\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 72, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/72\hfil Uniform decay of solutions] {Uniform decay of solutions for coupled viscoelastic wave equations} \author[J. Hao, L. Cai \hfil EJDE-2016/72\hfilneg] {Jianghao Hao, Li Cai} \address{Jianghao Hao (corresponding author)\newline School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{hjhao@sxu.edu.cn} \address{Li Cai \newline School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{2829742149@qq.com} \thanks{Submitted August 21, 2015. Published March 15, 2016.} \subjclass[2010]{35L05, 35L20, 35L70, 93D15} \keywords{Coupled viscoelastic wave equations; relaxation functions; \hfill\break\indent uniform decay} \begin{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. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we study the coupled system of quasi-linear viscoelastic equations \begin{gather} |u_{t}|^{\rho}u_{tt}-\Delta u-\Delta u_{tt} +\int^{t}_{0}g_1(t-s)\Delta u(s)ds+f_1(u,v)=0, \quad (x, t)\in \Omega\times (0,\infty), \nonumber \\ |v_{t}|^{\rho}v_{tt}-\Delta v-\Delta v_{tt} +\int^{t}_{0}g_2(t-s)\Delta v(s)ds+f_2(u,v)=0, \quad (x, t)\in \Omega\times (0,\infty), \nonumber\\ u=v=0, \quad (x, t)\in \partial\Omega\times(0,\infty), \label{e1.1}\\ u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_1(x),\quad x\in \bar{\Omega}, \nonumber\\ v(x,0)=v_{0}(x),\quad v_{t}(x,0)=v_1(x),\quad x\in \bar{\Omega}, \nonumber \end{gather} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ $(n\geq 1)$ with smooth boundary $\partial\Omega$, $\rho$ satisfies \begin{equation} \begin{gathered} 0<\rho\leq\frac{2}{n-2},\quad n\geq3,\\ \rho>0,\quad n=1,2. \end{gathered} \label{e1.2} \end{equation} The functions $u_{0},u_1,v_{0},v_1$ are given initial data. The functions $g_1$, $g_2$, $f_1$, $f_2$ 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 \cite{D} studied a one-dimensional viscoelastic problem, established some existence and asymptotic stability results for smooth monotone decreasing relaxation functions. Mu\~noz Rivera \cite{R} considered equations for linear isotropic viscoelastic solids of integral type, and established exponential decay and polynomial decay in a bounded domain and in the whole space respectively. Messaoudi \cite{M3} considered a nonlinear viscoelastic wave equation with source and damping terms \begin{equation} u_{tt}- \Delta u+\int_0^t g(t-s)\Delta u(s) ds+u_t|u_t|^{m-1}=u|u|^{p-1}. \label{e1.3} \end{equation} He established blow-up result for solutions with negative initial energy and $m0$, $\gamma\geq 0$, $p\geq 2$, $b=0$. An exponential decay result for $\gamma>0$ and $b=0$ has been obtained. For $\gamma=0$ and $b>0$, Messaoudi and Tatar \cite{MT2}, \cite{MT-2} 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 \cite{K, M-5, PK}. Han and Wang \cite{HW} studied the initial-boundary value problem for a coupled system of nonlinear viscoelastic equations \begin{gather} u_{tt}-\Delta u+\int_0^t g_1(t-\tau)\Delta u(\tau)d\tau+|u_t|^{m-1}u_t=f_1(u,v), \quad (x,t)\in\Omega\times(0,T), \nonumber \\ v_{tt}-\Delta v+\int_0^t g_2(t-\tau)\Delta v(\tau)d\tau+|v_t|^{m-1}v_t=f_2(u,v), \quad (x,t)\in\Omega\times(0,T), \nonumber \\ u=v=0, \quad (x,t)\in\partial\Omega\times(0,T), \label{e1.8}\\ u(x,0)=u_0(x),u_t(x,0)=u_1(x), \quad x\in\Omega, \nonumber \\ v(x,0)=v_0(x),v_t(x,0)=v_1(x), \quad x\in\Omega. \nonumber \end{gather} Existence of local and global solutions, uniqueness, and blow up in finite time were obtained when $f_1$, $f_2$, $g_1$, $g_2$ and the initial values satisfy some conditions. Messaoudi and Said-Houari \cite{MH} dealt with the problem \eqref{e1.8} 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 \cite{SMG-3} discussed \eqref{e1.8} and proved a general decay result. Liu \cite{L-3} studied the coupled equations \begin{equation} \begin{gathered} u_{tt}-\Delta u+\int_{0}^{t}g(t-s)\Delta u(x,s)ds+f_1(u,v)=0, \\ v_{tt}-\Delta v+\int_{0}^{t}h(t-s)\Delta v(x,s)ds+f_2(u,v)=0, \end{gathered} \label{e1.9} \end{equation} 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 \cite{AR, M--}. Motivated by the above researches, we consider the system \eqref{e1.1}. Liu \cite{L-4} already considered the system \eqref{e1.1}, and obtained the exponential or polynomial decay of the solutions energy depending on the decay rate of the relaxation functions. In \cite{L-4}, the relaxation functions $g_i(t)$ $(i=1,2)$ satisfy $g_i'(t)\leq -\xi_i g_i^{p_i}(t)$ for all $t\geq 0$, $p_i\in [1, 3/2)$ and some constants $\xi_1$, $\xi_2$. In this paper, the conditions have been replaced by $g_i'(t)\leq -\xi_i(t) g_i(t)$ where $\xi_i(t)$ 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 2. In Section 3, we give the proof of our main result. \section{Preliminaries and statement of main results} We denote the norm in $L^{\rho}(\Omega)$ by $\|\cdot\|_{\rho}, 1\leq \rho< \infty$. The Dirichlet norm in $H_{0}^{1}(\Omega)$ is $\|\nabla\cdot\|_2$. $C$ and $C_i$ denote general constants, which may be different in different estimates. Throughout this paper, we use the following notation, $$ (\phi\circ\psi )(t)=\int^{t}_{0}\phi(t-\tau)\|\psi(t)-\psi(\tau)\|^2_2d\tau. $$ To state our main result, we need the following assumptions. \begin{itemize} \item[(A1)] $g_{i}: R^{+}\to R^{+},i=1,2$, are differentiable functions such that $$ g_{i}(0)>0,\quad 1-\int^{+\infty}_{0}g_{i}(s)ds=l_{i}>0, $$ and there exist non-increasing functions $\xi_1,\xi_2: R^{+}\to R^{+}$ satisfying $$ g_{i}'(t)\leq-\xi_{i}(t)g_{i}(t),\quad t\geq 0. $$ \item[(A2)] There exists nonnegative function $F(u,v)$ such that $$ f_1(u,v)=\frac{\partial F(u, v)}{\partial u},\quad f_2(u,v)=\frac{\partial F(u, v)}{\partial v}, $$ and there exist constants $C,d>0$ such that \begin{gather*} uf_1(u,v)+vf_2(u,v)\geq CF(u,v),\\ |f_1(u,v)|\leq d(|u+v|^{p-1}+|u|^{\frac{p}{2}-1}|v|^{\frac{p}{2}}), \\ |f_2(u,v)|\leq d(|u+v|^{p-1}+|u|^{\frac{p}{2}}|v|^{\frac{p}{2}-1}), \end{gather*} where $p> 2$ if $n=1,2$ and $2< p\leq\frac{2(n-1)}{n-2}$ if $n\geq 3$. \end{itemize} By using the Galerkin method, as in \cite{LW}, we can obtain the existence of a local weak solution to \eqref{e1.1}. We omit the proof here. \begin{theorem} \label{thm2.1} Assume that {\rm (A1), (A2)} hold. For the initial data $(u_0, v_0, u_1, v_1)\in (H_0^1(\Omega))^4$, there exists at least one weak local solution $(u,v)$ such that for some $T>0$, $$ u, v\in L^\infty (0,T; H_0^1(\Omega)), \quad u_t, v_t\in L^\infty (0,T; H_0^1(\Omega)), \quad u_{tt}, v_{tt}\in L^2 (0,T; H_0^1(\Omega)). $$ \end{theorem} We introduce the energy functional of system \eqref{e1.1}, \begin{equation} \begin{aligned} E(t)&=\frac{1}{\rho+2}\|u_{t}\|^{\rho+2}_{\rho+2} +\frac{1}{\rho+2}\|v_{t}\|^{\rho+2}_{\rho+2} +\frac{1}{2}\|\nabla u_{t}\|^2_2+\frac{1}{2}\|\nabla v_{t}\|^2_2\\ &\quad+\frac{1}{2}(g_1\circ \nabla u) +\frac{1}{2}(g_2\circ \nabla v)+\frac{1}{2}\Big(1-\int^{t}_{0}g_1(s)ds\Big) \|\nabla u\|^2_2\\ &\quad +\frac{1}{2}\Big(1-\int^{t}_{0}g_2(s)ds\Big)\|\nabla v\|^2_2 +\int_{\Omega}F(u,v)dx. \end{aligned}\label{e2.1} \end{equation} It is easy to prove that \begin{equation} E'(t)=\frac{1}{2}(g'_1\circ\nabla u)(t )+\frac{1}{2}(g'_2\circ\nabla v)(t)-\frac{1}{2}g_1(t)\|\nabla u(t)\|^2_2 -\frac{1}{2}g_2(t)\|\nabla v\|^2_2\leq 0.\label{e2.2} \end{equation} Then we have \begin{equation} \|\nabla u_{t}\|^2_2+\|\nabla v_{t}\|^2_2 +l_1\|\nabla u\|^2_2+l_2\|\nabla v\|^2_2\leq 2E(0).\label{e2.3} \end{equation} Our main result reads as follows. \begin{theorem} \label{thm2.2} Assume that {\rm (A1), (A2)} hold. Let $(u_{0}, v_{0}, u_1, v_1)\in (H^{1}_{0}(\Omega))^4$ be given, and $(u, v)$ be the solution to \eqref{e1.1}. Then for any $t_1>0$ there exist positive constants $C$ and $\alpha$ such that for all $t\geq t_1$, $$ E(t)\leq Ce^{-\alpha\int_{t_1}^{t}\xi(\tau)d\tau}, $$ where $\xi(t)=\min\left\{\xi_1(t),\xi_2(t)\right\}$. \end{theorem} \section{Decay result} To prove the general decay result, we define the perturbed modified energy functional $$ L(t)=ME(t)+\varepsilon I(t)+J(t), $$ where $M$ and $\varepsilon$ are positive constants to be specified later and \begin{gather*} I(t)=\frac{1}{\rho+1}\int_{\Omega}|u_{t}|^{\rho}u_{t}udx +\frac{1}{\rho+1}\int_{\Omega}|v_{t}|^{\rho}v_{t}vdx +\int_{\Omega}\nabla u_{t}\nabla udx+\int_{\Omega}\nabla v_{t}\nabla vdx, \\ J(t)=J_1(t)+J_2(t), \end{gather*} where \begin{gather*} J_1(t)=\int_{\Omega}\Big(\Delta u_{t}-\frac{|u_{t}|^{\rho}u_{t}}{\rho+1}\Big) \int_{0}^{t}g_1(t-\tau)\left(u(t)-u(\tau)\right)\,d\tau\,dx, \\ J_2(t)=\int_{\Omega}\Big(\Delta v_{t}-\frac{|v_{t}|^{\rho}v_{t}}{\rho+1}\Big) \int_{0}^{t}g_2(t-\tau)\big(v(t)-v(\tau)\big)\,d\tau\,dx. \end{gather*} Firstly, we have the following lemmas. \begin{lemma}[\cite{FQZ}] \label{lem3.1} Under assumption {\rm (A1)}, if $(u,v)$ is the solution of \eqref{e1.1}, then the following hold for $i=1,2$: \begin{gather} \int_{\Omega}\Big(\int^{t}_{0}g_{i}(t-\tau) \left(\nabla u(t)-\nabla u(\tau)\right)d\tau\Big)^2dx \leq C_{i}(g_{i}\circ\nabla u),\label{e3.1}\\ \int_{\Omega}\Big(\int^{t}_{0}-g_{i}^{'}(t-\tau)\left(\nabla u(t)-\nabla u(\tau) \right)d\tau\Big)^2dx \leq -C_{i}(g_{i}^{'}\circ\nabla u).\label{e3.2} \end{gather} \end{lemma} \begin{lemma} \label{lem3.2} Let {\rm (A1), (A2)} hold and $(u, v)$ be the solution of \eqref{e1.1}. Then \begin{equation} \begin{aligned} I'(t)&\leq-\frac{l_1}{2}\|\nabla u\|^2_2+\frac{C_1}{4\delta} (g_1\circ\nabla u)+\frac{1}{\rho+1}\|u_{t}\|^{\rho+2}_{\rho+2}\\ &\quad +\|\nabla u_{t}\|^2_2 -\frac{l_2}{2}\|\nabla v\|^2_2+\frac{C_2}{4\delta}(g_2\circ\nabla v) +\frac{1}{\rho+1}\|v_{t}\|^{\rho+2}_{\rho+2}+\|\nabla v_{t}\|^2_2\\ &\quad -C\int_{\Omega}F(u,v)dx, \end{aligned}\label{e3.3} \end{equation} in which $\delta=\min\{\frac{l_1}{2},\frac{l_2}{2}\}$. \end{lemma} \begin{proof} Differentiating $I(t)$ and using \eqref{e1.1}, we obtain \begin{equation} \begin{aligned} I'(t)&=\frac{1}{\rho+1}\|u_{t}\|^{\rho+2}_{\rho+2}-\|\nabla u\|^2_2 +\int_{\Omega}\nabla u(t)\int_{0}^{t}g_1(t-\tau)\nabla u(\tau)\,d\tau\,dx\\ &\quad+ \frac{1}{\rho+1}\|v_{t}\|^{\rho+2}_{\rho+2}-\|\nabla v\|^2_2 +\int_{\Omega}\nabla v(t)\int_{0}^{t}g_2(t-\tau)\nabla v(\tau)\,d\tau\,dx\\ &\quad -\int_{\Omega}(uf_1+vf_2)dx+\|\nabla u_{t}\|^2_2+\|\nabla v_{t}\|^2_2. \end{aligned}\label{e3.4} \end{equation} Using \eqref{e3.1} and Young's inequality, we can estimate the third term of \eqref{e3.4} as follows \begin{equation} \begin{aligned} &\int_{\Omega}\nabla u(t)\int_{0}^{t}g_1(t-\tau)\nabla u(\tau)\,d\tau\,dx\\ &=\int_{\Omega}\nabla u\int_{0}^{t}g_1(t-\tau) \left(\nabla u(\tau)-\nabla u(t)+\nabla u(t)\right)\,d\tau\,dx\\ &=\|\nabla u\|^2_2\int_{0}^{t}g_1(\tau)d\tau +\int_{\Omega}\nabla u\int^{t}_{0}g(t-\tau) \left(\nabla u(\tau)-\nabla u(t)\right)\,d\tau\,dx\\ &\leq \|\nabla u\|^2_2\int_{0}^{t}g_1(\tau)d\tau+\delta\|\nabla u\|^2_2 +\frac{1}{4\delta}\int_{\Omega} \Big(\int^{t}_{0}g_1(t-\tau)|\nabla u(\tau)-\nabla u(t)|d\tau\Big)^2dx\\ &\leq \|\nabla u\|^2_2\int_{0}^{t}g_1(\tau)d\tau+\delta\|\nabla u\|^2_2 +\frac{C_1}{4\delta}(g_1\circ\nabla u). \end{aligned}\label{e3.5} \end{equation} Similarly, we obtain \begin{equation} \begin{aligned} &\int_{\Omega}\nabla v(t)\int_{0}^{t}g_2(t-\tau)\nabla v(\tau)\,d\tau\,dx\\ &\leq\|\nabla v\|^2_2\int_{0}^{t}g_2(\tau)d\tau+\delta\|\nabla v\|^2_2 +\frac{C_2}{4\delta}(g_2\circ\nabla v). \end{aligned} \label{e3.6} \end{equation} From \eqref{e3.4}--\eqref{e3.6}, we obtain \begin{equation} \begin{aligned} I'(t) &\leq-\Big(1-\int_{0}^{t}g_1(\tau)d\tau-\delta\Big)\|\nabla u\|^2_2 +\frac{C_1}{4\delta}(g_1\circ\nabla u) +\frac{1}{\rho+1}\|u_{t}\|^{\rho+2}_{\rho+2}\\ &\quad +\|\nabla u_{t}\|^2_2 -\Big(1-\int_{0}^{t}g_2(\tau)d\tau-\delta\Big)\|\nabla v\|^2_2 +\frac{C_2}{4\delta}(g_2\circ\nabla v)\\ &\quad +\frac{1}{\rho+1}\|v_{t}\|^{\rho+2}_{\rho+2}+\|\nabla v_{t}\|^2_2 -\int_{\Omega}(uf_1+vf_2)dx. \end{aligned}\label{e3.7} \end{equation} We can choose $\delta=\min\{\frac{l_1}{2},\frac{l_2}{2}\}$. From \eqref{e3.7} and (A1), \eqref{e3.3} follows. \end{proof} \begin{lemma} \label{lem3.3} Under assumptions {\rm (A1), (A2)}, we have \begin{equation} \begin{aligned} J'(t) &\leq\left(\delta+2\delta(1-l_2)^2+2C\delta\right) \|\nabla v\|^2_2+\left(\delta+2\delta(1-l_1)^2+2C\delta\right)\|\nabla u\|^2_2\\ &\quad+\left(\frac{3C_2}{4\delta}+2\delta C_2\right)(g_2\circ\nabla v) +\left(\frac{3C_1}{4\delta}+2\delta C_1\right)(g_1\circ\nabla u)\\ &\quad+\left(\frac{C_2}{4\delta}+\frac{C_2}{4\delta(\rho+1)}\right) (g_2'\circ\nabla v)+\left(\frac{C_1}{4\delta}+\frac{C_1}{4\delta(\rho+1)}\right) (g_1'\circ\nabla u)\\ &\quad-\Big(\int_{0}^{t}g_2(s)ds-\delta-\frac{\delta}{\rho+1}\left(2E(0) \right)^{\rho}\Big)\|\nabla v_{t}\|^2_2 \\ &\quad -\Big(\int_{0}^{t}g_1(s)ds-\delta-\frac{\delta}{\rho+1}\left(2E(0) \right)^{\rho}\Big)\|\nabla u_{t}\|^2_2\\ &\quad-\frac{1}{\rho+1}\Big(\int_{0}^{t}g_2(s)ds\Big)\|v_{t}\|^{\rho+2}_{\rho+2} -\frac{1}{\rho+1}\Big(\int_{0}^{t}g_1(s)ds\Big)\|u_{t}\|^{\rho+2}_{\rho+2}. \end{aligned}\label{e3.8} \end{equation} \end{lemma} \begin{proof} Differentiating $J_1(t)$ and using \eqref{e1.1}, we obtain \begin{align} J_1'(t) &=\int_{\Omega}\nabla