\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 78, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/78\hfil Blow-up of solutions]
{Blow-up of solutions to a coupled quasilinear viscoelastic wave system
with nonlinear damping and source}

\author[X. Zhang, S. Chai, J. Wu \hfil EJDE-2017/78\hfilneg]
{Xiaoying Zhang, Shugen Chai, Jieqiong Wu}

\address{Xiaoying Zhang \newline
School of Mathmatical Sciences, Shanxi University,
Taiyuan, Shanxi  030006, China. \newline
Department of Mathematics, Shanxi Agriculture University,
Taigu, Shanxi 030800,  China}
\email{zxybetter@163.com}

\address{Shugen Chai (corresponding author)\newline
School of Mathmatical Sciences, Shanxi University,
Taiyuan, Shanxi 030006,  China}
\email{sgchai@sxu.edu.cn, Phone +86-351-7010555, Fax +86-351-7010979}

\address{Jieqiong Wu \newline
School of Mathmatical Sciences, Shanxi University,
Taiyuan, Shanxi 030006,  China}
\email{ jieqiong@sxu.edu.cn}

\dedicatory{Communicated by Goong Chen}

\thanks{Submitted February 26, 2016. Published March 21, 2017.}
\subjclass[2010]{35A01, 35L53}
\keywords{Blow up; quasilinear wave system; viscoelasticity}

\begin{abstract}
 We study the blow-up of the solution to a quasilinear  viscoelastic wave
 system coupled by nonlinear sources. The system is of
 homogeneous Dirichlet boundary condition. The nonlinear damping and source
 are added to the equations.  We  assume that  the relaxation functions
 are non-negative non-increasing functions and the initial energy is
 negative. The competition relations among the nonlinear principal parts
 are not constant functions, the  viscoelasticity terms, dampings and
 sources are analyzed by using perturbed energy method.
 The blow-up result is proved under some conditions on the nonlinear principal
 parts, viscoelasticity terms,  dampings and sources by a contradiction argument.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Let  $\Omega$ be a bounded domain of $R^{n}(n\geq 1)$ with a smooth 
boundary $\partial \Omega$.  Consider the following nonlinear viscoelastic system
\begin{equation}\label{1.1}
\begin{gathered}
\begin{aligned}
&|u_t|^{\rho}u_{tt}-\operatorname{div}(\rho_1(|\nabla u|^2)\nabla u)
+\int_0^{t}g(t-\tau)\Delta u(x,\tau)d\tau+u_t+|u_t|^{m-1}u_t\\
&=f_1(u,v),\quad \Omega \times(0,T),\\
&|v_t|^{\rho}v_{tt}-\operatorname{div}(\rho_2(|\nabla v|^2)\nabla v)
+\int_0^{t}h(t-\tau)\Delta v(x,\tau)d\tau+v_t+|v_t|^{r-1}v_t \\
&=f_2(u,v),\quad \Omega \times(0,T),
\end{aligned}\\
u(x, t) =v(x,t)=0, \quad x\in \partial\Omega\times[0,T],\\
u(x, 0) = u_0(x), \quad  u_t(x, 0) = u_1(x),\quad x\in \Omega,\\
v(x, 0) =v_0(x), \quad  u_t(x, 0) =v_1(x),\quad x\in \Omega,
\end{gathered}
\end{equation}
where $\rho>0$, $m,r>1$ and $\rho_1,\rho_2,f_1,f_2,g,h$ are 
functions satisfying the following assumptions:
\begin{itemize}
\item[(A1)] $\rho_{i}(s)=b_1+b_2s^{q_{i}}$ with
$q_{i}\geq  0$ and $b_1, b_2>0$;  $\rho_{i}(s)>0$, for $s>0.$

\item[(A2)] The relaxation functions $g$ and $h$ are of class $C^{1}$ 
and satisfy, for $s\geq 0$,
\begin{gather*}
g(s)\geq 0,\quad b_1-\int_0^{\infty}g(s)ds=l>0,\quad g'(s)\leq  0,\\
h(s)\geq 0,\quad b_1-\int_0^{\infty}h(s)ds=k>0,\quad h'(s)\leq  0.
\end{gather*}

\item[(A3)] Let $F(u,v)=a|u+v|^{p+1}+2b|uv|^{\frac{p+1}{2}}$ with 
$a,b>0$, $1<p<\infty$ if $n=1,2$  and  $1<p<\frac{n}{n-2}$ if $n\geq  3$.
Assume that
$$
f_1(u,v)=\frac{\partial F}{\partial u},\quad
f_2(u,v)=\frac{\partial F}{\partial v},
$$
and that there are  positive constants $c_0, c_1$ such that
$$
c_0(|u|^{p+1}+|v|^{p+1})\leq F(u,v)\leq c_1(|u|^{p+1}+|v|^{p+1}).
$$
\end{itemize}
Many studies concerning existence of global solutions or their blow-up 
to system \eqref{1.1} with $\rho_i\equiv 1$  are available in the literature. 
 Georgiev and Todorova \cite{g1} considered the  single equation
\begin{equation}\label{1.2}
u_{tt}-\Delta u+u_t|u_t|^{m-1}=|u|^{p-1}u,\quad\text{in }
 \Omega \times(0,\infty),
\end{equation}
and  the interaction between the nonlinear damping and nonlinear source 
term. The authors showed that the solutions of the system with sufficient 
large initial data blow up in finite time if $p>m$.  
Messaoudi \cite{m1} extended the results of \cite{g1} to the case that the 
initial energy is negative.  Agre and Rammaha \cite{a1} extended the results of 
\cite{g1} by considering an initial-boundary value problem to the coupled 
wave equations.

