GLOBAL ATTRACTOR FOR REACTION-DIFFUSION EQUATIONS WITH SUPERCRITICAL NONLINEARITY IN UNBOUNDED DOMAINS

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1 Electroic Joural of Differetial Equatios, Vol (2016), No. 63, pp ISSN: URL: or ftp ejde.math.txstate.edu GLOBAL ATTRACTOR FOR REACTION-DIFFUSION EQUATIONS WITH SUPERCRITICAL NONLINEARITY IN UNBOUNDED DOMAINS JIN ZHANG, CHANG ZHANG, CHENGKUI ZHONG Abstract. We cosider the existece of global attractor for the ihomogeeous reactio-diffusio equatio u t u V (x)u + u p 2 u = g, i R R +, u(0) = u 0 L 2 (R ) L p (R ), where p > 2 is supercritical ad V (x) satisfies suitable assumptios. Sice 2 is ot positive defiite i H 1 (R ), the Growall iequality ca ot be derived ad the correspodig semigroup does ot possess bouded absorbig sets i L 2 (R ). Thus, by a special method, we prove that the equatio has a global attractor i L p (R ), which attracts ay bouded subset i L 2 (R ) L p (R ). 1. Itroductio We cosider the existece of global attractor for the ihomogeeous reactiodiffusio equatio i the whole space: u t u V (x)u + u p 2 u = g, i R R +, u(0) = u 0 L 2 (R ), (1.1) where p > 2 2 is a supercritical expoet, g L1 (R ) L (R ) is give ad the fuctio V (x) satisfies V L /2 (R ) L (R ). (1.2) The log-time behavior of solutio for the reactio-diffusio equatio u t u + λu + f(u) = g, (1.3) i ubouded domai has bee studied by may authors, where λ > 0 ad f(u) satisfyig some growth coditio. I the pioeerig work [3], Babai ad Vishik proved the existece of global attractor i some weighted space. I paper [11], uder some structural assumptios o the oliearity f, Wag proved the existece of global attractor i usual space L 2 (R ) istead of the weighted space. Other 2010 Mathematics Subject Classificatio. 35B41, 35K57, 37L99. Key words ad phrases. Global attractor; ihomogeeous reactio-diffusio equatio; ubouded domai; supercritical oliearity. c 2016 Texas State Uiversity. Submitted July 20, Published March 4,

2 2 J. ZHANG, C. ZHANG, C. ZHONG EJDE-2016/63 ivestigatios of global attractor for equatio (1.3) i ubouded domai ca be foud i [1, 7, 9]. I geeral, Growall iequality is utilized to prove the existece of absorbig set i L 2 (R ) for (1.3) whe λ > 0. However, it will be difficult i the case λ = 0 or λ < 0. Zelik [13] cosidered the existece of global attractor for the real Gizburg- Ladau equatio u t u u + u 3 = g (1.4) i R. For this equatio, because of the ifiiteess of the eergy fuctioal, the global attractor ca ot be obtaied i usual spaces, thus Zelik cosidered the existece of the locally compact global attractors for the semigroup associated with the equatio (1.4) i uiformly local spaces. More detailed iformatio ca be foud i [6, 10]. Arrieta, Cholewa, Dlotko ad Rodríguez-Beral [2] cosider the reactio-diffusio equatio u t u = f(x, u) + g with f(x, s)s C(x) s 2 + D(x) s i stadard Lebesgue space. They prove that for some suitable fuctios C(x) ad D(x), the existece of global solutios ca be obtaied. Furthermore, if the operator + C(x)I geerates a aalytic semigroup which decay expoetially, the this equatio has a global attractor. Motivated by the above works, we cosider the existece of a global attractor for (1.1) (which is ihomogeeous type of equatio (1.4) but u 3 is replaced by u p 2 u). Followig the proof i [2], we ca obtai the existece ad uiqueess of the solutios. We ecouter difficulties whe provig the existece of global attractor, sice the operator + V (x)i may ot be able to geerate a aalytic semigroup, ad the Growall iequality ca ot be applied to obtai the absorbig set i L 2 (R ). To overcome the difficulties, we assume that V (x) satisfies some suitable coditios, ad use the method of mootoicity of the eergy fuctioal to obtai a absorbig set i D 1,2 (R ) L p (R ). Furthermore, i order to establish the ω-limit compactess of the correspodig semigroup, we use the Sobolev embeddigs i iterior, ad estimate the L p orm of solutios is arbitrarily small uiformly for large time i exterior. Our mai result reads as follows. Theorem 1.1. Assume p > 2 2, g L1 (R ) L (R ), V (x) satisfies coditios (1.2). The the semigroup {S(t)} t 0 geerated by the equatio (1.1) has a global attractor A i L p (R ), which attracts ay bouded subset i L 2 (R ) L p (R ). Remark 1.2. Zelik [13] proved the existece of global attractor for the Gizburg- Ladau equatio u t u u + u 3 = g (1.5) i R, ad the attractor is oly locally compact i a uiformly local phase space. To obtai a global attractor which is compact i usual space L 2 (R ) L p (R ), we assume that the oliear term is supercritical growth ad the coditio (1.2) holds. 2. Prelimiaries I this sectio, we first review the basic cocept about the Kuratowski measure of ocompactess, which will be used to establish the ω-limit compactess of semigroup. See [5, 8, 12] for its some basic properties.