u(t)\Big(\int_{0}^{t}g_1(t-\tau) \left(\nabla u(t)-\nabla u(\tau)\right)d\tau\Big)dx \nonumber \\ &\quad-\int_{\Omega}\Big(\int_{0}^{t}g_1(t-\tau)\nabla u(\tau)d\tau\Big) \Big(\int_{0}^{t}g_1(t-\tau)\left(\nabla u(t)-\nabla u(\tau)\right)d\tau\Big)dx \nonumber \\ &\quad+\int_{\Omega}f_1(u,v)\int_{0}^{t}g_1(t-\tau)\left(u(t)-u(\tau)\right) \,d\tau\,dx -\Big(\int_{0}^{t}g_1(s)ds\Big)\|\nabla u_{t}\|^2_2 \nonumber \\ &\quad-\int_{\Omega}\nabla u_{t}\int_{0}^{t}g_1'(t-\tau) \left(\nabla u(t)-\nabla u(\tau)\right)\,d\tau\,dx \label{e3.9}\\ &\quad-\frac{1}{\rho+1}\int_{\Omega}|u_{t}|^{\rho}u_{t} \int_{0}^{t}g'_1(t-\tau)\left(u(t)-u(\tau)\right)\,d\tau\,dx \nonumber\\ &\quad -\frac{1}{\rho+1}\Big(\int_{0}^{t}g_1(s)ds\Big)\|u_{t}\|^{\rho+2}_{\rho+2}. \nonumber \end{align} By using Young's inequality and \eqref{e3.1}, we obtain that for some $\delta>0$, \begin{equation} \int_{\Omega}\nabla u(t) \Big(\int_{0}^{t}g_1(t-\tau)\left(\nabla u(t)-\nabla u(\tau)\right)d\tau\Big)dx \leq\delta\|\nabla u\|^2_2+\frac{C_1}{4\delta}(g_1\circ\nabla u). \label{e3.10} \end{equation} For the second term of \eqref{e3.9}, employing Young's inequality, (A1) and \eqref{e3.1}, we have for some $\delta>0$, \begin{equation} \begin{aligned} &\int_{\Omega}\Big(\int_{0}^{t}g_1(t-\tau)\nabla u(\tau)d\tau\Big) \Big(\int_{0}^{t}g_1(t-\tau)\left(\nabla u(t)-\nabla u(\tau)\right)d\tau\Big)dx\\ &\leq \delta\int_{\Omega}\Big(\int_{0}^{t}g_1(t-\tau)\left(\nabla u(\tau) -\nabla u(t)\right)d\tau+\int_{0}^{t}g_1(t-\tau)|\nabla u(t)|d\tau\Big)^2dx\\ &\quad+\frac{1}{4\delta}\int_{\Omega}\left(\int_{0}^{t}g_1(t-\tau)\left(\nabla u(t)-\nabla u(\tau)\right)d\tau\right)^2dx\\ &\leq (2\delta+\frac{1}{4\delta})\int_{\Omega} \Big(\int_{0}^{t}g_1(t-\tau)|\nabla u(t)-\nabla u(\tau)|d\tau\Big)^2dx\\ & \quad+2\delta\int_{\Omega} \Big(\int_{0}^{t}g_1(t-\tau)d\tau\Big)^2|\nabla u(t)|^2dx\\ &\leq (2\delta+\frac{1}{4\delta})C_1(g_1\circ\nabla u)+2\delta(1-l_1)^2 \|\nabla u\|^2_2. \end{aligned}\label{e3.11} \end{equation} Thanks to Young's inequality, Sobolev embedding theorem and \eqref{e3.1}, for some $\delta>0$ we have \begin{align*} &\int_{\Omega}f_1(u,v)\int_{0}^{t}g_1(t-\tau)\left(u(t)-u(\tau)\right)\,d\tau\,dx\\ &\leq \delta\int_{\Omega}f_1^2(u,v)dx +\frac{1}{4\delta}\int_{\Omega} \Big(\int^{t}_{0}g(t-\tau)\left(u(t)-u(\tau)\right)d\tau\Big)^2dx\\ &\leq\delta\int_{\Omega}f_1^2(u,v)dx+\frac{C_1}{4\delta}(g_1\circ u)\\ &\leq \delta\int_{\Omega}f_1^2(u,v)dx+\frac{C_1}{4\delta}(g_1\circ\nabla u). \end{align*} Using (A2) and the Sobolev embedding theorem and \eqref{e2.3}, we have \begin{align*} \int_{\Omega}f_1^2(u,v)dx &\leq C\Big(\int_{\Omega}|u+v|^{2(p-1)}dx+\int_{\Omega}|u|^{p-2}|v|^{p}dx\Big)\\ &\leq C\Big(\|u\|^{2(p-1)}_{2(p-1)}+\|v\|^{2(p-1)}_{2(p-1)}+\|u\|^{2(p-2)}_{n(p-2)} +\|v\|^{2p}_{\frac{np}{n-1}}\Big)\\ &\leq C\Big(\left(\|\nabla u\|^2_2\right)^{p-2}\|\nabla u\|^2_2 +\left(\|\nabla v\|^2_2\right)^{p-2}\|\nabla v\|^2_2\Big)\\ &\quad +C\Big(\left(\|\nabla u\|^2_2\right)^{p-3}\|\nabla u\|^2_2 +\left(\|\nabla v\|^2_2\right)^{p-1}\|\nabla v\|^2_2\Big)\\ &\leq C\Big(\Big(\frac{2E(0)}{l_1}\Big)^{p-2}\|\nabla u\|^2_2 +\Big(\frac{2E(0)}{l_2}\Big)^{p-2}\|\nabla v\|^2_2\Big)\\ &\quad +C\Big(\Big(\frac{2E(0)}{l_1}\Big)^{p-3}\|\nabla u\|^2_2 +\Big(\frac{2E(0)}{l_2}\Big)^{p-1}\|\nabla v\|^2_2\Big)\\ &\leq C\left(\|\nabla u\|^2_2+\|\nabla v\|^2_2\right). \end{align*} Then we obtain \begin{equation} \begin{aligned} &\int_{\Omega}f_1(u,v)\int_{0}^{t}g_1(t-\tau)\left(u(t)-u(\tau)\right)\,d\tau\,dx\\ &\leq C\delta\left(\|\nabla u\|^2_2+\|\nabla v\|^2_2\right) +\frac{C_1}{4\delta}(g_1\circ\nabla u). \end{aligned}\label{e3.12} \end{equation} The fifth term of \eqref{e3.9} yields \begin{equation} \int_{\Omega}\nabla u_{t}\int_{0}^{t}g_1' \left(\nabla u(t)-\nabla u(\tau)\right)\,d\tau\,dx \leq\delta\|\nabla u_{t}\|^2_2+\frac{C_1}{4\delta}(g_1'\circ\nabla u).\label{e3.13} \end{equation} We estimate the sixth term of \eqref{e3.9} by using Young's inequality, Sobolev embedding theorem and \eqref{e2.3} as follows \begin{equation} \begin{aligned} &\frac{1}{\rho+1}\int_{\Omega}|u_{t}|^{\rho}u_{t}\int_{0}^{t}g'_1(t-\tau) \left(u(t)-u(\tau)\right)\,d\tau\,dx\\ &\leq \frac{\delta}{\rho+1}\|u_{t}\|^{2(\rho+1)}_{2(\rho+1)} +\frac{C_1}{4\delta(\rho+1)}(g_1'\circ\nabla u)\\ &\leq \frac{\delta}{\rho+1}\left(2E(0)\right)^{\rho}\|\nabla u_{t}\|^2_2 +\frac{C_1}{4\delta(\rho+1)}(g_1'\circ\nabla u). \end{aligned}\label{e3.14} \end{equation} Inserting \eqref{e3.10}--\eqref{e3.14} into \eqref{e3.9}, we obtain \begin{equation} \begin{aligned} J_1'(t) &\leq\left(\delta+2\delta(1-l_1)^2+C\delta\right)\|\nabla u\|^2_2 +C\delta\|\nabla v\|^2_2\\ &\quad+\Big(\frac{3C_1}{4\delta}+2\delta C_1\Big) (g_1\circ\nabla u)+\Big(\frac{C_1}{4\delta}+\frac{C_1}{4\delta(\rho+1)}\Big) (g_1'\circ\nabla u)\\ &\quad-\Big(\int_{0}^{t}g_1(s)ds-\delta-\frac{\delta}{\rho+1}\left(2E(0) \right)^{\rho}\Big)\|\nabla u_{t}\|^2_2\\ &\quad -\frac{1}{\rho+1}\Big(\int_{0}^{t}g_1(s)ds\Big)\|u_{t}\|^{\rho+2}_{\rho+2}. \end{aligned}\label{e3.15} \end{equation} In the same way, we conclude that \begin{equation} \begin{aligned} J_2'(t) &\leq\left(\delta+2\delta(1-l_2)^2+C\delta\right)\|\nabla v\|^2_2 +C\delta\|\nabla u\|^2_2\\ &\quad+\Big(\frac{3C_2}{4\delta}+2\delta C_2\Big)(g_2\circ\nabla v) +\Big(\frac{C_2}{4\delta}+\frac{C_2}{4\delta(\rho+1)}\Big)(g_2'\circ\nabla v)\\ &\quad-\Big(\int_{0}^{t}g_2(s)ds-\delta-\frac{\delta}{\rho+1}\left(2E(0) \right)^{\rho}\Big)\|\nabla v_{t}\|^2_2 \\ &\quad -\frac{1}{\rho+1}\Big(\int_{0}^{t}g_2(s)ds\Big)\|v_{t}\|^{\rho+2}_{\rho+2}. \end{aligned}\label{e3.16} \end{equation} Combining the estimates \eqref{e3.15} and \eqref{e3.16}, we can obtain \eqref{e3.8}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.1}] It is not difficult to find positive constants $a_1,a_2$ such that $a_1E(t)\leq L(t)\leq a_2E(t)$. Differentiating $L(t)$, we have \begin{equation} \begin{aligned} L'(t) &\leq-\frac{1}{\rho+1}\Big(\int_{0}^{t}g_1(s)ds-\varepsilon\Big) \|u_{t}\|^{\rho+2}_{\rho+2} \\ &\quad -\Big(\int_{0}^{t}g_1(s)ds-\delta-\frac{\delta}{\rho+1} \left(2E(0)\right)^{\rho} -\varepsilon\Big)\|\nabla u_{t}\|^2_2\\ &\quad-\frac{1}{\rho+1}\Big(\int_{0}^{t}g_2(s)ds-\varepsilon\Big) \|v_{t}\|^{\rho+2}_{\rho+2}\\ &\quad -\Big(\int_{0}^{t}g_2(s)ds-\delta-\frac{\delta}{\rho+1} \left(2E(0)\right)^{\rho} -\varepsilon\Big)\|\nabla v_{t}\|^2_2\\ &\quad+\Big(\frac{3C_1}{4\delta}+2\delta C_1+\frac{C_1\varepsilon}{4\delta}\Big) (g_1\circ\nabla u) \\ &\quad +\Big(\frac{3C_2}{4\delta}+2\delta C_2+\frac{C_2\varepsilon}{4\delta}\Big) (g_2\circ\nabla