In the presence of the viscoelastic term, Messaoudi \cite{m2} considered 
the nonlinear viscoelastic equation
\begin{equation}\label{1.3}
u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau+au_t|u_t|^{m-1}=b|u|^{p-1}u,
\quad \Omega \times(0,\infty),
\end{equation}
with initial conditions and Dirichlet boundary conditions. He proved that 
the weak solution with negative initial energy blew up if $p>m$  when $g$ 
satisfied some conditions.  Messaoudi \cite{m3} considered the blow-up solution
 of \eqref{1.3} with $a=1$, $b=1$ and  with small positive initial energy.  
Song \cite{s1} extended the results of \cite{m3} to the case that the initial 
energy is arbitrarily positive. For other related works on  the viscoelastic 
wave equation, we refer the reader to \cite{c1,c3,z1}.

Problem \eqref{1.1} with  $\rho>0$ has also been extensively studied.  
Song \cite{s2} investigated the nonexistence of global solutions to the
initial-boundary value problem of the following equation with positive 
initial energy
\begin{equation}\label{1.4}
|u_t|^{\rho}u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau+u_t|u_t|^{m-2}
=|u|^{p-2}u,\quad \Omega \times(0,\infty).
\end{equation}
Liu \cite{l1} studied  the general decay for the global solution  and blow-up 
of solution to the  equation
\begin{equation}\label{1.5}
|u_t|^{\rho}u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau
-\Delta u_{tt}=|u|^{p-2}u,\quad \Omega \times(0,\infty).
\end{equation}
Cavalcanti et al.\ \cite{c2} studied the energy decay for the nonlinear 
viscoelastic problem
\begin{equation}\label{1.6}
|u_t|^{\rho}u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau-\Delta u_{tt}
-\gamma \Delta u_t=0,\quad \Omega \times(0,\infty).
\end{equation}
A global existence result for $\gamma\geq  0$ as well as an exponential
decay for $\gamma>0$ was  established in \cite{c2}. When the source term 
$b|u|^{p-2}u$ appeared on the right side of system \eqref{1.6},
Messaoudi et al.\ \cite{m4} proved that the viscoelastic term was enough 
to ensure  existence and uniform decay of global solutions provided that 
the initial data were in some stable set.

For $\rho_{i}(s)=b_1+b_2s^{q_{i}}$ with $q_{i}\geq  0$ and
$b_1, b_2>0$, Wu et al.\ \cite{w1} and \cite{w2}  considered the
blow-up of the initial boundary value problem (spatial dimension $n=1,2,3$) 
for the  system
\begin{equation}\label{1.7}
\begin{gathered}
u_{tt}-\operatorname{div}(\rho_1(|\nabla u|^2)\nabla u)+u_t
+|u_t|^{m-1}u_t=f(u,v),\quad \Omega \times(0,T),\\
v_{tt}-\operatorname{div}(\rho_2(|\nabla v|^2)\nabla v)+v_t+|v_t|^{r-1}v_t=g(u,v),
\quad \Omega \times(0,T).
\end{gathered}
\end{equation}
For a single wave equation with $\rho_{i}(s)\geq b_1+b_2s^{q_{i}}$,
$q_{i}\geq  0$, $b_1, b_2>0$, Hao et al.\ \cite{h1}  studied the global
existence and blow up of the solutions.

We note that, in the literature mentioned above,  only viscoelastic term 
was included in the equation or only nonlinear principal part 
(i.e. $\rho_i, i=1,2$, are not constant functions) was included. 
To the best of our knowledge, there are no papers considering the blow-up 
of the equation with both viscoelastic term and  nonlinear principal part.
 The main goal of our paper is to prove that  
for $\rho_{i}(s)=b_1+b_2s^{q_{i}}$ the  nonlinear coupled source
terms still leads to blow-up of the solutions though there are viscoelastic 
terms in the equations. To be more precise, we prove that when 
$p>\max\{2q_1+1,2q_2+1\}$ and the relaxation functions satisfy that
$\max\{\int_0^{\infty}g(s)ds,\int_0^{\infty}h(s)ds\}<\frac{q}{q+1}b_1$, 
the solutions of the system will blow up. Our method  is borrowed partly 
from  \cite{l1,w1}, but  we must  overcome some additional difficulty  
caused by the complex interaction among the nonlinear viscoelastic terms,  
the nonlinear principal parts, the  coupled source terms and 
the nonlinear damping.

\section{Preliminaries}

In this section, we present some other assumptions and existence result of 
local solution.   We use the following assumptions: 
\begin{itemize}
\item[(A4)]  $\rho>0$  if $n=1,2$ and $0<\rho<\frac{2}{n-2}$  if $n\geq 3$.

\item[(A5)] $m<p$, $r<p$ and $\rho+2<p$.
\end{itemize}
Define the  energy function of the system \eqref{1.1} by
\begin{equation}\label{2.1}
\begin{aligned}
E(t)
&=\frac{1}{\rho+2}\Big(\|u_t\|_{\rho+2}^{\rho+2}+\|v_t\|_{\rho+2}^{\rho+2}\Big)
 +\frac{1}{2}\Big(b_1-\int_0^{t}g(s)ds\Big)\|\nabla u\|^2\\
&\quad +\frac{1}{2}\Big(b_1-\int_0^{t}h(s)ds\Big)\|\nabla v\|^2
 +\frac{1}{2}(g\circ\nabla u)(t) 
 +\frac{1}{2}(h\circ\nabla v)(t) \\
&\quad  +\frac{b_2}{2(q_1+1)}\|\nabla u\|_{2(q_1+1)}^{2(q_1+1)}
 +\frac{b_2}{2(q_2+1)}\|\nabla u\|_{2(q_2+1)}^{2(q_2+1)}
-\int_{\Omega}F(u,v)\,dx.
\end{aligned}
\end{equation}

Combining the arguments in  \cite{g1} and \cite{c2}, and
 making some slight modification, we  have the following existence of
local weak solutions.

