UNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS

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1 Electronic Journal of Differential Equations, Vol , No. 7, pp ISSN: URL: or ftp ejde.math.txstate.edu UNIFORM DECAY OF SOLUTIONS FOR COUPLED VISCOELASTIC WAVE EQUATIONS JIANGHAO HAO, LI CAI Abstract. In this article, we consider a system of two coupled viscoelastic equations with Dirichlet boundary conditions. By using the perturbed energy method, we obtain a general decay result which depends on the behavior of the relaxation functions and source terms. 1. Introduction In this article, we study the coupled system of quasi-linear viscoelastic equations u t ρ u tt u u tt v t ρ v tt v v tt g 1 t s usds f 1 u, v =, g t s vsds f u, v =, x, t,, x, t,, u = v =, x, t,, 1.1 ux, = u x, u t x, = u 1 x, x, vx, = v x, v t x, = v 1 x, x, where is a bounded domain in R n n 1 with smooth boundary, ρ satisfies < ρ n, n 3, ρ >, n = 1,. 1. The functions u, u 1, v, v 1 are given initial data. The functions g 1, g, f 1, f will be specified later. The study of the asymptotic behavior of viscoelastic problems has attracted lots of interest of researchers. The pioneer work of Dafermos [4] studied a onedimensional viscoelastic problem, established some existence and asymptotic stability results for smooth monotone decreasing relaxation functions. Muñoz Rivera [18] considered equations for linear isotropic viscoelastic solids of integral type, and established exponential decay and polynomial decay in a bounded domain and in 1 Mathematics Subject Classification. 35L5, 35L, 35L7, 93D15. Key words and phrases. Coupled viscoelastic wave equations; relaxation functions; uniform decay. c 16 Texas State University. Submitted August 1, 15. Published March 15, 16. 1

2 J. HAO, L. CAI EJDE-16/7 the whole space respectively. Messaoudi [1] considered a nonlinear viscoelastic wave equation with source and damping terms u tt u gt s usds u t u t m 1 = u u p He established blow-up result for solutions with negative initial energy and m < p, and gave a global existence result for arbitrary initial if m p. This work was later improved by Messaoudi [13]. Liu [1] considered the equation with initial-boundary value conditions u tt u gt τ uτdτ ax u t m u t b u r u =, 1.4 he established exponential or polynomial decay result which depends on the rate of the decay of the relaxation function g. Song et al [3] studied the problem 1.4 with replacing ax u t m u t by axu t, they obtained general decay result. Cavalcanti et al [3] discussed the wave equation u tt u gt τdiv[ax uτ]dτ bxfu t = 1.5 on a compact Riemannian manifold M, g subject to a combination of locally distributed viscoelastic and frictional dissipations. It is shown that the solutions decay according to the law dictated by the decay rates corresponding to the slowest damping. Muñoz Rivera and Naso [19] studied a viscoelastic systems with nondissipative kernels, and showed that if the kernel function decays exponentially to zero, then the solution decays exponentially to zero. On the other hand, if the kernel function decays polynomially as t p, then the corresponding solution also decays polynomially to zero with the same rate of decay. Wu [4] considered the equation u t ρ u tt u u tt gt s usds u t = u p u, 1.6 he also improved some results to obtain the decay rate of the energy under the suitable conditions. Cavalcanti et al [] discussed a quasilinear initial-boundary value problem of equation u t ρ u tt u u tt gt s uτdτ γ u t = bu u p, 1.7 with Dirichlet boundary condition, where ρ >, γ, p, b =. An exponential decay result for γ > and b = has been obtained. For γ = and b >, Messaoudi and Tatar [16], [17] showed that there exists an appropriate set, called stable set, such that if the initial data are in stable set, the solution continuous to live there forever, and the solution approaches zero with an exponential or polynomial rate depending on the decay rate of relaxation function. For other related single wave equation, we refer the reader to [7, 14, 1]. Han and Wang [6] studied the initial-boundary value problem for a coupled system of nonlinear viscoelastic equations u tt u g 1 t τ uτdτ u t m 1 u t = f 1 u, v, x, t, T,