v)\\ &\quad-\Big(\Big(\frac{M}{2}g_1(t)+\frac{l_1}{2}\varepsilon\Big)-\delta -2\delta(1-l_1)^2-2C\delta\Big)\|\nabla u\|^2_2\\ &\quad-\Big(\Big(\frac{M}{2}g_2(t)+\frac{l_2}{2}\varepsilon\Big)-\delta -2\delta(1-l_2)^2-2C\delta\Big)\|\nabla v\|^2_2\\ &\quad -C\varepsilon\int_{\Omega}F(u,v)dx. \end{aligned}\label{e3.17} \end{equation} For any $t_{0}>0$ we can pick $\varepsilon,\delta>0$ small enough, $M$ so large such that for $t>t_{0}$ there exist constants $\eta_1, \eta_2, \eta_{3}, \eta_{4}>0$, and \begin{equation} \begin{aligned} L'(t) &\leq-\eta_1(\|u_{t}\|^{\rho+2}_{\rho+2}+\|v_{t}\|^{\rho+2}_{\rho+2}) -\eta_2(\|\nabla u_{t}\|^2_2+\|\nabla v_{t}\|^2_2)\\ &\quad +\eta_{3} \left((g_1\circ\nabla u)+(g_2\circ\nabla v)\right) -\eta_{4}(\|\nabla u\|^2_2+\|\nabla v\|^2_2)-\varepsilon C\int_{\Omega}F(u,v)dx. \end{aligned}\label{e3.18} \end{equation} Then, we can choose $t_1>t_{0}$ such that $\eta, C>0$ and \eqref{e3.18} takes the form \begin{equation} L'(t)\leq -\eta E(t)+C\left((g_1\circ\nabla u)+(g_2\circ\nabla v)\right), \quad t\geq t_1.\label{e3.19} \end{equation} Multiplying \eqref{e3.19} by $\xi(t)$, by using (A1) we have \begin{equation} \begin{aligned} &\xi(t)L'(t)\\ &\leq C\int_{\Omega}\int_{0}^{t}\xi_1(t-\tau)g_1(t-\tau)|\nabla u(t) -\nabla u(\tau)|^2\,d\tau\,dx\\ &\quad+C\int_{\Omega}\int_{0}^{t}\xi_2(t-\tau)g_2(t-\tau)|\nabla v(t) -\nabla v(\tau)|^2\,d\tau\,dx-\eta\xi(t)E(t)\\ &\leq-C\int_{\Omega}\int_{0}^{t}g_1'(t-\tau)|\nabla u(t) -\nabla u(\tau)|^2\,d\tau\,dx\\ &\quad-C\int_{\Omega}\int_{0}^{t}g_2'(t-\tau)|\nabla v(t) -\nabla v(\tau)|^2\,d\tau\,dx-\eta\xi(t)E(t)\\ &\leq-CE'(t)-\eta\xi(t)E(t). \end{aligned}\label{e3.20} \end{equation} where $\xi(t)=\min\left\{\xi_1(t),\xi_2(t)\right\}$. Thanks to (A1), we obtain \begin{equation} \frac{d}{dt}\left(\xi(t)L(t)+CE(t)\right)\leq-\eta\xi(t)E(t),\quad t\geq t_1.\label{e3.21} \end{equation} By defining the functional \begin{equation} F(t):=\xi(t)L(t)+CE(t)\sim E(t),\label{e3.22} \end{equation} we have \begin{equation} F'(t)\leq-\alpha\xi(t)F(t).\label{e3.23} \end{equation} Then integrating over $(t_1,t)$, we have $$ F(t)\leq F(t_1)e^{-\alpha\int_{t_1}^{t}\xi(\tau)d\tau}. $$ By using \eqref{e3.22} again, the decay result follows. \end{proof} \subsection*{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 (61374089), NSF of Shanxi Province (2014011005-2), Shanxi Scholarship council of China (2013-013), Shanxi international science and technology cooperation projects (2014081026). \begin{thebibliography}{00} \bibitem{AR} K. Agre, M. Rammaha; \emph{Systems of nonlinear wave equations with damping and source terms}, Differ. Integral Equ. 19 (2006), 1235-1270. \bibitem{CCF} M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira; \emph{Existence and uniform decay for nonlinear viscoelastic equation with strong damping}, Math. Meth. Appl. Sci. 24(2001), 1043-1053. \bibitem{CDLN} M. M. Cavalcanti, V. N. Domingos Cavalcanti, I. Lasiecka, F. A. F. Nascimento; \emph{Intrinsic decay rate estimates for the wave equation with competing viscoelastic and frictional dissipative effects}, Discrete Contin. Dyn. Syst. Ser. B 19(7) (2014), 1987-2012. \bibitem{D} C.M. Dafermos; \emph{Asymptotic stability in viscoelasticy}, Arch. Ration. Mech. Anal. 37 (1970), 297-308. \bibitem{FQZ} B. Feng, Y. Qin, M. Zhang; \emph{General decay for a system of nonlinear viscoelastic wave equations with weak damping}, Bound. Value Probl. 