\begin{theorem} \label{thm2.1}
 Let {\rm (A1)--(A4)} hold. Then for any initial data 
$u_0\in W_0^{1,2q_1+2}(\Omega)\cap L^{p+1}(\Omega)$, 
$v_0\in W_0^{1,2q_2+2}(\Omega)\cap L^{p+1}(\Omega)$, there exists a unique
local weak solution $(u,v)$ to the system \eqref{1.1} defined on $[0, T)$ 
for some $T>0$, and
\begin{gather*}
u\in L^{\infty}([0,T); W_0^{1,2q_1+2}(\Omega)\cap L^{p+1}(\Omega)), \\
v\in L ^{\infty}([0,T];W_0^{1,2q_2+2}(\Omega)\cap L^{p+1}(\Omega)),\\
u_t\in L^{\infty}([0,T); W_0^{1,2q_1+2}(\Omega)\cap L^{p+1}(\Omega)), \\
v_t\in L ^{\infty}([0,T];W_0^{1,2q_2+2}(\Omega)\cap L^{p+1}(\Omega))\\
u_{tt}\in L^{\infty}([0,T); L^2(\Omega)),\quad
v_{tt}\in L^{\infty}([0,T); L^2(\Omega))
\end{gather*}
\end{theorem}

Combining  the  arguments of \cite{g1,m3}, the following lemma  can be 
proved easily.

\begin{lemma}\label{lem2.1} 
Let {\rm (A1)--(A4)} hold.  And let $(u,v)$ be a solution of \eqref{1.1}.
 Then $E(t)$ satisfies the  inequality
\begin{equation} \label{2.2}
\begin{aligned}
E'(t) &= -\|u_t\|^2-\|u_t\|_{m+1}^{m+1}-\|v_t\|^2-\|v_t\|_{r+1}^{r+1}
 +\frac{1}{2}(g'\circ\nabla u)(t) \\
&\quad +\frac{1}{2}(h'\circ\nabla v)(t) -\frac{1}{2}g(t)\|\nabla u\|^2
 -\frac{1}{2}h(t)\|\nabla v\|^2\leq 0.
\end{aligned}
\end{equation}
\end{lemma}

\begin{lemma}[\cite{m1}] \label{lem2.2} 
Suppose $p$ satisfies {\rm (A3)}. Then there exists a positive constant 
$C(|\Omega|,p)$ such that
$$
\|u\|_{p+1}^{s}\leq C(|\Omega|,p)\Big(\|\nabla u\|^2+\|u\|_{p+1}^{p+1}\Big), \quad
 \forall u\in H_0^{1}(\Omega),
$$
where $2\leq s\leq p+1$.
\end{lemma}

In this article, we use $\|\cdot\|$ and $\|\cdot\|_{p}$ denote the usual 
$L^2(\Omega)$ norm and $L^{p}(\Omega)$ norm, respectively.
$B_1$ is the optimal constant of the Sobolev embedding
$H_0^{1}(\Omega)\hookrightarrow L^2(\Omega)$.

\section{Blow-up results}

In this section, we state and prove our main result.

\begin{theorem}\label{thm3.1} 
Let {\rm (A1)--(A5)} hold. $q=\max\{q_1,q_2\}$. Assume the initial energy
$E(0)<0$ and
$$
\max\Big\{\int_0^{\infty}g(s)ds,\int_0^{\infty}h(s)ds\Big\}<\frac{q}{q+1}b_1,\quad
 p>\max\{2q_1+1,2q_2+1\}.
$$
Then the solution of \eqref{1.1}  blows up at finite time.
\end{theorem}

