Exponential Growth of Solution for a Class of Reaction Diffusion Equation with Memory and Multiple Nonlinearities

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1 Research in Alied Mahemaics vol. 1 (217), Aricle ID 11258, 9 ages doi: /217/11258 AgiAl Publishing House h:// Research Aricle Exonenial Growh of Soluion for a Class of Reacion Diffusion Equaion wih Memory and Mulile Nonlineariies Jian Dang, Qingying Hu, Suxia Xia, and Hongwei Zhang Dearmen of Mahemaics, Henan Universiy of Technology, Zhengzhou 451, China Absrac. In his aer, we consider he iniial boundary value roblem of a class of reacion diffusion equaion wih memory and mulile nonlineariies. We show he exonenial growh of soluion wih L -norm using a differenial inequaliy. Keywords: reacion diffusion equaion, exonenial growh, memory erm, mulile nonlineariies. Mahemaics Subjec Classificaion: 35K57,35B44. Corresonding Auhor Qingying Hu slxhqy@163.com Edior Zhouhong Li Daes Received 8 November 216 Acceed 19 Augus 217 Coyrigh 217 Jian Dang e al. This is an oen access aricle disribued under he Creaive Commons Aribuion License, which ermis unresriced use, disribuion, and reroducion in any medium, rovided he original work is roerly cied. 1. Inroducion In his aer, we sudy he following iniial boundary value roblem for a class of reacion diffusion equaion wih memory and mulile nonlineariies u Δu + g( s)δu(s)ds + u k 2 u = u 2 u, (1.1) u(x, ) =, x Ω, (1.2) u(x, ) = u (x), x Ω, (1.3) where k > 2, > 2 are real numbers and Ω is bounded domain in R n wih smooh boundary Ω so ha he divergence heorem can be alied. Here, g reresens he kernel of memory erm and Δ denoes he Lalace oeraor in Ω. This ye of roblem no only is imoran from he heoreical oin of view, bu also arises in many hysical alicaions and describes a grea deal of models in alied science. I aears in he models of chemical reacions, hea ransfer, oulaion dynamics, and so on (see [1] and references herein). One of he mos imoran field of such roblems arises in he models of nonlinear viscoelasiciy. In absence of he nonlinear diffusion erm u k 2 u, and when he funcion g vanishes idenically (i.e. g = ), hen he equaion (1.1) can be reduced o he following equaion u Δu = u 2 u, (1.4) A relaed roblem o he equaion (1.4) has araced grea aenions in he las wo decades, and many resuls aeared on he exisence, blow-u and asymoic behavior of soluions. I How o cie his aricle: Jian Dang, Qingying Hu, Suxia Xia, and Hongwei Zhang, Exonenial Growh of Soluion for a Class of Reacion Diffusion Equaion wih Memory and Mulile Nonlineariies, Research in Alied Mahemaics, vol. 1, Aricle ID 11258, 9 ages, 217. doi: /217/11258 Page 1

2 Research in Alied Mahemaics is well known ha he nonlinear reacion erm u 2 u drives he soluion of (1.4) o blow u in finie ime and he diffusion erm is known o yield exisence of global soluion if he reacion erm is removed from he equaion [2]. The more general equaion u div ( u m 2 u) = f(u), (1.5) has also araced a grea deal of eole. The obained resuls show ha global exisence and nonexisence deend roughly on m, he degree of nonlineariy in f, he dimension n, and he size of he iniial daa. In his regard, see he works of Levine [3], Kalanarov and Ladyzhenskaya [4], Levine e al. [5], Messaoudi [6], Liu e al. [7], Ayouch [8] and references herein. Pucci and Serrin [9] discussed he sabiliy of he following equaion u l 2 u div ( u m 2 u) = f(u), (1.6) Levine e al. [5] go he global exisence and nonexisence of soluion for (1.6). Pang e. al [1, 11] and Berrimi [12] gave he sufficien condiion of blow-u resul for cerain soluions of (1.6) wih osiive or negaive iniial energy. When g =, he class of equaion (1.1) can also be as a secial case of doubly nonlinear arabolic-ye equaions (or he orous medium equaion) [5, 13] β(u) Δu = u 2 u (1.7) if we ake β(u) = u + u m 2 u. Such equaion lay an imoran role in hysics and biology. I should be noed ha quesions of solvabiliy, local and global in ime, asymoic behavior and blow-u of iniial boundary value roblems and iniial value roblems for equaion of he ye (1.7) were invesigaed by many auhors. We only menion he work [13, 14] for his class equaion. We should also oin ou ha Pola [15] esablished a blow-u resul for he soluion wih vanishing iniial energy of he following iniial boundary value roblem u u xx + u m 2 u = u 2 u. They also gave deailed resuls of he necessary and sufficien blow-u condiions ogeher wih blow-u rae esimaes for he osiive soluion of he roblem (u m ) Δu = f(u), subjeced o various boundary condiions. Korusov [16, 17] have been obained sufficien condiions for he blowu of a finie ime and solvabiliy for he following generalized Boussinesq equaion u Δu Δu + u m 2 u = u(u α)(u β), (1.8) wih iniial boundary value (1.2) and (1.3) in R 3 for α, β > by concaviy mehod [3, 4]. The resul is exended by recen aer [18 21]. In absence of he nonlinear diffusion erm u k 2 u, and wih he resence of he viscoelasic erm (i.e. g ), he equaion (1.1) becomes u Δu + g( s)δu(s)ds = u 2 u, (1.9) doi: /217/11258 Page 2