3 EJDE-16/7 UNIFORM DECAY OF SOLUTIONS 3 v tt v g t τ vτdτ v t m 1 v t = f u, v, x, t, T, u = v =, x, t, T, 1.8 ux, = u x, u t x, = u 1 x, x, vx, = v x, v t x, = v 1 x, x. Existence of local and global solutions, uniqueness, and blow up in finite time were obtained when f 1, f, g 1, g and the initial values satisfy some conditions. Messaoudi and Said-Houari [15] dealt with the problem 1.8 and proved a global nonexistence of solutions for a large class of initial data for which the initial energy takes positive values. Also, Said-Houari et al [] discussed 1.8 and proved a general decay result. Liu [9] studied the coupled equations u tt u v tt v gt s ux, sds f 1 u, v =, ht s vx, sds f u, v =, 1.9 he proved that the decay rate of the solution energy is similar to that of relaxation functions which is not necessarily of exponential or polynomial type. Others similar problems were considered in [1, ]. Motivated by the above researches, we consider the system 1.1. Liu [8] already considered the system 1.1, and obtained the exponential or polynomial decay of the solutions energy depending on the decay rate of the relaxation functions. In [8], the relaxation functions g i t i = 1, satisfy g i t ξ ig pi i t for all t, p i [1, 3/ and some constants ξ 1, ξ. In this paper, the conditions have been replaced by g i t ξ itg i t where ξ i t are positive non-increasing functions. This allow us to obtain a general decay rate than just exponential or polynomial type. We use the perturbed energy method to obtain a general decay of solutions energy. The rest of this article is organized as follows. Some preparation and main result are given in Section. In Section 3, we give the proof of our main result.. Preliminaries and statement of main results We denote the norm in L ρ by ρ, 1 ρ <. The Dirichlet norm in H 1 is. C and C i denote general constants, which may be different in different estimates. Throughout this paper, we use the following notation, φ ψt = φt τ ψt ψτ dτ. To state our main result, we need the following assumptions. A1 g i : R R, i = 1,, are differentiable functions such that g i >, 1 g i sds = l i >, and there exist non-increasing functions ξ 1, ξ : R R satisfying g it ξ i tg i t, t.

4 4 J. HAO, L. CAI EJDE-16/7 A There exists nonnegative function F u, v such that F u, v f 1 u, v =, f u, v = u and there exist constants C, d > such that uf 1 u, v vf u, v CF u, v, F u, v, v f 1 u, v d u v p 1 u p 1 v p, f u, v d u v p 1 u p v p 1, where p > if n = 1, and < p n 1 n if n 3. By using the Galerkin method, as in [11], we can obtain the existence of a local weak solution to 1.1. We omit the proof here. Theorem.1. Assume that A1, A hold. For the initial data u, v, u 1, v 1 H 1 4, there exists at least one weak local solution u, v such that for some T >, u, v L, T ; H 1, u t, v t L, T ; H 1, u tt, v tt L, T ; H 1. We introduce the energy functional of system 1.1, Et = 1 ρ u t ρ ρ 1 ρ v t ρ ρ 1 u t 1 v t 1 g 1 u 1 g v 1 1 It is easy to prove that 1 1 g sds v F u, vdx. g 1 sds u.1 E t = 1 g 1 ut 1 g vt 1 g 1t ut 1 g t v.. Then we have Our main result reads as follows. u t v t l 1 u l v E..3 Theorem.. Assume that A1, A hold. Let u, v, u 1, v 1 H 1 4 be given, and u, v be the solution to 1.1. Then for any t 1 > there exist positive constants C and α such that for all t t 1, where ξt = min {ξ 1 t, ξ t}. Et Ce α R t t 1 ξτdτ, 3. Decay result To prove the general decay result, we define the perturbed modified energy functional Lt = MEt εit Jt, where M and ε are positive constants to be specified later and It = 1 u t ρ u t udx 1 v t ρ v t vdx u t udx v t vdx,

5 EJDE-16/7 UNIFORM DECAY OF SOLUTIONS 5 where J 1 t = J t = Firstly, we have the following lemmas. Jt = J 1 t J t, u t u t ρ u t g 1 t τ ut uτ dτ dx, v t v t ρ v t g t τ vt vτ dτ dx. Lemma 3.1 [5]. Under assumption A1, if u, v is the solution of 1.1, then the following hold for i = 1, : dx g i t τ ut uτ dτ Ci g i u, 3.1 dx g it τ ut uτ dτ Ci g i u. 3. Lemma 3.. Let A1, A hold and u, v be the solution of 1.1. Then I t l 1 u C 1 4 g 1 u 1 u t ρ ρ u t l v C 4 g v 1 v t ρ ρ v t C F u, vdx, in which = min{ l1, l }. Proof. Differentiating It and using 1.1, we obtain I t = 1 u t ρ ρ u ut g 1 t τ uτ dτ dx 1 v t ρ ρ v vt g t τ vτ dτ dx uf 1 vf dx u t v t Using 3.1 and Young s inequality, we can estimate the third term of 3.4 as follows ut g 1 t τ uτ dτ dx = u = u u u g 1 t τ uτ ut ut dτ dx g 1 τdτ u g 1 τdτ u 1 4 g 1 τdτ u C 1 4 g 1 u. gt τ uτ ut dτ dx dx g 1 t τ uτ ut dτ 3.5

