	on	on	ADP
ma-28	130	9	∂ω	∂ω	PROPN
ma-28	130	10	,	,	PUNCT
ma-28	130	11	}	}	PUNCT
ma-28	130	12	(	(	PUNCT
ma-28	130	13	3.1	3.1	NUM
ma-28	130	14	)	)	PUNCT
ma-28	130	15	with	with	ADP
ma-28	130	16	a(x	a(x	NOUN
ma-28	130	17	)	)	PUNCT
ma-28	130	18	>	>	X
ma-28	130	19	0	0	PUNCT
ma-28	131	1	for	for	ADP
ma-28	131	2	all	all	DET
ma-28	131	3	x	x	SYM
ma-28	131	4	∈	∈	NOUN
ma-28	131	5	ω.here	ω.here	X
ma-28	131	6	p	p	X
ma-28	131	7	(	(	PUNCT
ma-28	131	8	x	x	NOUN
ma-28	131	9	)	)	PUNCT
ma-28	131	10	=	=	SYM
ma-28	131	11	1	1	NUM
ma-28	131	12	,	,	PUNCT
ma-28	131	13	(	(	PUNCT
ma-28	131	14	k	k	X
ma-28	131	15	/	/	SYM
ma-28	131	16	d	d	NOUN
ma-28	131	17	)	)	PUNCT
ma-28	131	18	=	=	SYM
ma-28	131	19	λ	λ	PROPN
ma-28	131	20	,	,	PUNCT
ma-28	131	21	p	p	NOUN
ma-28	131	22	=	=	NOUN
ma-28	131	23	2	2	X
ma-28	131	24	.	.	PUNCT
ma-28	131	25	then	then	ADV
ma-28	131	26	according	accord	VERB
ma-28	131	27	to	to	ADP
ma-28	131	28	theorem	theorem	NOUN
ma-28	131	29	1	1	NUM
ma-28	131	30	.	.	NUM
ma-28	131	31	,	,	PUNCT
ma-28	131	32	every	every	DET
ma-28	131	33	positive	positive	ADJ
ma-28	131	34	weak	weak	ADJ
ma-28	131	35	solution	solution	NOUN
ma-28	131	36	of	of	ADP
ma-28	131	37	(	(	PUNCT
ma-28	131	38	3.1	3.1	NUM
ma-28	131	39	)	)	PUNCT
ma-28	131	40	is	be	AUX
ma-28	131	41	unstable	unstable	ADJ
ma-28	131	42	.	.	PUNCT
ma-28	131	43	example	example	NOUN
ma-28	131	44	3.2	3.2	NUM
ma-28	131	45	.	.	PUNCT
ma-28	132	1	consider	consider	VERB
ma-28	132	2	the	the	DET
ma-28	132	3	population	population	NOUN
ma-28	132	4	denisty	denisty	VERB
ma-28	132	5	steady	steady	ADJ
ma-28	132	6	-	-	PUNCT
ma-28	132	7	state	state	NOUN
ma-28	132	8	problem	problem	NOUN
ma-28	132	9	of	of	ADP
ma-28	132	10	degree	degree	NOUN
ma-28	132	11	two	two	NUM
ma-28	133	1	[	[	X
ma-28	133	2	19	19	NUM
ma-28	133	3	]	]	PUNCT
ma-28	133	4	,	,	PUNCT
ma-28	133	5	−∆pu	−∆pu	NOUN
ma-28	133	6	=	=	SYM
ma-28	134	1	λa(x)[u	λa(x)[u	NOUN
ma-28	134	2	−	−	PROPN
ma-28	134	3	u2	u2	PROPN
ma-28	134	4	]	]	PUNCT
ma-28	134	5	in	in	ADP
ma-28	134	6	ω	ω	PROPN
ma-28	134	7	,	,	PUNCT
ma-28	134	8	bu	bu	ADP
ma-28	134	9	=	=	NOUN
ma-28	134	10	0	0	NUM
ma-28	134	11	on	on	ADP
ma-28	134	12	∂ω	∂ω	PROPN
ma-28	134	13	,	,	PUNCT
ma-28	134	14	}	}	PUNCT
ma-28	134	15	(	(	PUNCT
ma-28	134	16	3.2	3.2	NUM
ma-28	134	17	)	)	PUNCT
ma-28	134	18	with	with	ADP
ma-28	134	19	a(x	a(x	NOUN
ma-28	134	20	)	)	PUNCT
ma-28	134	21	>	>	X
ma-28	134	22	0	0	PUNCT
ma-28	134	23	for	for	ADP
ma-28	134	24	all	all	DET
ma-28	134	25	x	x	SYM
ma-28	134	26	∈	∈	PROPN
ma-28	134	27	ω.hence	ω.hence	NOUN
ma-28	134	28	,	,	PUNCT
ma-28	134	29	according	accord	VERB
ma-28	134	30	to	to	ADP
ma-28	134	31	theorem	theorem	NOUN
ma-28	134	32	1	1	NUM
ma-28	134	33	.	.	NUM
ma-28	134	34	,	,	PUNCT
ma-28	134	35	every	every	DET
ma-28	134	36	positive	positive	ADJ
ma-28	134	37	weak	weak	ADJ
ma-28	134	38	solution	solution	NOUN
ma-28	134	39	of	of	ADP
ma-28	134	40	(	(	PUNCT
ma-28	134	41	3.1	3.1	NUM
ma-28	134	42	)	)	PUNCT
ma-28	134	43	is	be	AUX
ma-28	134	44	stable	stable	ADJ
ma-28	134	45	.	.	PUNCT
ma-28	134	46	example	example	NOUN
ma-28	134	47	3.3	3.3	NUM
ma-28	134	48	.	.	PUNCT
ma-28	135	1	consider	consider	VERB
ma-28	135	2	the	the	DET
ma-28	135	3	chemotaxis	chemotaxis	ADJ
ma-28	135	4	steady	steady	ADJ
ma-28	135	5	-	-	PUNCT
ma-28	135	6	state	state	NOUN
ma-28	135	7	problem	problem	NOUN
ma-28	135	8	of	of	ADP
ma-28	135	9	degree	degree	NOUN
ma-28	135	10	two	two	NUM
ma-28	136	1	[	[	X
