\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 $m<p$,
and gave a global existence result for arbitrary initial if $m\geq p$.
This work was later improved by Messaoudi \cite{M-1}.

Liu \cite{L-1} considered the  equation with initial-boundary value
conditions
\begin{equation}
u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau)\Delta u(\tau)d\tau+a(x)|u_{t}|^{m}u_{t}
+b|u|^{r}u=0,\label{e1.4}
\end{equation}
he established exponential or polynomial decay result which depends on the
 rate of the decay of the relaxation function $g$. Song et al \cite{SZM}
studied the problem \eqref{e1.4} with replacing $a(x)|u_{t}|^{m}u_{t}$
by $a(x)u_{t}$, they obtained general decay result.

Cavalcanti et al \cite{CDLN} discussed the wave equation
\begin{equation}
u_{tt}-\Delta u+\int_0^t g(t-\tau)div[a(x)\nabla u(\tau)]d\tau+b(x)f(u_t)=0
\label{e1.5}
\end{equation}
on a compact Riemannian manifold $(M, \mathbf{g})$ 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.

Mu\~noz Rivera and Naso \cite{RN} 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 \cite{W} considered the equation
\begin{equation}
|u_{t}|^{\rho}u_{tt}-\Delta u-\Delta u_{tt}+\int_{0}^{t}g(t-s)\Delta u(s)ds+u_{t}
=|u|^{p-2}u,\label{e1.6}
\end{equation}
he also improved some results to obtain the decay rate of the energy under
the suitable conditions.

 Cavalcanti et al \cite{CCF} discussed a quasilinear initial-boundary value
problem of equation
\begin{equation}
{|u_t|}^\rho u_{tt}-\Delta u-\Delta u_{tt}
+\int_0^t g(t-s)\Delta u(\tau)d\tau-\gamma\Delta u_t=bu|u|^{p-2},
\label{e1.7}
\end{equation}
with Dirichlet boundary condition, where $\rho>0$, $\gamma\geq 0$,
$p\geq 2$, $b=0$. 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).


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\end{document}