6 6 J. HAO, L. CAI EJDE-16/7 Similarly, we obtain vt v From , we obtain I t 1 u t g t τ vτ dτ dx g τdτ v C 4 g v. g 1 τdτ u C v t ρ ρ v t 4 g 1 u 1 u t ρ ρ g τdτ v C 4 g v uf 1 vf dx. We can choose = min{ l1, l }. From 3.7 and A1, 3.3 follows Lemma 3.3. Under assumptions A1, A, we have J t 1 l C v 1 l 1 C u 3C 4 C 3C1 g v 4 C 1 g 1 u C 4 C g C1 v 4 4 C 1 g 1 u 4 g sds Eρ v t g 1 sds Eρ u t 1 g sds v t ρ ρ 1 g 1 sds u t ρ ρ. Proof. Differentiating J 1 t and using 1.1, we obtain J 1t = ut g 1 t τ ut uτ dτ dx g 1 t τ uτdτ f 1 u, v g 1 t τ ut uτ dτ dx g 1 t τ ut uτ dτ dx g 1 sds u t 3.8 u t g 1t τ ut uτ dτ dx u t ρ u t g 1t τ ut uτ dτ dx 1 g 1 sds u t ρ ρ.

7 EJDE-16/7 UNIFORM DECAY OF SOLUTIONS 7 By using Young s inequality and 3.1, we obtain that for some >, ut g 1 t τ ut uτ dτ dx u C 1 4 g 1 u. 3.1 For the second term of 3.9, employing Young s inequality, A1 and 3.1, we have for some >, g 1 t τ uτdτ g 1 t τ ut uτ dτ dx 1 4 g 1 t τ uτ ut dτ 1 4 g 1 t τ ut uτ dτ dx dx g 1 t τ ut uτ dτ ut g 1 t τdτ dx 1 4 C 1g 1 u 1 l 1 u. g 1 t τ ut dτ dx 3.11 Thanks to Young s inequality, Sobolev embedding theorem and 3.1, for some > we have f 1 u, v g 1 t τ ut uτ dτ dx f1 u, vdx 1 dx gt τ ut uτ dτ 4 f1 u, vdx C 1 4 g 1 u f1 u, vdx C 1 4 g 1 u. Using A and the Sobolev embedding theorem and.3, we have f1 u, vdx C u v p 1 dx u p v p dx C u p 1 p 1 v p 1 p 1 u p np v p np n 1 u p C u v p v u p 3 C u v p 1 v E p C u E p l v 1 l E p 3 C u E p 1 l v 1 l C u v.

8 8 J. HAO, L. CAI EJDE-16/7 Then we obtain f 1 u, v g 1 t τ ut uτ dτ dx C u v C 1 4 g 1 u. 3.1 The fifth term of 3.9 yields u t g 1 ut uτ dτ dx u t C 1 4 g 1 u We estimate the sixth term of 3.9 by using Young s inequality, Sobolev embedding theorem and.3 as follows 1 t u t ρ u t g 1t τ ut uτ dτ dx u t ρ1 ρ1 C 1 4 g 1 u Eρ u t C 1 4 g 1 u Inserting into 3.9, we obtain J 1t 1 l 1 C u C v 3C1 4 C C1 1 g 1 u 4 C 1 g 1 u 4 g 1 sds Eρ u t 1 g 1 sds u t ρ ρ In the same way, we conclude that J t 1 l C v C u 3C 4 C C g v 4 C g v 4 g sds Eρ v t 1 g sds v t ρ ρ. Combining the estimates 3.15 and 3.16, we can obtain