ma-28	136	2	9	9	NUM
ma-28	136	3	,	,	PUNCT
ma-28	136	4	19	19	NUM
ma-28	136	5	]	]	PUNCT
ma-28	136	6	,	,	PUNCT
ma-28	136	7	−∆pu	−∆pu	X
ma-28	137	1	=	=	PUNCT
ma-28	137	2	λa(x)[−u	λa(x)[−u	PROPN
ma-28	138	1	+	+	NUM
ma-28	138	2	u2	u2	PROPN
ma-28	138	3	]	]	PUNCT
ma-28	138	4	in	in	ADP
ma-28	138	5	ω	ω	PROPN
ma-28	138	6	,	,	PUNCT
ma-28	138	7	bu	bu	ADP
ma-28	138	8	=	=	NOUN
ma-28	138	9	0	0	NUM
ma-28	138	10	on	on	ADP
ma-28	138	11	∂ω	∂ω	PROPN
ma-28	138	12	,	,	PUNCT
ma-28	138	13	}	}	PUNCT
ma-28	138	14	(	(	PUNCT
ma-28	138	15	3.3	3.3	NUM
ma-28	138	16	)	)	PUNCT
ma-28	138	17	with	with	ADP
ma-28	138	18	a(x	a(x	NOUN
ma-28	138	19	)	)	PUNCT
ma-28	138	20	>	>	X
ma-28	138	21	0	0	PUNCT
ma-28	139	1	for	for	ADP
ma-28	139	2	all	all	DET
ma-28	139	3	x	x	SYM
ma-28	139	4	∈	∈	PROPN
ma-28	139	5	ω.hence	ω.hence	NOUN
ma-28	139	6	,	,	PUNCT
ma-28	139	7	according	accord	VERB
ma-28	139	8	to	to	ADP
ma-28	139	9	theorem	theorem	NOUN
ma-28	139	10	1	1	NUM
ma-28	139	11	.	.	NUM
ma-28	139	12	,	,	PUNCT
ma-28	139	13	every	every	DET
ma-28	139	14	positive	positive	ADJ
ma-28	139	15	weak	weak	ADJ
ma-28	139	16	solution	solution	NOUN
ma-28	139	17	of	of	ADP
ma-28	139	18	(	(	PUNCT
ma-28	139	19	3.1	3.1	NUM
ma-28	139	20	)	)	PUNCT
ma-28	139	21	is	be	AUX
ma-28	139	22	unstable	unstable	ADJ
ma-28	139	23	.	.	PUNCT
ma-28	140	1	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	140	2	eur	eur	PROPN
ma-28	140	3	.	.	PUNCT
ma-28	141	1	j.	j.	PROPN
ma-28	141	2	math	math	PROPN
ma-28	141	3	.	.	PUNCT
ma-28	142	1	anal	anal	PROPN
ma-28	142	2	.	.	PUNCT
ma-28	143	1	10.28924	10.28924	NUM
ma-28	143	2	/	/	SYM
ma-28	143	3	ada	ada	PROPN
ma-28	143	4	/	/	SYM
ma-28	143	5	ma.2.8	ma.2.8	PROPN
ma-28	143	6	7references	7reference	NOUN
ma-28	143	7	[	[	X
ma-28	143	8	1	1	NUM
ma-28	143	9	]	]	X
ma-28	143	10	g.	g.	PROPN
ma-28	143	11	afrouzi	afrouzi	PROPN
ma-28	143	12	,	,	PUNCT
ma-28	143	13	s.	s.	PROPN
ma-28	143	14	rasouli	rasouli	PROPN
ma-28	143	15	,	,	PUNCT
ma-28	143	16	stability	stability	NOUN
ma-28	143	17	properties	property	NOUN
ma-28	143	18	of	of	ADP
ma-28	143	19	non	non	ADJ
ma-28	143	20	-	-	ADJ
ma-28	143	21	negative	negative	ADJ
ma-28	143	22	solutions	solution	NOUN
ma-28	143	23	to	to	ADP
ma-28	143	24	a	a	DET
ma-28	143	25	non	non	ADJ
ma-28	143	26	-	-	ADJ
ma-28	143	27	autonomous	autonomous	ADJ
ma-28	143	28	p	p	ADJ
ma-28	143	29	-	-	PUNCT
ma-28	143	30	laplacian	laplacian	ADJ
ma-28	143	31	equation	equation	NOUN
ma-28	143	32	,	,	PUNCT
ma-28	143	33	chaos	chaos	NOUN
ma-28	143	34	solitons	soliton	NOUN
ma-28	143	35	fractals	fractal	NOUN
ma-28	143	36	.	.	PUNCT
ma-28	144	1	29	29	NUM
ma-28	144	2	(	(	PUNCT
ma-28	144	3	2006	2006	NUM
ma-28	144	4	)	)	PUNCT
ma-28	144	5	1095	1095	NUM
ma-28	144	6	-	-	SYM
ma-28	144	7	1099	1099	NUM
ma-28	144	8	.	.	PUNCT
ma-28	145	1	https://doi.org/10.1016/j.chaos.2005.08.165.[2	https://doi.org/10.1016/j.chaos.2005.08.165.[2	NOUN
ma-28	145	2	]	]	X
ma-28	145	3	g.	g.	PROPN
ma-28	145	4	afrouzi	afrouzi	PROPN
ma-28	145	5	,	,	PUNCT
ma-28	145	6	z.	z.	PROPN
ma-28	145	7	sadeeghi	sadeeghi	PROPN
ma-28	145	8	,	,	PUNCT
ma-28	145	9	stability	stability	NOUN
ma-28	145	10	results	result	VERB
ma-28	145	11	for	for	ADP
ma-28	145	12	a	a	DET
ma-28	145	13	class	class	NOUN
ma-28	145	14	of	of	ADP
ma-28	145	15	elliptic	elliptic	ADJ
ma-28	145	16	problems	problem	NOUN
ma-28	145	17	,	,	PUNCT
ma-28	145	18	int	int	NOUN
ma-28	145	19	.	.	PUNCT
ma-28	146	1	j.	j.	PROPN