\begin{proof}
 We use the contradiction method. 
Suppose that the solution  $(u,v)$ of \eqref{1.1} is global. Then
\begin{equation}\label{3.1}
\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2  +\|u\|_{p+1}^{p+1}
 +\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2  +\|v\|_{p+1}^{p+1}\leq C,
\quad \forall t\geq 0.
\end{equation}
Set $M_1=\max_{t\in [0,T]}\|u\|_{p+1}^{p+1}$, 
$M_2=\max_{t\in [0,T]}\|v\|_{p+1}^{p+1}$,
$M=M_1+M_2$. Let  $H(t)=-E(t)$.  Then by Lemma \ref{lem2.1}, the function 
$H(t)$ is increasing. Moreover,
from $E(0)<0$ and (A3), we obtain
\begin{equation}\label{3.2}
\begin{aligned}
0<H(0)&\leq H(t)\leq\int_{\Omega}F(u,v)\,dx\\
&\leq c_1\int_{\Omega}(|u|^{p+1}+|v|^{p+1})\,dx\\
&\leq c_1 \max_{t\in [0,T]}\int_{\Omega}|u|^{p+1}+|v|^{p+1}\,dx
 =c_1M.
\end{aligned}
\end{equation}
Let us introduce the  auxiliary function 
\begin{equation}\label{3.3}
L(t)=H^{1-\sigma}(t)+\frac{\varepsilon}{\rho+1}
\Big(\int_{\Omega}|u_t|^{\rho}u_tu\,dx+\int_{\Omega}|v_t|^{\rho}v_tv\,dx\Big),
\end{equation}
where $0<\varepsilon\ll 1$  and
\begin{equation}\label{3.4}
0<\sigma<\min\Big\{\frac{1}{\rho+2}-\frac{1}{p},\frac{p-m}{m(p+1)},
\frac{p-r}{r(p+1)}\Big\}.
\end{equation}
By differentiating $L(t)$, we obtain
\begin{equation}\label{3.5}
\begin{aligned}
&L'(t) \\
&=(1-\sigma)H^{\sigma}(t)H'(t)+\frac{\varepsilon}{\rho+1}
 \Big(\int_{\Omega}|u_t|^{\rho+2}\,dx+\int_{\Omega}|v_t|^{\rho+2}\,dx\Big)\\
&\quad +\varepsilon\Big(\int_{\Omega}|u_t|^{\rho}u_{tt}u\,dx
 +\int_{\Omega}|v_t|^{\rho}v_{tt}v\,dx\Big)\\
&=(1-\sigma)H^{\sigma}(t)H'(t)+\frac{\varepsilon}{\rho+1}
 \Big(\|u_t\|_{\rho+2}^{\rho+2}+\|u_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad -\varepsilon\int_{\Omega}(\rho_1(|\nabla u|^2)|\nabla u|^2
 +\rho_2(|\nabla v|^2)|\nabla v|^2)\,dx\\
&\quad +\varepsilon\int_{\Omega}\int_0^{t}g(t-s)\nabla u(s)\cdot\nabla u(t)ds\,dx
 +\varepsilon\int_{\Omega}\int_0^{t}h(t-s)\nabla v(s)\cdot\nabla v(t)ds\,dx\\
&\quad -\varepsilon\int_{\Omega}(uu_t+vv_t+|u_t|^{m-1}u_tu+|v_t|^{r-1}v_tv)\,dx
+\varepsilon(p+1)\int_{\Omega}F(u,v)\,dx\\
&=(1-\sigma)H^{\sigma}(t)H'(t)+\frac{\varepsilon}{\rho+1}
 \Big(\|u_t\|_{\rho+2}^{\rho+2}+\|v_t\|_{\rho+2}^{\rho+2}\Big)
-\varepsilon b_1(\|\nabla u\|^2+\|\nabla v\|^2)\\
&\quad -\varepsilon b_2\Big(\|\nabla u\|_{2(q_1+1)}^{2(q_1+1)}+
\|\nabla v\|_{2(q_2+1)}^{2(q_2+1)}\Big)\\
&\quad +\varepsilon\int_{\Omega}\int_0^{t}g(t-s)\nabla u(s)\cdot\nabla u(t)ds\,dx
 +\varepsilon\int_{\Omega}\int_0^{t}h(t-s)\nabla v(s)\cdot\nabla v(t)ds\,dx\\
&\quad -\varepsilon\int_{\Omega}(uu_t+vv_t+|u_t|^{m-1}u_tu+|v_t|^{r-1}v_tv)\,dx
+\varepsilon(p+1)\int_{\Omega}F(u,v)\,dx.
\end{aligned}
\end{equation}
Now, we estimate the fourth term on the right hand of \eqref{3.5}.
Let  $\mu=\min\{l,k\}$. From the the definition of $H(t)$,
it follows that
\begin{equation}\label{3.6}
\begin{aligned}
&- b_2 \|\nabla u\|_{2(q_1+1)}^{2(q_1+1)}-b_2\|\nabla v\|_{2(q_2+1)}^{2(q_2+1)}\\
&\geq - b_2\frac{(q+1)}{q_1+1}\|\nabla u\|_{2(q_1+1)}^{2(q_1+1)}-b_2\frac{(q+1)}{q_2+1}\|\nabla v\|_{2(q_2+1)}^{2(q_2+1)}\\
&= (q+1)\Big(2H(t)-2\int_{\Omega}F(u,v)\,dx
 +\frac{2}{\rho+2}\Big(\int_{\Omega}|u_t|^{\rho+2}\,dx
 +\int_{\Omega}|v_t|^{\rho+2}\,dx\Big)\\
&\quad +\Big(b_1-\int_0^{t}g(s)ds\Big)\|\nabla u\|^2
 +\Big(b_1-\int_0^{t}h(s)ds\Big)\|\nabla v\|^2 \\
&\quad +(g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big)\\
&\geq   2(q+1)H(t)+\frac{2(q+1)}{\rho+2}
 \Big(\|u_t\|_{\rho+2}^{\rho+2}+\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad -2(q+1)\int_{\Omega}F(u,v)\,dx +(q+1)\mu(\|\nabla u\|^2
 +\|\nabla v\|^2) \\
&\quad +(q+1)\Big((g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big).
\end{aligned}
\end{equation}
By H$\ddot{\rm {o}}$lder's  and Young's inequalities, we estimate the fifth 
term on the right hand of \eqref{3.5}. It yields 

\begin{equation}\label{3.7}
\begin{aligned}
&\int_{\Omega}\int_0^{t}g(t-s)\nabla u(s)\cdot\nabla u(t)ds\,dx\\
&=\int_{\Omega}\int_0^{t}g(t-s)\nabla u(t)\cdot(\nabla u(s)-\nabla u(t))ds\,dx
 +\int_0^{t}g(t-s)\|\nabla u(t)\|^2\\
&\geq-(g\circ\nabla u)(t)+\frac{3}{4}\int_0^{t}g(t-s)\|\nabla u(t)\|^2.
\end{aligned}
\end{equation}
Similarly, we obtain
\begin{equation}\label{3.8}
\int_{\Omega}\int_0^{t}h(t-s)\nabla v(s)\cdot\nabla v(t)ds\,dx
\geq-(h\circ\nabla v)(t)+\frac{3}{4}\int_0^{t}h(t-s)\|\nabla v(t)\|^2.
\end{equation}
Therefore, based on \eqref{3.6}, \eqref{3.7} and \eqref{3.8}, we conclude that