3 EJDE-2016/63 GLOBAL ATTRACTOR 3 Defiitio 2.1. Let (M, d) be a metric space ad let A be a bouded subset of M. The measure of ocompactess κ(a) is defied by κ(a) = if{δ > 0 A admits a fiite cover by sets of diameter δ}. The properties of the measure of ocompactess κ(a) are provided i the followig lemmas. Lemma 2.2. Let (M, d) be a complete metric space ad κ be the measure of ocompactess. The (i) κ(b) = 0, if ad oly if B is compact; (ii) if M is a Baach space, the κ(b 1 + B 2 ) κ(b 1 ) + κ(b 2 ) ; (iii) κ(b 1 ) κ(b 2 ) wheever B 1 B 2 ; (iv) κ(b 1 B 2 ) = max{κ(b 1 ), κ(b 2 )} ; (v) κ(b) = κ(b). Lemma 2.3. Let M be a ifiite dimesioal Baach space ad let B(ε) be a ball of radius ε. The κ(b(ε)) = 2ε. The cocept of ω-limit compactess of a semigroup, which is a importat ecessary ad sufficiet coditio for the existece of global attractor (see [8]). Defiitio 2.4. A semigroup {S(t)} t 0 i a complete metric space (M, d) is called a C 0 or cotiuous semigroup if it satisfies: S(0) = I, S(t)S(s) = S(s)S(t) = S(t + s), S(t)x 0 is cotiuous i x 0 M ad t R. Defiitio 2.5. A cotiuous semigroup {S(t)} t 0 i a complete metric space (M, d) is called ω-limit compact, if for ay bouded subset B ad ay ε > 0, there exists a time t 0 such that ( ) κ t t S(t)B ε. Lemma 2.6. Let {S(t)} t 0 be a cotiuous semigroup i a complete metric space (M, d). The S(t) has a global attractor A i M if ad oly if (1) there is a bouded absorbig set B M, ad (2) {S(t)} t 0 is ω-limit compact. Now, we give the geeral existece ad uiqueess of solutios which ca be obtaied as i [2]. Theorem 2.7. Let p > 2, g L 1 (R ) L (R ), V (x) satisfies coditios (1.2). The for ay u 0 L 2 (R ) L p (R ) ad T > 0, there exists a uique weak solutio u(x, t) of (1.1) satisfies u C([0, T ], L 2 (R ) L p (R )) L 2 (0, T, D 1,2 (R )). Furthermore, u 0 u(t) is cotiuous o L 2 (R ) L p (R ). For coveiece, here ad subsequetly, we ca assume V (x) l sice V L (R ). I additio, for ay R > 0, we deote Ω R := {x R : x R}.