ma-28	146	2	nonlinear	nonlinear	PROPN
ma-28	146	3	sci	sci	PROPN
ma-28	146	4	.	.	PROPN
ma-28	146	5	6	6	NUM
ma-28	146	6	(	(	PUNCT
ma-28	146	7	2008	2008	NUM
ma-28	146	8	)	)	PUNCT
ma-28	146	9	114	114	NUM
ma-28	146	10	-	-	SYM
ma-28	146	11	117	117	NUM
ma-28	146	12	.	.	PUNCT
ma-28	147	1	http://www.internonlinearscience.org/upload/papers/20110307063941556.pdf.[3	http://www.internonlinearscience.org/upload/papers/20110307063941556.pdf.[3	PROPN
ma-28	147	2	]	]	X
ma-28	147	3	i.	i.	PROPN
ma-28	147	4	ali	ali	PROPN
ma-28	147	5	,	,	PUNCT
ma-28	147	6	a.	a.	NOUN
ma-28	147	7	castro	castro	PROPN
ma-28	147	8	,	,	PUNCT
ma-28	147	9	r	r	NOUN
ma-28	147	10	,	,	PUNCT
ma-28	147	11	shivaji	shivaji	NOUN
ma-28	147	12	,	,	PUNCT
ma-28	147	13	uniqueness	uniqueness	NOUN
ma-28	147	14	and	and	CCONJ
ma-28	147	15	stability	stability	NOUN
ma-28	147	16	of	of	ADP
ma-28	147	17	nonnegative	nonnegative	ADJ
ma-28	147	18	solutions	solution	NOUN
ma-28	147	19	for	for	ADP
ma-28	147	20	semipositone	semipositone	NOUN
ma-28	147	21	problems	problem	NOUN
ma-28	147	22	in	in	ADP
ma-28	147	23	a	a	DET
ma-28	147	24	ball	ball	NOUN
ma-28	147	25	,	,	PUNCT
ma-28	147	26	proc	proc	NOUN
ma-28	147	27	.	.	PUNCT
ma-28	148	1	amer	amer	PROPN
ma-28	148	2	.	.	PUNCT
ma-28	148	3	math	math	PROPN
ma-28	148	4	.	.	PUNCT
ma-28	149	1	soc	soc	PROPN
ma-28	149	2	.	.	PUNCT
ma-28	150	1	117	117	NUM
ma-28	150	2	(	(	PUNCT
ma-28	150	3	1993	1993	NUM
ma-28	150	4	)	)	PUNCT
ma-28	150	5	775	775	NUM
ma-28	150	6	-	-	SYM
ma-28	150	7	782	782	NUM
ma-28	150	8	.	.	PUNCT
ma-28	151	1	https://doi.org/10.1090/s0002-9939-1993-1116249-5.[4	https://doi.org/10.1090/s0002-9939-1993-1116249-5.[4	PROPN
ma-28	151	2	]	]	PUNCT
ma-28	151	3	c.	c.	PROPN
ma-28	151	4	atkinson	atkinson	PROPN
ma-28	151	5	,	,	PUNCT
ma-28	151	6	k.	k.	PROPN
ma-28	151	7	ali	ali	PROPN
ma-28	151	8	.	.	PROPN
ma-28	151	9	,	,	PUNCT
ma-28	151	10	some	some	DET
ma-28	151	11	boundary	boundary	ADJ
ma-28	151	12	value	value	NOUN
ma-28	151	13	problems	problem	NOUN
ma-28	151	14	for	for	ADP
ma-28	151	15	the	the	DET
ma-28	151	16	bingham	bingham	PROPN
ma-28	151	17	model	model	NOUN
ma-28	151	18	,	,	PUNCT
ma-28	151	19	j.	j.	PROPN
ma-28	151	20	non	non	PROPN
ma-28	151	21	-	-	PROPN
ma-28	151	22	newton	newton	PROPN
ma-28	151	23	.	.	PUNCT
ma-28	152	1	fluid	fluid	ADJ
ma-28	152	2	mech	mech	NOUN
ma-28	152	3	.	.	PUNCT
ma-28	153	1	41	41	NUM
ma-28	153	2	(	(	PUNCT
ma-28	153	3	1992)339	1992)339	PROPN
ma-28	153	4	-	-	PUNCT
ma-28	153	5	363	363	NUM
ma-28	153	6	.	.	PUNCT
ma-28	154	1	https://doi.org/10.1016/0377-0257(92)87006-w.[5	https://doi.org/10.1016/0377-0257(92)87006-w.[5	PROPN
ma-28	154	2	]	]	PUNCT
ma-28	154	3	k.	k.	PROPN
ma-28	154	4	brown	brown	PROPN
ma-28	154	5	,	,	PUNCT
ma-28	154	6	r.	r.	PROPN
ma-28	154	7	shivaji	shivaji	PROPN
ma-28	154	8	,	,	PUNCT
ma-28	154	9	instability	instability	NOUN
ma-28	154	10	of	of	ADP
ma-28	154	11	nonnegative	nonnegative	ADJ
ma-28	154	12	solutions	solution	NOUN
ma-28	154	13	for	for	ADP
ma-28	154	14	a	a	DET
ma-28	154	15	class	class	NOUN
ma-28	154	16	of	of	ADP
ma-28	154	17	semipositone	semipositone	NOUN
ma-28	154	18	problems	problem	NOUN
ma-28	154	19	,	,	PUNCT
ma-28	154	20	proc	proc	PROPN
ma-28	154	21	.	.	PUNCT
ma-28	155	1	amer	amer	PROPN
ma-28	155	2	.	.	PUNCT
ma-28	156	1	math.soc	math.soc	X
ma-28	156	2	.	.	PROPN
ma-28	156	3	112	112	NUM
ma-28	156	4	(	(	PUNCT
ma-28	156	5	1991	1991	NUM
ma-28	156	6	)	)	PUNCT