\begin{equation}\label{3.9}
\begin{aligned}
&L'(t)  \\
&\geq (1-\sigma)H^{\sigma}(t)H'(t)+\frac{\varepsilon}{\rho+1}
 \Big(\|u_t\|_{\rho+2}^{\rho+2}+\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad -\varepsilon b_1(\|\nabla u\|^2+\|\nabla v\|^2)
 +2\varepsilon(q+1)H(t)+\frac{2\varepsilon(q+1)}{\rho+1}
 \Big(\|u_t\|_{\rho+2}^{\rho+2}+\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad +\mu\varepsilon(q+1)(\|\nabla u\|^2+\|\nabla v\|^2)
 +\varepsilon(q+1)\Big((g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big) \\
&\quad +\varepsilon(p-2q-1)\int_{\Omega}F(u,v)\,dx
 -\varepsilon(g\circ\nabla u)(t)\\
&\quad +\frac{3}{4}\varepsilon\int_0^{t}g(s)ds
 \|\nabla u(t)\|^2-\varepsilon(h\circ\nabla v)(t)
 +\frac{3}{4}\varepsilon\int_0^{t}h(s)ds\|\nabla v(t)\|^2\\
&\quad -\varepsilon\int_{\Omega}(uu_t+vv_t+|u_t|^{m-1}u_tu+|v_t|^{r-1}v_tv)\,dx.
\end{aligned}
\end{equation}
Now we use Young's inequality and \eqref{2.2} to obtain the inequality
\begin{gather}\label{3.10}
\int_{\Omega}|u||u_t|\,dx\leq \frac{\varepsilon_1^2}{2}\|u\|^2
 +\frac{1}{2\varepsilon_1^2}\|u_t\|^2
\leq \frac{\varepsilon_1^2B_1}{2}\|\nabla u\|^2+\frac{1}{2\varepsilon_1^2}H'(t),\\
\label{3.11}
\int_{\Omega}|v||v_t|\,dx\leq \frac{\varepsilon_1^2}{2}\|v\|^2
 +\frac{1}{2\varepsilon_1^2}\|v_t\|^2
\leq \frac{\varepsilon_1^2B_1}{2}\|\nabla v\|^2+\frac{1}{2\varepsilon_1^2}H'(t), \\
\label{3.12}
\begin{aligned}
\int_{\Omega}|u_t|^{m-1}u_tu\,dx    
&\leq \frac{\delta_1^{m+1}}{m+1}\|u\|_{m+1}^{m+1}
 +\frac{m\delta_1^{-\frac{m+1}{m}}}{m+1}\|u_t\|_{m+1}^{m+1}\\
&\leq \frac{\delta_1^{m+1}}{m+1}\|u\|_{m+1}^{m+1}
 +\frac{m\delta_1^{-\frac{m+1}{m}}}{m+1}H'(t),
\end{aligned}\\
\label{3.13}
\begin{aligned}
\int_{\Omega}|v_t|^{r-1}v_tv\,dx    
&\leq \frac{\delta_2^{r+1}}{r+1}\|v\|_{r+1}^{r+1}
 +\frac{r\delta_2^{-\frac{r+1}{r}}}{r+1}\|v_t\|_{r+1}^{r+1}\\
&\leq \frac{\delta_2^{r+1}}{r+1}\|v\|_{r+1}^{r+1}
 +\frac{r\delta_2^{-\frac{r+1}{r}}}{r+1}H'(t),
\end{aligned}
\end{gather}
where $\varepsilon_1,\delta_1,\delta_2$ are constants depending on the time $t$ 
and are specified later.

Since $g$ and $h$ are positive, we have, for any $t>t_0>0$,
$$
\int_0^{t}g(s)ds\geq\int_0^{t_0}g(s)ds=:g_0>0, \quad
\int_0^{t}h(s)ds\geq\int_0^{t_0}h(s)ds=:h_0>0.
$$
Let $\chi=\min\Big\{\frac{3}{4}g_0,\frac{3}{4}h_0\Big\}$. 
Then $\chi>0$. By \eqref{3.10}--\eqref{3.13}, we obtain
\begin{equation}\label{3.14}
\begin{aligned}
L'(t)    
&\geq  \Big((1-\sigma)H^{\sigma}(t)
 -\frac{\varepsilon m\delta_1^{-\frac{m+1}{m}}}{m+1}
 -\frac{\varepsilon r\delta_2^{-\frac{r+1}{r}}}{r+1}
 -\frac{\varepsilon}{\varepsilon_1^2}\Big)H'(t)\\
&\quad +2\varepsilon(q+1)H(t)+\varepsilon\Big(\frac{1}{\rho+1}
 +\frac{2(q+1)}{\rho+2}\Big)\Big(\|u_t\|_{\rho+2}^{\rho+2}
 +\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad +\varepsilon\Big(\mu(q+1)-b_1-\frac{B_1\varepsilon_1^2}{2}
 +\chi\Big)(\|\nabla u\|^2+\|\nabla v\|^2)\\
&\quad +\varepsilon(p-2q-1)\int_{\Omega}F(u,v)\,dx
 -\varepsilon\Big(\frac{\delta_1^{m+1}}{m+1}\|u\|_{m+1}^{m+1}
 +\frac{\delta_2^{r+1}}{r+1}\|v\|_{r+1}^{r+1}\Big)\\
&\quad +\varepsilon q\Big((g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big).
\end{aligned}
\end{equation}
Let $\varepsilon_1^{-2}=K_1H^{-\sigma}$, 
$\delta_1^{-\frac{m+1}{m}}=K_2H^{-\sigma}$,
$\delta_2^{-\frac{r+1}{r}}=K_{3}H^{-\sigma}$,  where $K_1, K_2, K_{3}>0$ will be
chosen later. Then, by \eqref{3.2}, we obtain
\begin{gather}\label{3.15}
\delta_1^{m+1}=K_2^{-m}H^{\sigma m}(t)\leq K_2^{-m}c_1^{\sigma m}
(\|u\|_{p+1}^{p+1}+\|v\|_{p+1}^{p+1})^{\sigma m}, \\
\label{3.16}
\delta_2^{r+1}=K_{3}^{-r}H^{\sigma r}(t)\leq K_{3}^{-r}c_1^{\sigma r}
 (\|u\|_{p+1}^{p+1}+\|v\|_{p+1}^{p+1})^{\sigma r}.
\end{gather}
Hence,
\begin{equation}\label{3.17}
\begin{aligned}
L'(t)  
&\geq \Big((1-\sigma)H^{\sigma}(t)-\frac{\varepsilon m K_2H^{-\sigma}}{m+1}
 -\frac{\varepsilon r K_{3}H^{-\sigma}}{r+1}-\varepsilon K_1H^{-\sigma}\Big)H'(t)\\
&\quad +2\varepsilon(q+1)H(t)+\varepsilon\Big(\frac{1}{\rho+1}
 +\frac{2(q+1)}{\rho+2}\Big)\Big(\|u_t\|_{\rho+2}^{\rho+2}
 +\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad +\varepsilon\Big(\mu(q+1)-b_1-\frac{B_1\varepsilon_1^2}{2}
 +\chi\Big)(\|\nabla u\|^2+\|\nabla v\|^2)\\
&\quad +\varepsilon(p-2q-1)\int_{\Omega}F(u,v)\,dx 
 -\varepsilon\Big(\frac{K_2^{-m}c_1^{\sigma m}}{m+1}
 \big(\|u\|_{p+1}^{p+1} \\
&\quad +\|v\|_{p+1}^{p+1}\big)^{\sigma m}\|u\|_{m+1}^{m+1}
 +\frac{K_{3}^{-r}c_1^{\sigma r}}{r+1}(\|u\|_{p+1}^{p+1}
 +\|v\|_{p+1}^{p+1})^{\sigma r}\|v\|_{r+1}^{r+1}\Big)\\
&\quad +\varepsilon q\Big((g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big).
\end{aligned}
\end{equation}
By (A5) and the Sobolev embedding theorem, we have
\begin{gather}\label{3.18}
\|u\|_{m+1}^{m+1}\leq B_2\|u\|_{p+1}^{m+1}\leq B_2(\|u\|_{p+1}+\|v\|_{p+1})^{m+1},\\
\label{3.19}
\|v\|_{r+1}^{r+1}\leq B_{3}\|v\|_{p+1}^{r+1}\leq B_{3}(\|u\|_{p+1}+\|v\|_{p+1})^{r+1}.
\end{gather}
Using the inequality $(a+b)^{\lambda}\leq B_{4}(a^{\lambda}+b^{\lambda})$, we have
\begin{equation}\label{3.20}
\begin{aligned}
L'(t)    
&\geq  \Big((1-\sigma)H^{\sigma}(t)-\frac{\varepsilon m K_2H^{-\sigma}}{m+1}
 -\frac{\varepsilon r K_{3}H^{-\sigma}}{r+1}-\varepsilon K_1H^{-\sigma}\Big)H'(t)\\
&\quad +2\varepsilon(q+1)H(t)+\varepsilon\Big(\frac{1}{\rho+1}
 +\frac{2(q+1)}{\rho+2}\Big)\Big(\|u_t\|_{\rho+2}^{\rho+2}
 +\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad +\varepsilon\Big(\mu(q+1)-b_1-\frac{B_1\varepsilon_1^2}{2}
 +\chi\Big)(\|\nabla u\|^2+\|\nabla v\|^2)\\
&\quad +\varepsilon(p-2q-1)\int_{\Omega}F(u,v)\,dx \\
&\quad -\varepsilon\Big(\frac{K_2^{-m}B_{5}c_1^{\sigma m}}{m+1}
 (\|u\|_{p+1}+\|v\|_{p+1})^{\sigma m(p+1)+m+1}\\
&\quad +\frac{K_{3}^{-r}B_{6}c_1^{\sigma r}}{r+1}
 (\|u\|_{p+1}+\|v\|_{p+1})^{\sigma r(p+1)+r+1}\Big)\\
&\quad +\varepsilon q\Big((g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big)
\end{aligned}
\end{equation}
where $B_{5}=B_2B_{4},B_{6}=B_{3}B_{4}$.