4 4 J. ZHANG, C. ZHANG, C. ZHONG EJDE-2016/63 3. Bouded absorbig set i D 1,2 (R ) L p (R ) ad L 2p 2 (R ) By Theorem 2.7, we ca defie the operator semigroup {S(t)} t 0 i L 2 (R ) L p (R ) as S(t)u = u(t) : L 2 (R ) L p (R ) L 2 (R ) L p (R ), which is geerated by the weak solutios of (1.1) with iitial data u 0 L 2 (R ) L p (R ). Theorem 3.1. There exist costats ρ 1 > 0 ad t 1 ( u 0 2 ) such that, for the solutio u(t) of (1.1), u(t) 2 dx + u(t) p dx ρ 1, for all t t 1. R R Proof. first, we multiply (1.1) by u ad itegrate over R, 1 d 2 dt u u 2 dx V u 2 dx + u p dx = gudx. (3.1) R R R R Applyig Hölder iequality ad Youg iequality, we estimate the right-had side as gu dx 1 u p dx + C ( ) g p R 4 R p 1. (3.2) The, we divide the third term o the left-had side ito V u 2 dx V u 2 dx + V u 2 dx := I 1 + I 2, R Ω R0 R \Ω R0 where the costat R 0 is sufficietly large such that ( ) 2/ V /2 S dx R \Ω R0 2, ad S is the Sobolev costat satisfyig S u 2 2 u 2 2. Therefore, utilizig 2 Hölder ad Youg iequality, the two terms I 1 ad I 2 ca be estimated as I 1 1 u p dx + C(p, l,, R 0 ), I 2 S 4 R 2 u u 2 dx. (3.3) 2 2 R Combiig the estimates (3.1), (3.2) ad (3.3) yields 1 d 2 dt u u 2 dx + 1 u p dx C. (3.4) 2 R 2 R Itegratig this iequality betwee 0 ad t gives t ( u(s) 2 + u(s) p ) dx ds tc + u(0) 2 2, 0 R it follows from u(0) 2 2 is bouded that there exists a sufficietly large time t 1 such that u(t 1 ) 2 dx + u(t 1 ) p dx 2C. (3.5) R R Meawhile, deotig E(u(t)) := 1 u(t) 2 dx 1 V u(t) 2 dx + 1 u(t) p dx gu(t)dx, 2 R 2 R p R R

5 EJDE-2016/63 GLOBAL ATTRACTOR 5 ad multiplyig the equatio (1.1) by u t ad itegratig over R, this yields d dt E(u(t)) = u t 2 2 0, thus E(u(t)) E(u(t 1 )), for all t t 1. (3.6) Utilizig the similar techiques i (3.2) ad (3.3), the followig two estimates gu dx 1 u p dx + C, R 4p R V u 2 dx 1 u 2 dx + 1 u p dx + C R 2 R 4p R are also valid, ad yield E(u(t)) 1 u(t) 2 dx + 1 u(t) p dx 2C, (3.7) 4 R 2p R E(u(t 1 )) 3 u(t 1 ) 2 dx + 3 u(t) p dx + 2C. (3.8) 4 R 2p R Combiig the estimates (3.5), (3.6), (3.7) ad (3.8), it is obvious that there exists ρ 1 > 0 such that 1 u(t) 2 dx + 1 u(t) p dx ρ 1 4 R 2p R 2p. The coclusio is cosequetly obtaied. Next, we show the semigroup also has a absorbig set i the space L 2p 2 (R ). Theorem 3.2. There exist costats ρ 2 ad t 2 ( u 0 2 ) such that, for the solutio u(t) of the equatio(1.1), R u(t) 2p 2 dx < ρ 2, for all t t 2. Proof. Similar techiques ca be used for (3.4), whe multiplyig (1.1) by u p 2 u ad u 2p 4 u respectively. We have the followig two estimates: 1 d p dt u p p + 1 u 2p 2 dx C, (3.9) 2 R d ( ) dt u 2p 2 C 1 + u 2p 2 2p 2. (3.10) We ca itegrate (3.9) betwee t ad t + 1 to obtai 1 t+1 u(s) 2p 2 dx ds C t R p u(t) p p. Recallig the fact that u p p is bouded for all t t 1, therefore there exists a costat C such that t+1 u(s) 2p 2 dx ds C for all t t 1. (3.11) t R Now, itegratig (3.10) betwee s ad t + 1 (t s < t + 1)gives u(t + 1) 2p 2 2p 2 C (1 + t+1 s u(ξ) 2p 2 2p 2 dξ ) + u(s) 2p 2 2p 2,