ma-28	156	7	121	121	NUM
ma-28	156	8	-	-	SYM
ma-28	156	9	124	124	NUM
ma-28	156	10	.	.	PUNCT
ma-28	157	1	https://doi.org/10.1090/s0002-9939-1991-1043405-5.[6	https://doi.org/10.1090/s0002-9939-1991-1043405-5.[6	PROPN
ma-28	157	2	]	]	PUNCT
ma-28	157	3	p.	p.	PROPN
ma-28	157	4	drabek	drabek	PROPN
ma-28	157	5	,	,	PUNCT
ma-28	157	6	a.	a.	NOUN
ma-28	157	7	kufner	kufner	PROPN
ma-28	157	8	,	,	PUNCT
ma-28	157	9	f.	f.	PROPN
ma-28	157	10	nicolosi	nicolosi	PROPN
ma-28	157	11	,	,	PUNCT
ma-28	157	12	quasilinear	quasilinear	NOUN
ma-28	157	13	elliptic	elliptic	ADJ
ma-28	157	14	equation	equation	NOUN
ma-28	157	15	with	with	ADP
ma-28	157	16	degenerations	degeneration	NOUN
ma-28	157	17	and	and	CCONJ
ma-28	157	18	singularities	singularity	NOUN
ma-28	157	19	,	,	PUNCT
ma-28	157	20	walter	walter	NOUN
ma-28	157	21	degruyter	degruyter	PROPN
ma-28	157	22	,	,	PUNCT
ma-28	157	23	bertin	bertin	PROPN
ma-28	157	24	,	,	PUNCT
ma-28	157	25	new	new	PROPN
ma-28	157	26	york	york	PROPN
ma-28	157	27	,	,	PUNCT
ma-28	157	28	1997	1997	NUM
ma-28	157	29	.	.	PUNCT
ma-28	158	1	https://doi.org/10.1515/9783110804775.[7	https://doi.org/10.1515/9783110804775.[7	X
ma-28	158	2	]	]	X
ma-28	158	3	g.	g.	PROPN
ma-28	158	4	farkas	farkas	PROPN
ma-28	158	5	,	,	PUNCT
ma-28	158	6	p.	p.	PROPN
ma-28	158	7	simon	simon	PROPN
ma-28	158	8	,	,	PUNCT
ma-28	158	9	stability	stability	NOUN
ma-28	158	10	properties	property	NOUN
ma-28	158	11	of	of	ADP
ma-28	158	12	positive	positive	ADJ
ma-28	158	13	solutions	solution	NOUN
ma-28	158	14	to	to	ADP
ma-28	158	15	partial	partial	ADJ
ma-28	158	16	differential	differential	ADJ
ma-28	158	17	equations	equation	NOUN
ma-28	158	18	with	with	ADP
ma-28	158	19	delay	delay	NOUN
ma-28	158	20	,	,	PUNCT
ma-28	158	21	electron	electron	PROPN
ma-28	158	22	j.diff	j.diff	PROPN
ma-28	158	23	.	.	PUNCT
ma-28	159	1	eqn	eqn	PROPN
ma-28	159	2	.	.	PUNCT
ma-28	160	1	64	64	NUM
ma-28	160	2	(	(	PUNCT
ma-28	160	3	2001	2001	NUM
ma-28	160	4	)	)	PUNCT
ma-28	160	5	1	1	NUM
ma-28	160	6	-	-	SYM
ma-28	160	7	8	8	NUM
ma-28	160	8	.	.	PUNCT
ma-28	161	1	https://ejde.math.txstate.edu/volumes/2001/64/farkas.pdf.[8	https://ejde.math.txstate.edu/volumes/2001/64/farkas.pdf.[8	PROPN
ma-28	161	2	]	]	PUNCT
ma-28	161	3	i.	i.	PROPN
ma-28	161	4	flores	flores	PROPN
ma-28	161	5	,	,	PUNCT
ma-28	161	6	a	a	DET
ma-28	161	7	resonance	resonance	NOUN
ma-28	161	8	phenomenon	phenomenon	NOUN
ma-28	161	9	for	for	ADP
ma-28	161	10	ground	ground	NOUN
ma-28	161	11	states	state	NOUN
ma-28	161	12	of	of	ADP
ma-28	161	13	an	an	DET
ma-28	161	14	elliptic	elliptic	ADJ
ma-28	161	15	equation	equation	NOUN
ma-28	161	16	of	of	ADP
ma-28	161	17	emden	emden	ADJ
ma-28	161	18	-	-	PUNCT
ma-28	161	19	fowler	fowler	PROPN
ma-28	161	20	type	type	NOUN
ma-28	161	21	,	,	PUNCT
ma-28	161	22	j.	j.	PROPN
ma-28	161	23	diff	diff	PROPN
ma-28	161	24	.	.	PUNCT
ma-28	162	1	eqn	eqn	PROPN
ma-28	162	2	.	.	PUNCT
ma-28	163	1	198(2004	198(2004	NUM
ma-28	163	2	)	)	PUNCT
ma-28	163	3	1	1	NUM
ma-28	163	4	-	-	SYM
ma-28	163	5	15	15	NUM
ma-28	163	6	.	.	PUNCT
ma-28	164	1	https://doi.org/10.1016/s0022-0396(02)00015-3.[9	https://doi.org/10.1016/s0022-0396(02)00015-3.[9	ADP
ma-28	164	2	]	]	X
ma-28	164	3	j.	j.	PROPN
ma-28	164	4	goddard	goddard	PROPN
ma-28	164	5	ii	ii	PROPN
ma-28	164	6	,	,	PUNCT
ma-28	164	7	r.	r.	PROPN
ma-28	164	8	shivaji	shivaji	PROPN
ma-28	164	9	,	,	PUNCT
ma-28	164	10	diffusive	diffusive	ADJ
ma-28	164	11	logistic	logistic	ADJ