If we set $s=\sigma m(p+1)+m+1$ and
$\sigma r(p+1)+r+1$,  then by Lemma \ref{lem2.2}, there exist two positive 
constants $B_{7},B_{8}$ depending on $|\Omega|,m, r$ such that
\begin{gather}\label{3.21}
\|u\|_{p+1}^{\sigma m(p+1)+m+1}\leq B_{7}(\|\nabla u\|^2+\|u\|_{p+1}^{p+1}),\\
\label{3.22}
\|v\|_{p+1}^{\sigma r(p+1)+r+1}\leq B_{8}(\|\nabla v\|^2+\|v\|_{p+1}^{p+1}).
\end{gather}
Thus
\begin{equation}\label{3.23}
\begin{aligned}
&L'(t)  \\
&\geq \Big((1-\sigma)H^{\sigma}(t)-\frac{\varepsilon m K_2H^{-\sigma}}{m+1}
 -\frac{\varepsilon r K_{3}H^{-\sigma}}{r+1}-\varepsilon K_1H^{-\sigma}\Big)H'(t)\\
&\quad +2\varepsilon(q+1)H(t)+\varepsilon\Big(\frac{1}{\rho+1}
 +\frac{2(q+1)}{\rho+2}\Big)\Big(\|u_t\|_{\rho+2}^{\rho+2}
 +\|v_t\|_{\rho+2}^{\rho+2}\Big)\\
&\quad +\varepsilon\Big(\mu(q+1)-b_1-\frac{B_1\varepsilon_1^2}{2}+\chi
 -\frac{K_2^{-m}B_{5}B_{7}c_1^{\sigma m}}{m+1}
 -\frac{K_{3}^{-r}B_{6}B_{8}c_1^{\sigma r}}{r+1}\Big) \\
&\quad\times \left(\|\nabla u\|^2+\|\nabla v\|^2\right)
 +\varepsilon\Big((p-2q-1)c_0-\frac{K_2^{-m}B_{5}B_{7}c_1^{\sigma m}}{m+1}\\
&\quad -\frac{K_{3}^{-r}B_{6}B_{8}c_1^{\sigma r}}{r+1}\Big) 
 (\|u\|_{p+1}^{p+1}+\|v\|_{p+1}^{p+1})
 +\varepsilon q\Big((g\circ\nabla u)(t)+(h\circ\nabla v)(t)\Big).
\end{aligned}
\end{equation}