9 EJDE-16/7 UNIFORM DECAY OF SOLUTIONS 9 Proof of Theorem.1. It is not difficult to find positive constants a 1, a such that a 1 Et Lt a Et. Differentiating Lt, we have L t 1 g 1 sds ε u t ρ ρ g 1 sds Eρ ε u t 1 g sds ε v t ρ ρ g sds Eρ ε v t 3C1 4 C 1 C 1ε 3.17 g 1 u 4 3C 4 C C ε g v 4 M g 1t l 1 ε 1 l 1 C u M g t l ε 1 l C v Cε F u, vdx. For any t > we can pick ε, > small enough, M so large such that for t > t there exist constants η 1, η, η 3, η 4 >, and L t η 1 u t ρ ρ v t ρ ρ η u t v t η 3 g 1 u g v η 4 u v εc Then, we can choose t 1 > t such that η, C > and 3.18 takes the form F u, vdx L t ηet C g 1 u g v, t t Multiplying 3.19 by ξt, by using A1 we have ξtl t C C C C ξ 1 t τg 1 t τ ut uτ dτ dx CE t ηξtet. ξ t τg t τ vt vτ dτ dx ηξtet g 1t τ ut uτ dτ dx g t τ vt vτ dτ dx ηξtet where ξt = min {ξ 1 t, ξ t}. Thanks to A1, we obtain 3. d dt ξtlt CEt ηξtet, t t

10 1 J. HAO, L. CAI EJDE-16/7 By defining the functional we have Then integrating over t 1, t, we have F t := ξtlt CEt Et, 3. F t αξtf t. 3.3 F t F t 1 e α R t t 1 ξτdτ. By using 3. again, the decay result follows. Acknowledgements. The authors want to thank the anonymous referees for their valuable comments and suggestions which lead to the improvement of this paper. This work was partially supported by NNSF of China , NSF of Shanxi Province , Shanxi Scholarship council of China 13-13, Shanxi international science and technology cooperation projects References [1] K. Agre, M. Rammaha; Systems of nonlinear wave equations with damping and source terms, Differ. Integral Equ. 19 6, [] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira; Existence and uniform decay for nonlinear viscoelastic equation with strong damping, Math. Meth. Appl. Sci. 41, [3] M. M. Cavalcanti, V. N. Domingos Cavalcanti, I. Lasiecka, F. A. F. Nascimento; Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects, Discrete Contin. Dyn. Syst. Ser. B , [4] C.M. Dafermos; Asymptotic stability in viscoelasticy, Arch. Ration. Mech. Anal , [5] B. Feng, Y. Qin, M. Zhang; General decay for a system of nonlinear viscoelastic wave equations with weak damping, Bound. Value Probl , [6] X. Han, M. Wang; Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal. TMA, 71 9, [7] M. Kafini; On the uniform decay in Cauchy viscoelastic problems, Afr. Mat. 3 1, [8] W. Liu; Uniform decay of solutions for a quasilinear system of viscoelastic equations, Nonlinear Anal. TMA. 71 9, [9] W. Liu; General decay of solutions of a nonlinear system of viscoelastic equations, Acta. Appl. Math. 11 1, [1] W. Liu; Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping, Appl. Math. Comput. 3 1, [11] W. Liu, J. Wu; Global existence and uniform decay of solutions for a coupled system of nonlinear viscoelastic wave equations with not necessarily differentiable relaxation functions, Appl. Math , [1] S. A. Messaoudi; Blow up and global existence in a nonlinear viscoelsatic wave equation, Math. Nachr. 6 3, [13] S. A. Messaoudi; Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl. 3 6, [14] S. A. Messaoudi,; General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng , [15] S. A. Messaoudi, B. Said-Houari; Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms, J. Math. Anal. Appl , [16] S. A. Messaoudi, N. E. Tatar; Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Math. Meth. Appl. Sci. 3 7, [17] S. A. Messaoudi, N. E. Tatar; Exponential and Polynomial Decay for a Quasilinear Viscoelastic Equation, Nonlinear Anal. TMA. 68 7,

11 EJDE-16/7 UNIFORM DECAY OF SOLUTIONS 11 [18] J. E. Muñoz Rivera; Asymptotic behavior in linear viscoelasticity, Quart. Appl. Math , [19] J. E. Muñoz Rivera, M.G. Naso; On the decay of the energy for systems with memory and indefinite dissipation, Asymptot. Anal , [] M. L. Mustafa; Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations, Nonlinear Anal. RWA. 13 1, [1] J. Y. Park, J. R. Kang; Global existence and uniform decay for a nonlinear viscoelastics equation with damping, Acta. Appl. Math, 11 1, [] B. Said-Houari, S. A. Messaoudi, A. Guesmia; General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonlinear Diff. Equ. Appl , [3] X. Song, R. Zeng, C. Mu; General decay of solutions in a viscoelastic equation with nonlinear localized damping, Math. Res. Appl. 3 1, [4] S. Wu; General decay of energy for a viscoelastic equation with damping and source terms, Taiwan. J. Math. 16 1, Jianghao Hao corresponding author School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 36, China address: hjhao@sxu.edu.cn Li Cai School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 36, China address: @qq.com