ma-28	164	12	equation	equation	NOUN
ma-28	164	13	with	with	ADP
ma-28	164	14	constantyield	constantyield	ADJ
ma-28	164	15	harvesting	harvesting	NOUN
ma-28	164	16	and	and	CCONJ
ma-28	164	17	negative	negative	ADJ
ma-28	164	18	density	density	NOUN
ma-28	164	19	dependentemigration	dependentemigration	NOUN
ma-28	164	20	on	on	ADP
ma-28	164	21	the	the	DET
ma-28	164	22	boundary	boundary	NOUN
ma-28	164	23	,	,	PUNCT
ma-28	164	24	j.	j.	PROPN
ma-28	164	25	math	math	PROPN
ma-28	164	26	.	.	PUNCT
ma-28	165	1	anal	anal	PROPN
ma-28	165	2	.	.	PUNCT
ma-28	165	3	appl	appl	PROPN
ma-28	165	4	.	.	PUNCT
ma-28	166	1	4147	4147	NUM
ma-28	166	2	(	(	PUNCT
ma-28	166	3	2014	2014	NUM
ma-28	166	4	)	)	PUNCT
ma-28	166	5	561	561	NUM
ma-28	166	6	-	-	SYM
ma-28	166	7	573	573	NUM
ma-28	166	8	.	.	PUNCT
ma-28	167	1	https://doi.org/10.1016/j.jmaa.2014	https://doi.org/10.1016/j.jmaa.2014	NOUN
ma-28	167	2	.	.	PUNCT
ma-28	168	1	01.016.[10	01.016.[10	PRON
ma-28	168	2	]	]	X
ma-28	168	3	i.	i.	PROPN
ma-28	168	4	karatson	karatson	PROPN
ma-28	168	5	,	,	PUNCT
ma-28	168	6	p.simon	p.simon	NOUN
ma-28	168	7	,	,	PUNCT
ma-28	168	8	on	on	ADP
ma-28	168	9	the	the	DET
ma-28	168	10	stability	stability	NOUN
ma-28	168	11	properties	property	NOUN
ma-28	168	12	of	of	ADP
ma-28	168	13	nonnegative	nonnegative	ADJ
ma-28	168	14	solutions	solution	NOUN
ma-28	168	15	of	of	ADP
ma-28	168	16	semilinear	semilinear	ADJ
ma-28	168	17	problems	problem	NOUN
ma-28	168	18	with	with	ADP
ma-28	168	19	convex	convex	PROPN
ma-28	168	20	orconcave	orconcave	ADJ
ma-28	168	21	nonlinearity	nonlinearity	NOUN
ma-28	168	22	,	,	PUNCT
ma-28	168	23	j.	j.	PROPN
ma-28	168	24	comput	comput	PROPN
ma-28	168	25	.	.	PUNCT
ma-28	169	1	appl	appl	PROPN
ma-28	169	2	.	.	PROPN
ma-28	169	3	math	math	PROPN
ma-28	169	4	.	.	PUNCT
ma-28	170	1	131	131	NUM
ma-28	170	2	(	(	PUNCT
ma-28	170	3	2001	2001	NUM
ma-28	170	4	)	)	PUNCT
ma-28	170	5	497	497	NUM
ma-28	170	6	-	-	SYM
ma-28	170	7	501	501	NUM
ma-28	170	8	.	.	PUNCT
ma-28	171	1	https://doi.org/10.1016/s0377-0427(00	https://doi.org/10.1016/s0377-0427(00	PROPN
ma-28	171	2	)	)	PUNCT
ma-28	171	3	00714	00714	NUM
ma-28	171	4	-	-	SYM
ma-28	171	5	7.[11	7.[11	NUM
ma-28	171	6	]	]	PUNCT
ma-28	171	7	s.	s.	PROPN
ma-28	171	8	khafagy	khafagy	PROPN
ma-28	171	9	,	,	PUNCT
ma-28	171	10	existence	existence	NOUN
ma-28	171	11	results	result	VERB
ma-28	171	12	for	for	ADP
ma-28	171	13	weighted	weighted	ADJ
ma-28	171	14	(	(	PUNCT
ma-28	171	15	p	p	X
ma-28	171	16	,	,	PUNCT
ma-28	171	17	q)-laplacian	q)-laplacian	PUNCT
ma-28	171	18	nonlinear	nonlinear	ADJ
ma-28	171	19	system	system	NOUN
ma-28	171	20	,	,	PUNCT
ma-28	171	21	applied	apply	VERB
ma-28	171	22	mathematics	mathematic	NOUN
ma-28	171	23	e	e	NOUN
ma-28	171	24	-	-	NOUN
ma-28	171	25	notes	note	NOUN
ma-28	171	26	.	.	PUNCT
ma-28	172	1	17(2017	17(2017	NUM
ma-28	172	2	)	)	PUNCT
ma-28	172	3	242	242	NUM
ma-28	172	4	-	-	SYM
ma-28	172	5	250	250	NUM
ma-28	172	6	.	.	PUNCT
ma-28	173	1	https://www.emis.de/journals/amen/2017/amen-170214.pdf.[12	https://www.emis.de/journals/amen/2017/amen-170214.pdf.[12	PROPN
ma-28	173	2	]	]	X
ma-28	173	3	s.	s.	PROPN
ma-28	173	4	khafagy	khafagy	PROPN
ma-28	173	5	,	,	PUNCT
ma-28	173	6	non	non	ADJ
ma-28	173	7	-	-	NOUN
ma-28	173	8	existence	existence	NOUN
ma-28	173	9	of	of	ADP
ma-28	173	10	positive	positive	ADJ
ma-28	173	11	weak	weak	ADJ
ma-28	173	12	solutions	solution	NOUN