Using the condition of Theorem \ref{thm3.1}, we obtain $\mu(q+1)-b_1>0$. 
Now, we can choose $K_1, K_2,K_{3}$ large enough so that the following 
inequalities hold:
\begin{equation}\label{3.24}
\begin{aligned}
&\mu(q+1)-b_1+\chi-\frac{B_1\varepsilon_1^2}{2}
-\frac{K_2^{-m}B_{5}B_{7}c_1^{\sigma m}}{m+1}
-\frac{K_{3}^{-r}B_{6}B_{8}c_1^{\sigma r}}{r+1}\\
&\geq \mu(q+1)-b_1+\chi-\frac{B_1M^{\sigma}}{2K_1}
 -\frac{K_2^{-m}B_{5}B_{7}c_1^{\sigma m}}{m+1}
 -\frac{K_{3}^{-r}B_{6}B_{8}c_1^{\sigma r}}{r+1}\\
&\geq \frac{\mu(q+1)-b_1}{2}
\end{aligned}
\end{equation}
and
\begin{equation}\label{3.25}
(p-2q-1)c_0-\frac{K_2^{-m}B_{5}B_{7}c_1^{\sigma m}}{m+1}
-\frac{K_{3}^{-r}B_{6}B_{8}c_1^{\sigma r}}{r+1}\geq\frac{(p-2q-1)c_0}{2}\,.
\end{equation}

Furthermore, for fixed $K_1, K_2,K_{3}$, $T_0\geq t_0$,  we choose
 $\varepsilon$ small enough such that
\begin{gather}\label{3.26}
(1-\sigma)-\frac{\varepsilon m K_2}{m+1}-\frac{\varepsilon r K_{3}}{r+1}
-\varepsilon K_1\geq 0, \\
\label{3.27}
\begin{aligned}
L(T_0)&=H^{1-\sigma}(T_0)+\frac{\varepsilon}{\rho+1}
\Big(\int_{\Omega}|u_t(T_0)|^{\rho}u_t(T_0)u(T_0)\,dx \\
&\quad +\int_{\Omega}|v_t(T_0)|^{\rho}v_t(T_0)v(T_0)\,dx\Big)>0.
\end{aligned}
\end{gather}
From the condition of Theorem \ref{thm3.1}, for $t>T_0$, we have
\begin{gather}\label{3.28}
\begin{aligned}
L'(t)
&\geq \varepsilon \gamma\Big[H(t)+\|u_t\|_{\rho+2}^{\rho+2}
 +\|\nabla u\|^2+\|u\|_{p+1}^{p+1} \\ 
&\quad  +\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2+\|v\|_{p+1}^{p+1}\Big],
\end{aligned} \\
\label{3.29}
L(t)\geq L(T_0)>0,
\end{gather}
where
\begin{equation}\label{3.30}
\gamma=\min\Big\{2(q+1),\Big(\frac{1}{\rho+1}+\frac{2(q+1)}{\rho+2}\Big),
 \frac{\mu(q+1)-b_1}{2}, \frac{(p-2q-1)c_0}{2}\Big\}.
\end{equation}

We now estimate $L(t)^{\frac{1}{1-\sigma}}$.
By H\"older's inequality and the condition (A5), we obtain
\begin{equation}\label{3.31}
\begin{array}{l}
\big|\int_{\Omega}|u_t|^{\rho}u_tu\,dx\big|\leq\|u_t\|_{\rho+2}^{\rho+1}\|u\|_{\rho+2}\leq B_{9}\|u_t\|_{\rho+2}^{\rho+1}\|u\|_{p+1}.
\end{array}
\end{equation}
Therefore,
\begin{equation}\label{3.32}
\Big|\int_{\Omega}|u_t|^{\rho}u_tu\,dx\Big|^{\frac{1}{1-\sigma}}
\leq B_{9}\|u_t\|_{\rho+2}^{\frac{\rho+1}{1-\sigma}}
\|u\|_{p+1}^{\frac{1}{1-\sigma}}\leq B_{10}(\|u_t\|_{\rho+2}^{\frac{\rho+1}{1-\sigma}
\mu}+\|u\|_{p+1}^{\frac{\theta}{1-\sigma}}),
\end{equation}
where $\frac{1}{\mu}+\frac{1}{\theta}=1$. Choosing 
$\mu=\frac{(1-\sigma)(\rho+2)}{\rho+1}>1$, we have
\begin{equation}\label{3.33}
2<\frac{\theta}{1-\sigma}=\frac{\rho+2}{(1-\sigma)(\rho+2)-(\rho+1)}<p+1.
\end{equation}
By Lemma \ref{lem2.2}, taking $s=\frac{\theta}{1-\sigma}$, it follows that
\begin{equation}\label{3.34}
\|u\|_{p+1}^{\frac{\theta}{1-\sigma}}\leq B_{11}(\|\nabla u\|^2+\|u\|_{p+1}^{p+1}).
\end{equation}
Hence
\begin{equation}\label{3.35}
\big|\int_{\Omega}|u_t|^{\rho}u_tu\,dx\big|^{\frac{1}{1-\sigma}}
\leq B_{12}\Big[\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2+\|u\|_{p+1}^{p+1}\Big].
\end{equation}
Similarly,
\begin{equation}\label{3.36}
\big|\int_{\Omega}|v_t|^{\rho}v_tv\,dx\big|^{\frac{1}{1-\sigma}}
\leq B_{13}\Big[\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2+\|v\|_{p+1}^{p+1}\Big].
\end{equation}
Hence, combining \eqref{3.2}, \eqref{3.35} and \eqref{3.36}, we easily get
\begin{equation}\label{3.37}
\begin{aligned}
L(t)^{\frac{1}{1-\sigma}}    
&=\Big(H^{1-\sigma}(t)+\frac{\varepsilon}{\rho+1}
 \Big(\int_{\Omega}|u_t|^{\rho}u_tu\,dx
+\int_{\Omega}|v_t|^{\rho}v_tv\,dx\Big)\Big)^{\frac{1}{1-\sigma}}\\
&\leq  2^{\frac{1}{1-\sigma}}\Big(H(t)+\frac{\varepsilon}{\rho+1}
 \Big(\big|\int_{\Omega}|u_t|^{\rho}u_tu\,dx\big|
+\big|\int_{\Omega}|v_t|^{\rho}v_tv\,dx\big|\Big)^{\frac{1}{1-\sigma}}\Big)\\
& \leq  2^{\frac{1}{1-\sigma}}B_{14}\Big[H(t)
 +\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2+\|u\|_{p+1}^{p+1} \\
&\quad +\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2+\|v\|_{p+1}^{p+1}\Big]\\
& \leq \widetilde{C}\Big[\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2+\|u\|_{p+1}^{p+1}
+\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2+\|v\|_{p+1}^{p+1}\Big],
\end{aligned}
\end{equation}
where $\widetilde{C}$ depends on $c_1$, $B_9$---$B_{14}$.