6 6 J. ZHANG, C. ZHANG, C. ZHONG EJDE-2016/63 the we itegrate this equatio with respect to s betwee t ad t + 1, we obtai t+1 u(t + 1) 2p 2 2p 2 C + C u(s) 2p 2 dx ds. (3.12) t R It follow from (3.11) that there exists a costat ρ 2 > 0 such that u(t + 1) 2p 2 2p 2 ρ 2 t > t 1, the proof is complete because t 2 = t Remark 3.3. We observe that, i the proofs of Theorem 3.1 ad Theorem 3.2, we oly eed g L p p 1 (R ) ad g L 3p 4 p 1 (R ) respectively. Actually, we ca prove that the semigroup has a bouded absorbig set i L q (R ) for ay q [p, ) whe the fuctio g L 1 (R ) L (R ). 4. ω-limit compactess ad global attractor We defie a smooth fuctio θ : R + [0, 1], such that { 0 s 1, θ(s) = 1 s 2, with θ (s) 2. Let θ R (x) = θ R ( x ) = θ ( x ) 2 R. I this way, ay solutio u(t) of 2 equatio (1.1) ca be decomposed as u(t) = θ R u(t) + (1 θ R )u(t). Before the proof of ω-limit compactess ad global attractor for the correspodig semigroup, we first give the estimate the L p orm of solutios are arbitrarily small uiformly o exterior. Lemma 4.1. For arbitrary ε > 0, there exist costats t 3 ad R 0 > 0, such that for the solutio u(t), R θ 2 R 0 u(t) p dx < ε, for all t t 3. Proof. Multiplyig (1.1) by θ p R u p 2 u ad itegratig over R, 1 d θ p R p dt u p dx u θ p R u (p 2) u dx θ p R V u p dx R R R + θ p R u 2p 2 dx R = θ p R u p 2 ug dx. R We first cosider the estimate of the secod term i the left-had side, sice u θ p R u (p 2) dx R 4(p 1) = p 2 θ p R u p 2 2 dx + p θ p 1 R up 1 θ R udx R R 4(p 1) p 2 θ p R u p 2 2 dx 1 θ p R R 4 u 2p 2 dx p 2 θ R 2 u 2 dx. R R (4.1)

7 EJDE-2016/63 GLOBAL ATTRACTOR 7 Referrig to Theorem 3.1 ad assumptio of the fuctio θ R, we have u 2 2 is bouded ad θ R (x) 4 R. Thus, there exists a costat C > 0 such that for all t t 1, u θ p R u p 2 u dx R 4(p 1) p 2 θ p R u p 2 2 dx 1 θ p R R 4 u 2p 2 dx C (4.2) R R 2. The, the applicatio of Hölder ad Youg iequalities gives the followig two estimates θ p R V u p dx ( ) 2/ ( ) 2 V /2 dx θ R u p dx R R \Ω R R ( ) 2/ (4.3) S V /2 dx θr u 2 p 2 2 dx, R \Ω R R θ p R u p 2 ugdx 1 θ p R R 4 u 2p 2 dx + θ p R g 2 dx. (4.4) R R It is obvious that the terms C R, ( 2 R \Ω R V /2 dx ) 2/ ad θ R p R g 2 dx ca be sufficietly small whe R. Therefore, from (4.1)-(4.4) it follows that, for arbitrary ε > 0, there exists R 0, such that for all t t 1 ad R R 0, d θ p R dt u p dx + θ p R u 2p 2 dx < 1 R R 2 ε 2p 2 p(1 λ), (4.5) where λ (0, 1) satisfies 1 p = ( 2)λ 2 there exists a time t 3 t 1, such that + 1 λ 2p 2. Similarly to the proof of Theorem 3.1, R θ p R u(t 3) 2p 2 dx < ε 2p 2 p(1 λ). (4.6) Now, combiig (4.5) with (4.6), we ca prove that if t t 3, there exists a costat C (ρ 1, p), such that R θ p R u(t) p dx Cε. (4.7) Actually, applyig the iterpolatio iequality ad otice that u 2 2 ρ 1 S, we have ( ) 1/p θ p R u p dx ε 1/p u 2 + ( ε 1/p) ( λ 1 λ θ p R 2 R u 2p 2 dx R ε 1/p ρ1 S + ( ε 1/p) λ ( 1 λ 2 R θ p R u 2p 2 dx 1 S u 2 2 ) 1 2p 2 ) 1 2p 2 Therefore (4.7) is valid provided that θ p R R u(t) 2p 2 dx < ε 2p 2 p(1 λ). O the other had, if θ p R R u(t) 2p 2 dx ε 2p 2 p(1 λ), the referrig to the estimate (4.5), it follows that d dt θ p R u(t) p dx < 1 2 ε 2p 2 p(1 λ) < 0,.