ma-28	173	13	for	for	ADP
ma-28	173	14	some	some	DET
ma-28	173	15	weighted	weight	VERB
ma-28	173	16	p	p	PROPN
ma-28	173	17	-	-	PUNCT
ma-28	173	18	laplacian	laplacian	ADJ
ma-28	173	19	systems	system	NOUN
ma-28	173	20	,	,	PUNCT
ma-28	173	21	j.	j.	PROPN
ma-28	173	22	adv	adv	PROPN
ma-28	173	23	.	.	PUNCT
ma-28	174	1	res	res	PROPN
ma-28	174	2	.	.	PROPN
ma-28	174	3	dyn.control	dyn.control	PROPN
ma-28	175	1	syst	syst	PROPN
ma-28	175	2	.	.	PROPN
ma-28	175	3	7	7	NUM
ma-28	175	4	(	(	PUNCT
ma-28	175	5	2015	2015	NUM
ma-28	175	6	)	)	PUNCT
ma-28	175	7	71	71	NUM
ma-28	175	8	-	-	SYM
ma-28	175	9	77	77	NUM
ma-28	175	10	.	.	PUNCT
ma-28	176	1	https://www.jardcs.org/backissues/abstract.php?archiveid=296.[13	https://www.jardcs.org/backissues/abstract.php?archiveid=296.[13	PROPN
ma-28	176	2	]	]	PUNCT
ma-28	176	3	s.	s.	PROPN
ma-28	176	4	khafagy	khafagy	PROPN
ma-28	176	5	,	,	PUNCT
ma-28	176	6	on	on	ADP
ma-28	176	7	the	the	DET
ma-28	176	8	stabiblity	stabiblity	NOUN
ma-28	176	9	of	of	ADP
ma-28	176	10	positive	positive	ADJ
ma-28	176	11	weak	weak	ADJ
ma-28	176	12	solution	solution	NOUN
ma-28	176	13	for	for	ADP
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ma-28	177	7	)	)	PUNCT
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ma-28	177	9	-	-	SYM
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ma-28	177	11	.	.	PUNCT
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ma-28	178	15	(	(	PUNCT
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ma-28	179	5	(	(	PUNCT
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ma-28	179	7	)	)	PUNCT
ma-28	179	8	86	86	NUM
ma-28	179	9	-	-	SYM
ma-28	179	10	92	92	NUM
ma-28	179	11	.	.	PUNCT
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ma-28	180	15	solutions	solution	NOUN
ma-28	180	16	for	for	ADP
ma-28	180	17	nonlinear	nonlinear	ADJ
ma-28	180	18	system	system	NOUN
ma-28	180	19	involvingweighted	involvingweighte	VERB
ma-28	180	20	(	(	PUNCT
ma-28	180	21	p	p	X
ma-28	180	22	,	,	PUNCT
ma-28	180	23	q)-laplacian	q)-laplacian	PUNCT
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ma-28	180	26	asian	asian	ADJ
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ma-28	180	28	.	.	PUNCT
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ma-28	181	2	.	.	PUNCT
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ma-28	182	2	(	(	PUNCT
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ma-28	182	4	)	)	PUNCT
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ma-28	182	6	-	-	SYM
ma-28	182	7	364	364	NUM
ma-28	182	8	.	.	PUNCT
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ma-28	183	2	.	.	PUNCT
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ma-28	184	8	,	,	PUNCT
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ma-28	184	13	positive	positive	ADJ
ma-28	184	14	weak	weak	ADJ
ma-28	184	15	solution	solution	NOUN
ma-28	184	16	for	for	ADP
ma-28	184	17	(	(	PUNCT
ma-28	184	18	p	p	X
ma-28	184	19	;	;	PUNCT
ma-28	184	20	q)-laplacian	q)-laplacian	PUNCT
ma-28	184	21	nonlinear	nonlinear	ADJ
ma-28	184	22	system	system	NOUN
ma-28	184	23	,	,	PUNCT
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ma-28	184	25	.	.	PUNCT
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ma-28	185	2	-	-	PUNCT
ma-28	185	3	notes	note	NOUN
ma-28	185	4	.	.	PUNCT
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ma-28	186	2	(	(	PUNCT
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ma-28	186	4	)	)	PUNCT