Combining \eqref{3.28} and \eqref{3.37}, we have
\begin{equation}\label{3.38}
L'(t)>\frac{\varepsilon \gamma}{\widetilde{C}}L^{\frac{1}{1-\sigma}}(t), \quad
\text{for } t\geq T_0.
\end{equation}
The inequality above implies that $L(t)$ blows up at a finite time $T^*$ and
\begin{equation}\label{3.39}
T^{\ast}\leq\frac{\widetilde{C}(1-\sigma)}
{\varepsilon\gamma L^{\sigma/(1-\sigma)}(T_0)}.
\end{equation}
Furthermore,  from \eqref{3.37} we obtain
\begin{equation}\label{3.40}
\lim_{t\to T^{\ast-}}\Big[\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2+\|u\|_{p+1}^{p+1}
+\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2+\|v\|_{p+1}^{p+1}\Big]=+\infty.
\end{equation}
If we choose the 
$T>\frac{\widetilde{C}(1-\sigma)}{\varepsilon\gamma L^{\sigma/(1-\sigma)}(T_0)}$, 
obviously, \eqref{3.40} contradicts \eqref{3.1}. 
Thus, the solution of problem \eqref{1.1} blows up in finite time.
\end{proof}

\subsection*{Concluding remarks}

In this paper, we considered the blow-up of solutions to a coupled quasilinear 
system with the nonlinear viscoelastic terms,  
the nonlinear principal parts, the  coupled source terms and the nonlinear dampings.
A  sufficient condition under which the solutions of the system will blow up  
at finite time is given.  We show that the coupled sources are enough to 
lead to the blow-up when the relaxation functions and the nonlinear principle 
parts satisfy some conditions.


\subsection*{Acknowledgments}
This research supported by the National Natural Science
Foundation of China (11671240, 61403239, 61503230).



\begin{thebibliography}{00}

\bibitem{a1} K. Agre, M. A. Rammaha;
\emph{System of nonlinear wave equations with damping and source terms},
 Differential Intergral Equations, 2006, 19: 1235-1270.

\bibitem{c1} M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano;
\emph{Exponential decay for the solution of semilinear viscoelastic wave
equations with  localized damping}, Electron.J. Differential equations, 2002,
44: 1-14.

\bibitem{c2} 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., 2001, 24: 1043-1053.

\bibitem{c3} M. M. Cavalcanti, H. P. Oquendo;
\emph{Fractional versusviscoelastic damping in a semilinear wave equation},
SIAM J. Control Optim., 2003, 42(4): 1310-1324.

\bibitem{g1} V. Georgiev, G. Todorova;
\emph{Existence of solutions of the wave equation with nonlinear damping 
and source terms},  J. Differential equations, 1994, 109: 295-308.

\bibitem{h1} J. H. Hao, Y. J. Zhang, S. J. Li;
\emph{Global existence and blow up phenomena for a nonlinear wave equation},
 Nonlinear Anal., TMA, 2009, 71: 4823-4832.

\bibitem{l1} W. J. Liu;
\emph{General decay and blow up of solution for a quasilinear viscoelastic
 problem with nonlinear source}, Nonlinear Anal., TMA, 2010, 73: 1890-1904.

\bibitem{m1} S. A. Messaoudi;
\emph{Blow up in a nonlinearly damped wave equation}, math. Nachr., 2001, 231: 1-7.

\bibitem{m2} S. A. Messaoudi;
\emph{Blow up and global existence in a nonlinear viscoelastic wave equation}, 
math. Nachr., 2003, 260: 58-66.

\bibitem{m3} S. A. Messaoudi;
\emph{Blow up of positive-initial-energy solutions of a nonlinear viscoelastic
 hyperbolic equation}, J. Math .Anal. Appl., 2006, 320: 902-915.

\bibitem{m4} S. A. Messaoudi, N. E. Tatar;
\emph{Global existence and uniform stability of solutions for a quasilinear
viscoelastic problem}, Math. Meth. Appl. Sci., 2007, 30: 665-680.

\bibitem{s1} H. T. Song;
\emph{Blow up of arbitrarily positive initial energy solutions for 
a viscoelastic wave equation}, Nonlinear Anal., RWA, 2015, 26: 306-314.

\bibitem{s2} H. T. Song;
\emph{Global nonexistence of positive initial energy solutions for a viscoelastic
wave equation}, Nonlinear Anal., TMA, 2015, 125: 260-269.

\bibitem{w1} J. Q. Wu, S. J. Li;
\emph{Blow up for coupled nonlinear wave equations with damping and source}, 
Appl. Math. Lett., 2011, 24: 1093-1098.

\bibitem{w2} J. Q. Wu, S. J. Li, S. G. Chai;
\emph{Existence and nonexistence of a global solution for coupled nonlinear 
wave equations with damping and source}, Nonlinear Anal., TMA,
  2010, 72: 3969-3975.

\bibitem{z1} E. Zuazua;
\emph{Exponential decay for the semilinear wave equatiuon with locally distributed
damping}, Comm. Partial Differential Equations, 1990, 15: 205-235.


\end{thebibliography}

\end{document}