146 (2012), 1-11. \bibitem{HW} X. Han, M. Wang; \emph{Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source}, Nonlinear Anal. TMA, 71 (2009), 5427-5450. \bibitem{K} M. Kafini; \emph{On the uniform decay in Cauchy viscoelastic problems}, Afr. Mat. 23 (2012), 85-97. \bibitem{L-4} W. Liu; \emph{Uniform decay of solutions for a quasilinear system of viscoelastic equations}, Nonlinear Anal. TMA. 71 (2009), 2257-2267. \bibitem{L-3} W. Liu; \emph{General decay of solutions of a nonlinear system of viscoelastic equations}, Acta. Appl. Math. 110 (2010), 153-165. \bibitem{L-1} W. Liu; \emph{Exponential or polynomial decay of solutions to a viscoelastic equation with nonlinear localized damping}, Appl. Math. Comput. 32 (2010), 59-68. \bibitem{LW} W. Liu, J. Wu; \emph{Global existence and uniform decay of solutions for a coupled system of nonlinear viscoelastic wave equations with not necessarily differentiable relaxation functions}, Appl. Math. 127 (2011), 315-344. \bibitem{M3} S. A. Messaoudi; \emph{Blow up and global existence in a nonlinear viscoelsatic wave equation}, Math. Nachr. 260 (2003), 58-66. \bibitem{M-1} S. A. Messaoudi; \emph{Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation}, J. Math. Anal. Appl. 320 (2006), 902-915. \bibitem{M-5} S. A. Messaoudi,; \emph{General decay of solutions of a weak viscoelastic equation}, Arab. J. Sci. Eng. 36 (2011), 1569-1579. \bibitem{MH} S. A. Messaoudi, B. Said-Houari; \emph{Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms}, J. Math. Anal. Appl. 365 (2010), 277-287. \bibitem{MT2} S. A. Messaoudi, N. E. Tatar; \emph{Global existence and uniform stability of solutions for a quasilinear viscoelastic problem}, Math. Meth. Appl. Sci. 30 (2007), 665-680. \bibitem{MT-2} S. A. Messaoudi, N. E. Tatar; \emph{Exponential and Polynomial Decay for a Quasilinear Viscoelastic Equation}, Nonlinear Anal. TMA. 68 (2007), 785-793. \bibitem{R} J. E. Mu\~noz Rivera; \emph{Asymptotic behavior in linear viscoelasticity}, Quart. Appl. Math. 52(4) (1994), 628-648. \bibitem{RN} J. E. Mu\~noz Rivera, M.G. Naso; \emph{On the decay of the energy for systems with memory and indefinite dissipation}, Asymptot. Anal. 49(3-4) (2006), 189-204. \bibitem{M--} M. L. Mustafa; \emph{Well posedness and asymptotic behavior of a coupled system of nonlinear viscoelastic equations}, Nonlinear Anal. RWA. 13 (2012), 452-463. \bibitem{PK} J. Y. Park, J. R. Kang; \emph{Global existence and uniform decay for a nonlinear viscoelastics equation with damping}, Acta. Appl. Math, 110 (2010), 1393-1406. \bibitem{SMG-3} B. Said-Houari, S. A. Messaoudi, A. Guesmia; \emph{General decay of solutions of a nonlinear system of viscoelastic wave equations}, Nonlinear Diff. Equ. Appl. 18 (2011), 659-684. \bibitem{SZM} X. Song, R. Zeng, C. Mu; \emph{General decay of solutions in a viscoelastic equation with nonlinear localized damping}, Math. Res. Appl. 32 (2012), 53-62. \bibitem{W} S. Wu; \emph{General decay of energy for a viscoelastic equation with damping and source terms}, Taiwan. J. Math. 16 (2012), 113-128. \end{thebibliography} \end{document}