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ma-28	186	6	-	-	SYM
ma-28	186	7	114	114	NUM
ma-28	186	8	.	.	PUNCT
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ma-28	187	4	khafagy	khafagy	PROPN
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ma-28	187	6	h.	h.	PROPN
ma-28	187	7	serag	serag	PROPN
ma-28	187	8	,	,	PUNCT
ma-28	187	9	stability	stability	NOUN
ma-28	187	10	results	result	NOUN
ma-28	187	11	of	of	ADP
ma-28	187	12	positive	positive	ADJ
ma-28	187	13	weak	weak	ADJ
ma-28	187	14	solution	solution	NOUN
ma-28	187	15	for	for	ADP
ma-28	187	16	singular	singular	ADJ
ma-28	187	17	p	p	PROPN
ma-28	187	18	-	-	PUNCT
ma-28	187	19	laplacian	laplacian	ADJ
ma-28	187	20	nonlinear	nonlinear	ADJ
ma-28	187	21	system	system	NOUN
ma-28	187	22	,	,	PUNCT
ma-28	188	1	j.	j.	PROPN
ma-28	188	2	appl.math	appl.math	PROPN
ma-28	188	3	.	.	PUNCT
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ma-28	188	5	.	.	PUNCT
ma-28	189	1	36	36	NUM
ma-28	189	2	(	(	PUNCT
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ma-28	189	4	)	)	PUNCT
ma-28	189	5	173	173	NUM
ma-28	189	6	-	-	SYM
ma-28	189	7	179	179	NUM
ma-28	189	8	.	.	PUNCT
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ma-28	190	5	,	,	PUNCT
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ma-28	190	9	quasi	quasi	ADJ
ma-28	190	10	-	-	ADJ
ma-28	190	11	yield	yield	ADJ
ma-28	190	12	harvest	harvest	NOUN
ma-28	190	13	rates	rate	NOUN
ma-28	190	14	,	,	PUNCT
ma-28	190	15	math	math	NOUN
ma-28	190	16	.	.	PUNCT
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ma-28	191	2	.	.	PUNCT
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ma-28	192	2	.	.	PROPN
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ma-28	192	4	(	(	PUNCT
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ma-28	192	6	)	)	PUNCT
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ma-28	192	8	-	-	SYM
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ma-28	192	10	.	.	PUNCT
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ma-28	194	19	without	without	ADP
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ma-28	194	21	,	,	PUNCT
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ma-28	194	32	-	-	SYM
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ma-28	197	2	(	(	PUNCT
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ma-28	197	4	)	)	PUNCT
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ma-28	197	6	-	-	SYM
ma-28	197	7	128	128	NUM
ma-28	197	8	.	.	PUNCT
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ma-28	198	2	.	.	PUNCT
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ma-28	202	2	.	.	PUNCT
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ma-28	203	2	/	/	SYM
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ma-28	204	1	[	[	X
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ma-28	204	21	,	,	PUNCT
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ma-28	207	6	-	-	SYM
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ma-28	207	8	.	.	PUNCT
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ma-28	208	38	,	,	PUNCT
ma-28	208	39	diffusive	diffusive	ADJ
ma-28	208	40	logistic	logistic	ADJ
ma-28	208	41	equations	equation	NOUN
ma-28	208	42	with	with	ADP
ma-28	208	43	harvesting	harvesting	NOUN
ma-28	208	44	and	and	CCONJ
ma-28	208	45	heterogeneity	heterogeneity	NOUN
ma-28	208	46	understrong	understrong	ADJ