8 8 J. ZHANG, C. ZHANG, C. ZHONG EJDE-2016/63 which cocludes that θ p R u(t) p dx is decreasig with respect to variable t. Hece, i ay case as t t 3, we have R θ p R u(t 3) p dx Cε. Now, we prove that the semigroup geerated by the solutios of equatio(1.1) has a global attractor A, which attracts ay bouded subset B L 2 (R ) L p (R ) i the topology of L p (R ). Proof of Theorem 1.1. We oly eed to verify that the correspodig semigroup is ω-limit compact. For ay fixed R, it follows from Theorem 3.1 ad Theorem 3.2 that there exists a time t 2 such that t t2 u0 B (1 θ R )S(t)u 0 is bouded i H 1 (Ω 2R ) ad L 2p 2 (R ), the by the compactess of Sobolev embeddig H 1 (Ω 2R ) L 2 (Ω 2R ) ad iterpolatio iequality (2 < p < 2p 2), we obtai that t t2 u0 B (1 θ R )S(t)u 0 is compact i L p (Ω 2R ), thus ( ) κ t t2 u0 B(1 θ R )S(t)u 0 = 0, for ay R > 0. Lp O the other had, from Lemma 4.1, we kow for ay ε > 0, there exist costats t 3 ad R 0 > 0 such that t t3 u0 Bθ R0 S(t)u 0 p p < ε, by Lemma2.3, its measure of ocompactess is less tha 2ε, i.e., ( ) κ t t3 u0 Bθ R0 S(t)u 0 < 2ε. L p Thus takig t = max{t 2, t 3 }, we have ( ) κ t t u0 BS(t)u 0 L ( p ( )Lp ) κ t t u0 Bθ R0 S(t)u 0 + κ t t u0 B(1 θ R0 )S(t)u 0 < 2ε, L p which cocludes that the semigroup {S(t)} t 0 is ω-limit compact. Therefore, we obtai the existece of global attractor, which attracts ay bouded subset B L 2 (R ) L p (R ) i the topology of L p (R ). Ackowledgmets. We would like to express our sicere thaks to the aoymous referee for his(her) valuable commets ad suggestios which led to a importat improvemet of our origial mauscript. This work was partly supported by NSFC Grat (No ). Refereces [1] F. Abergel; Existece ad fiite dimesioality of the global attractor for evolutio equatios o ubouded domais, Joural of Differetial Equatios, 83(1) (1990), [2] J. Arrieta, J. Cholewa, T. Dlotko, A. Rodríguez-Beral; Asymptotic behavior ad attractors for reactio diffusio equatios i ubouded domais, Noliear Aalysis: Theory, Methods & Applicatios, 56(4) (2004), [3] A. V. Babi, M. I. Vishik; Attractors of partial differetial evolutio equatios i a ubouded domai, Proceedigs of the Royal Society of Ediburgh: Sectio A Mathematics, 116(3-4) (1990),

9 EJDE-2016/63 GLOBAL ATTRACTOR 9 [4] A. V. Babi, M. I. Vishik; Attractor of evolutio equatios, North-Hollad Publishig Co., Amsterdam (1992). [5] K. Deimlig; Noliear Fuctioal Aalysis, Spriger-Verlag, Berli, [6] M. Efediev, S. Zelik; Upper ad lower bouds for the Kolmogorov etropy of the attractor for the RDE i a ubouded domai, Joural of Dyamics ad Differetial Equatios, Spriger, 14(2) (2002), [7] E. Feireisl, Ph. Laurecot, F. Simodo; Global attractors for degeerate parabolic equatios o ubouded domais, J. Diff. Eqs. 129 (1996), [8] Q. F. Ma, S. H. Wag, C. K. Zhog; Necessary ad sufficiet coditios for the existece of global attractors for semigroups ad applicatios, Idiaa Uiv. Math. J. 51 (2002), [9] S. Merio; O the existece of the compact global attractor for semiliear reactio-diffusio systems o R, J. Diff. Eqs. 132 (1996), [10] A. Miraville, S. Zelik; Attractors for dissipative partial differetial equatios i bouded ad ubouded domais, Hadbook of Differetial Equatios: Evolutioary Equatios 4, Elsevier, North-Hollad, Amsterdam (2008), [11] B. X. Wag; Attractors for reactio diffusio equatios i ubouded domais, Physica D 128 (1999), [12] C. K. Zhog, M. H. Yag, C. Y. Su; The existece of global attractors for the orm-toweak cotiuous semigroup ad applicatio to the oliear reactio-diffusio equatios, J. Differetial Equatios, 223(2) (2006), Pages [13] S. Zelik; The attractor for a oliear reactio-diffusio system i the ubouded domai ad kolmogorove s -etropy, Mathematische Nachrichte 232(1) (2001), Ji Zhag Departmet of Mathematics, College of Sciece, Hohai Uiversity, Najig , Chia address: zhagji86@hhu.edu.c Chag Zhag Departmet of Mathematics, Najig Uiversity, Najig , Chia address: chzhju@126.com Chegkui Zhog (correspodig author) Departmet of Mathematics, Najig Uiversity, Najig , Chia address: ckzhog@ju.edu.c