3 Research in Alied Mahemaics i arises from he sudy of hea conducion in maerials wih memory. In his case, Messaoudi [22, 23] obained a blow-u resul for cerain soluions wih osiive or negaive iniial energy and Giorgi [24] go he asymoic behavior. Furhermore, for he quasilinear case u l 2 u Δu + g( s)δu(s)ds = u 2 u, (1.1) Messaoudi e al. [25, 26] esablished a general decay resul from which he usual exonenial and olynomial decay resuls are jus only secial cases for (1.1) wihou reacion erm u 2 u. Liu e al. [27] obained a general decay of he energy funcion for he global soluion and a blow-u resul for he soluion wih boh osiive and negaive iniial energy for (1.1). In his aer, we will invesigae roblem (1.1) (1.3), which has few resul of he roblem o our knowledge. We will rove ha L -norm of he soluion grows as an exonenial funcion. An essenial ool o he roof is an idea used in [28, 29], which based on an auxiliary funcion (a small erurbaion of he oal energy), using a differenial inequaliy and obaining he resul. This resul exend he early aer. This aricle is organized as follows. Secion 2 is concerned wih some noaions and saemen of assumions. In Secion 3, we give he main resul. 2. Preliminaries In his secion, we will give some noaions and saemen of assumions for m,, g. We denoe L (Ω) by L and H 1 (Ω) by H 1,he usual Soblev sace. The norm and inner of L (Ω) are denoed by = L (Ω) and (u, v) = Ω u(x)v(x)dx resecively. Esecially, = L 2 (Ω) for = 2. For he relaxaion funcion g and he number m and, we assume ha (A1) g R + R + is a differeniable funcion saisfying + g() >, 1 g(s)ds = l > ; (A2) here exiss a nonincreasing funcion ξ R + R + such ha (A3) we also assume ha 2 < k < g (s) ξ(s)g(s); 2(n 1), if n 3; 2 < k < < +, if n = 1, 2. n 2 Similar o [15, 27], we call u(x, ) a weak soluion o roblem (1.1) (1.3) on Ω [, T), if saisfying u(x, ) = u (x) and u C (, T; H 1 ) C 1 (, T; L 2 ), u k 2 u L 2 (Ω [, T)) s Ω [ u(s) v(s) g(s τ) u(τ) v(τ)dτ + u (s)v(s) + u k 2 u v u 2 uv dxds =, v C(, T; H 1 ] ), [, T). doi: /217/11258 Page 3

4 Research in Alied Mahemaics In his aer, we always assume ha roblem (1.1) (1.3) exiss a local soluion (see [16, 17]). Now, we inroduce wo funcionals E() = E(u) = 1 2 u (g u)() 1 2 g(s)ds u 2 1 u, (2.1) E() = 1 2 u 2 1 u, (2.2) where u H 1 and (g v)() = g( s) v(s) v() 2 ds. Mulilying Equaion (1.1) by u and inegraing over Ω, by (A2)(g () ), we have E () = u (g u)() u k 2 u 2 dx 1 Ω 2 g() u 2. (2.3) 3. Exonenial Growh of Soluion In his secion, we rove he main resul. Our echnique is differen from ha in [5, 15, 27 29] because of he resence of nonlinear erm u k 2 u and he memory erm. Theorem 3.1. Suose ha assumions (A1), (A2) and (A3) hold, u H 1 and u is a local soluion of he sysem (1.1) (1.3), and E() <. Furhermore, we assume ha Then he soluion of he sysem (1.1) (1.3) grows exonenially. Proof. We se By he definiion of H() and (2.3) Consequenly, by E() <, we have Noing ha g(s)ds < 2 1. (3.1) H() = E(). (3.2) H () = E (). (3.3) H() = E() >. (3.4) H() 1 1 u = [ 2 u (g u)() 1 g(s)ds u 2 <, (3.5) 2 ] hen (3.3), (3.4) and (3.5) imly < H() H() 1 u. (3.6) doi: /217/11258 Page 4