ma-28	208	47	growth	growth	NOUN
ma-28	208	48	rate	rate	NOUN
ma-28	208	49	,	,	PUNCT
ma-28	208	50	adv	adv	PROPN
ma-28	208	51	.	.	PUNCT
ma-28	209	1	nonlinear	nonlinear	PROPN
ma-28	209	2	anal	anal	PROPN
ma-28	209	3	.	.	PUNCT
ma-28	210	1	8	8	NUM
ma-28	210	2	(	(	PUNCT
ma-28	210	3	2019	2019	NUM
ma-28	210	4	)	)	PUNCT
ma-28	211	1	455	455	NUM
ma-28	211	2	-	-	SYM
ma-28	211	3	467	467	NUM
ma-28	211	4	.	.	PUNCT
ma-28	211	5	https://doi.org/10.1515/anona-2016-0208.[25	https://doi.org/10.1515/anona-2016-0208.[25	PROPN
ma-28	211	6	]	]	PUNCT
ma-28	211	7	a.	a.	PROPN
ma-28	211	8	tertikas	tertikas	PROPN
ma-28	211	9	,	,	PUNCT
ma-28	211	10	stability	stability	NOUN
ma-28	211	11	and	and	CCONJ
ma-28	211	12	instability	instability	NOUN
ma-28	211	13	of	of	ADP
ma-28	211	14	positive	positive	ADJ
ma-28	211	15	solutions	solution	NOUN
ma-28	211	16	of	of	ADP
ma-28	211	17	semilinear	semilinear	NOUN
ma-28	211	18	problems	problem	NOUN
ma-28	211	19	,	,	PUNCT
ma-28	211	20	proc	proc	PROPN
ma-28	211	21	.	.	PUNCT
ma-28	212	1	amer	amer	PROPN
ma-28	212	2	.	.	PUNCT
ma-28	212	3	math	math	PROPN
ma-28	212	4	.	.	PUNCT
ma-28	213	1	soc	soc	PROPN
ma-28	213	2	.	.	PUNCT
ma-28	214	1	114	114	NUM
ma-28	214	2	(	(	PUNCT
ma-28	214	3	1992)1035	1992)1035	NUM
ma-28	214	4	-	-	SYM
ma-28	214	5	1040	1040	NUM
ma-28	214	6	.	.	PUNCT
ma-28	215	1	https://doi.org/10.1090/s0002-9939-1992-1092928-2.[26	https://doi.org/10.1090/s0002-9939-1992-1092928-2.[26	PROPN
ma-28	215	2	]	]	X
ma-28	215	3	i.	i.	NOUN
ma-28	215	4	voros	voros	PROPN
ma-28	215	5	,	,	PUNCT
ma-28	215	6	stability	stability	NOUN
ma-28	215	7	properties	property	NOUN
ma-28	215	8	of	of	ADP
ma-28	215	9	nonnegative	nonnegative	ADJ
ma-28	215	10	solutions	solution	NOUN
ma-28	215	11	of	of	ADP
ma-28	215	12	semilinear	semilinear	PROPN
ma-28	215	13	symmetric	symmetric	ADJ
ma-28	215	14	cooperative	cooperative	ADJ
ma-28	215	15	systems	system	NOUN
ma-28	215	16	,	,	PUNCT
ma-28	215	17	electronic	electronic	ADJ
ma-28	215	18	j.	j.	PROPN
ma-28	215	19	diff.eqn	diff.eqn	PROPN
ma-28	215	20	.	.	PROPN
ma-28	215	21	105	105	NUM
ma-28	215	22	(	(	PUNCT
ma-28	215	23	2004	2004	NUM
ma-28	215	24	)	)	PUNCT
ma-28	215	25	1	1	NUM
ma-28	215	26	-	-	SYM
ma-28	215	27	6	6	NUM
ma-28	215	28	.	.	PUNCT
ma-28	215	29	https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf	https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf	PROPN
ma-28	215	30	.	.	PUNCT
ma-28	216	1	https://doi.org/10.28924/ada/ma.2.8	https://doi.org/10.28924/ada/ma.2.8	PROPN
ma-28	216	2	https://doi.org/10.1090/s0002-9947-02-03005-2	https://doi.org/10.1090/s0002-9947-02-03005-2	PROPN
ma-28	216	3	https://doi.org/10.1515/anona-2016-0208	https://doi.org/10.1515/anona-2016-0208	PROPN
ma-28	216	4	https://doi.org/10.1090/s0002-9939-1992-1092928-2	https://doi.org/10.1090/s0002-9939-1992-1092928-2	PROPN
ma-28	216	5	https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf	https://ejde.math.txstate.edu/volumes/2004/105/voros.pdf	PROPN
ma-28	216	6	1	1	NUM
ma-28	216	7	.	.	PUNCT
ma-28	217	1	introduction	introduction	NOUN
ma-28	217	2	:	:	PUNCT
ma-28	217	3	2	2	X
ma-28	217	4	.	.	X
ma-28	217	5	main	main	ADJ
ma-28	217	6	results	result	NOUN
ma-28	217	7	3	3	NUM
ma-28	217	8	.	.	PUNCT
ma-28	217	9	applications	application	NOUN
ma-28	217	10	and	and	CCONJ
ma-28	217	11	related	relate	VERB
ma-28	217	12	results	result	NOUN
ma-28	217	13	references	reference	NOUN