5 Research in Alied Mahemaics Le us define he funcion By aking he ime derivaive of (3.7) and by (1.1), we have L () = H () + ε Ω uu dx L() = H() + ε 2 u 2. (3.7) = u (g u)() + Ω u k 2 u 2 dx g() u 2 +ε u ε u 2 + ε Ω ε Ω u k 2 uu dx g( s) u() u(s)dsdx (3.8) u 2 ε u 2 + ε u + ε Ω + Ω u k 2 u 2 dx ε Ω u k 2 uu dx. g( s) u() u(s)dsdx To esimae he las erm in he righ-hand side of (3.8), by using he following Young s inequaliy we have Therefore, combining wih we have Ω ab δ 1 a 2 + δb 2, Ω u k 2 uu dx = Ω u k 2 2 u u k 2 2 udx g( s) u() u(s)dsdx δ 1 Ω u k 2 u 2 dx + δ Ω u k dx. = g(s)ds u() 2 + g( s) u() Ω ( u(s) u()) dxds 1 2 g(s)ds u() 2 1 (g u)(), 2 L () u ε g(s)ds 1 u 2 + ε u ( 2 ) (3.9) εδ u k k + (1 εδ 1 ) u k 2 u 2 dx ε Ω 2 (g u). By using u = H() + 2 ( 1 g(s)ds u 2 + (g u), ) 2 doi: /217/11258 Page 5

6 Research in Alied Mahemaics (3.9) becomes L () u ε g(s)ds 1 u 2 ( 2 ) +ε [ H() + 2 (g u) + 2 ( 1 g(s)ds ) u 2 ] εδ u k k + (1 εδ 1 ) u k 2 u 2 dx ε (g u) Ω 2 (3.1) u 2 + (1 εδ 1 ) Ω u k 2 u 2 dx +εa 1 u 2 + εa 2 (g u) + εh() εδ u k k, where a 1 = g(s)ds + 2 embedding heorem, we have > by (3.1), a 2 = 2 1 >. Noing ha > k > 2 and u k k C u k C ( u k ), (3.11) where C > is a osiive embedding consan. Since < k < 1, now alying he inequaliy x l (x + 1) (1 + 1 )(x + z), which holds for all x, l 1, z >, in aricular, aking z x = u, l = k, z = H(), we obain hen from (3.6) and (3.11) we have ( u k ) ( H()) ( u + H()), u k k C u k C 1 u, L () u 2 + (1 εδ 1 ) Ω u k 2 u 2 dx +εa 1 u 2 + εa 2 (g u) + εh() εδc 1 u, (3.12) Taking < a 3 < min{a 1, a 2 }, and by 2H() u 2 (g u) + 2 u, we have L () u 2 + (1 εδ 1 ) Ω u k 2 u 2 dx + ε(a 1 a 3 ) u 2 + εh() +ε (a 2 a 3) (g u) + ε ( 2 a 2 δc1 ) u +εa 3 [ u 2 + (g u) 2 u ] (3.13) = u 2 + (1 εδ 1 ) Ω u k 2 u 2 dx + ε (a 1 a 3) u 2 +ε (a 2 a 3) (g u) + ε ( 2 a 3 δc1 ) u + ε ( 2a 3) H(). doi: /217/11258 Page 6

7 Research in Alied Mahemaics Taking δ small enough such ha 2 a 3 δc 1 >, and hen aking ε small enough such ha 1 εδ 1 >, and noing ha 2a 3 >, hen L () C 2 (H() + u 2 + u 2 + (g u) + u ). (3.14) On he oher hand, by he definiion of H(), we ge L () H() + u 2 C 3 (H() + u 2 ) C 3 (H() + u 2 + u 2 + (g u) + u ). From (3.14) and (3.15), we obain he differenial inequaliy (3.15) Inegraion of (3.16) beween and gives us From (3.8) and wih ε small enough, we have By (3.17) and (3.18), we deduce L () rl(). (3.16) L() L()ex (r). (3.17) L() H() 1 u. (3.18) u Cex (r). Therefore, we conclude ha he soluion in he L -norm growhs exonenially. Remark. By he same mehod (similar o [29]), we can also ge he similar resul o roblem (1.1) (1.3) wih osiive iniial energy. Concluding In his aer, he iniial boundary value roblem of a class of reacion diffusion equaion wih memory and mulile nonlineariies is considered. Using a differenial inequaliy, exonenial growh of soluion wih L -norm is roved for negaive and osiive iniial energy. Comeing Ineress The auhors declare no comeing ineress. Acknowledgemen This work is suored by Naional Naural Science Foundaion of China (No ). References [1] Z. Jiang, S. Zheng, and X. Song, Blow-u analysis for a nonlinear diffusion equaion wih nonlinear boundary condiions, Alied Mahemaics Leers. An Inernaional Journal of Raid Publicaion, vol. 17, no. 2, , 24. doi: /217/11258 Page 7

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9 Research in Alied Mahemaics [28] S. Gerbi and B. Said-Houari, Local exisence and exonenial growh for a semilinear damed wave equaion wih dynamic boundary condiions, Advances in Differenial Equaions, vol. 13, no , , 28. [29] B. Said-Houari, Exonenial growh of osiive iniial-energy soluions of a sysem of nonlinear viscoelasic wave equaions wih daming and source erms, Z. Angew. Mah. Phys, vol. 62, no. 1, , 211. doi: /217/11258 Page 9