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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 242, pp. 1--10.\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-2017/242\hfil Blow up of solutions]
{Blow up of solutions for viscoelastic wave equations of Kirchhoff
type with arbitrary positive initial energy}

\author[E. Pi\c{s}kin, A. Fidan \hfil EJDE-2017/242\hfilneg]
{Erhan Pi\c{s}kin, Ay\c{s}e Fidan}

\address{Erhan Pi\c{s}kin \newline
 Dicle University,
Department of Mathematics,
21280 Diyarbak\i{r},
Turkey}
\email{episkin@dicle.edu.tr}

\address{Ay\c{s}e Fidan \newline
 Dicle University,
Department of Mathematics,
21280 Diyarbak\i{r},
Turkey}
\email{afidanmat@gmail.com}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted March 11, 2017. Published October 4, 2017.}
\subjclass[2010]{35B44, 35L05, 35L53}
\keywords{Blow up; viscoelastic wave equation; arbitrary positive initial energy}

\begin{abstract}
 In this article we consider the nonlinear
 Viscoelastic wave equations of Kirchhoff type
 \begin{gather*}
 u_{tt}-M( \| \nabla u\| ^2)
 \Delta u+\int_0^{t}g_1( t-\tau )\Delta u( \tau ) d\tau +u_t
 =( p+1)| v| ^{q+1}| u| ^{p-1}u, \\
 v_{tt}-M( \| \nabla v\| ^2) \Delta v+\int_0^{t}g_2( t-\tau )
 \Delta v( \tau ) d\tau +v_t=( q+1)
 | u| ^{p+1}| v| ^{q-1}v
 \end{gather*}
 with initial conditions and Dirichlet boundary conditions. We proved
 the global nonexistence of solutions by applying a lemma by Levine, and
 the concavity method. 
 \end{abstract}

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

\section{Introduction}

In this article we consider the  initial boundary value problem
\begin{equation}
\begin{gathered}
u_{tt}-M( \| \nabla u\| ^2)
\Delta u+\int_0^{t}g_1( t-\tau )
\Delta u( \tau ) d\tau +u_t=( p+1)
| v| ^{q+1}| u| ^{p-1}u,\\ (x,t) \in \Omega \times ( 0,T) , \\
v_{tt}-M( \| \nabla v\| ^2)
\Delta v+\int_0^{t}g_2( t-\tau )
\Delta v( \tau ) d\tau +v_t=( q+1)
| u| ^{p+1}| v| ^{q-1}v,\\ (x,t) \in \Omega \times ( 0,T) , \\
u( x,t) =v( x,t) =0,\quad (x,t) \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 v_t( x,0)=v_1( x) ,\quad x\in \Omega , 
\end{gathered}  \label{300}
\end{equation}
where $\Omega $ is a bounded domain with a smooth boundary $\partial \Omega $
in $R^{n}$ $( n=1,2,3)$, $p>1$, $q>1$ and $M( s) $ is a
nonnegative $C^{1}$ function such as
\begin{equation*}
M( s) =a+bs^{\gamma },\quad s\geq 0
\end{equation*}
for $s\geq 0$, $a>0$, $b\geq 0$, $a+b\geq 0$, $\gamma >0$. 
The function $g_{i}:R^{+}\to R^{+}$ represents the kernel of the memory
term and is a given positive function to be specified later.

The single viscoelastic wave equation of Kirchhoff type of the form
\begin{equation}
u_{tt}-M( \| \nabla u\| ^2)\Delta u+\int_0^{t}g( t-\tau ) \Delta
u( \tau ) d\tau +h( u_t) =| u|^{q-1}u,  \label{301}
\end{equation}
has been extensively studied and many results concerning nonexistence have
been proved. See in this regard \cite{LiHongLiu, JieFei}. When $M\equiv 1$,
the equation \eqref{301} reduces to 
\begin{equation}
u_{tt}-\Delta u+\int_0^{t}g( t-\tau )
\Delta u( \tau ) d\tau +h( u_t) 
=|u| ^{q-1}u.  \label{302}
\end{equation}
The existence, and blow up in finite time of solution for \eqref{302} were
established (see \cite{LuFei, Messaoudi1, Messaoudi2, YWang} and references
therein).

For the case $M\equiv 1$,  system \eqref{300} reduces to 
\begin{equation}
\begin{gathered}
u_{tt}-\Delta u+\int_0^{t}g_1( t-\tau )
\Delta u( \tau ) d\tau +u_t=f_1( u,v) , \\
v_{tt}-\Delta v+\int_0^{t}g_2( t-\tau )
\Delta v( \tau ) d\tau +v_t=f_1( u,v) .
\end{gathered}  \label{303}
\end{equation}
Han and Wang \cite{HanWang} obtained the existence and nonexistence of the
solution of  problem \eqref{303}. Messaoudi and Said Houari 
\cite{MessaoudiHouari} considered problem \eqref{303} and improved the blow up
result in \cite{HanWang}, for positive initial energy, using the same
techniques as in \cite{Said-Houari1}. Ma et al.\ \cite{MaMuZeng} studied the
blow up of the solution of the problem \eqref{303} with arbitrary positive
initial energy. For more information about \eqref{303}, see references
\cite{HouariMesGu, Kafini1, Kafini2, Piskin1, Piskin2}.

In this article, we consider problem \eqref{303} and prove the blow up result
by a concavity method with arbitrary positive initial energy.

This paper is organized as follows. In section 2, we present some lemmas. In
section 3, we show the blow up of solutions.

\section{Preliminaries}

In this section, we introduce some notation, assumptions and lemmas which
will be needed in this paper. Let $\| \cdot\| $ and $\|\cdot\| _p$ 
denote the usual $L^2( \Omega ) $ norm and $L^{p}( \Omega ) $ norm, respectively.

To state and prove our main results, we make the following assumptions:
\begin{itemize}
\item[(A1)] $g_{i}\in C^{1}[ 0,\infty ] $ $(i=1,2)$ is a
non-negative and non-increasing differentiable function satisfying
\begin{equation*}
1-\int_0^{\infty }g_{i}( s) ds=l_{i}>0,\quad  i=1,2.
\end{equation*}

\item[(A2)] $g_{i}( t) \geq 0,\ g_{i}'( t)\leq 0$, for all $t\geq 0$, $i=1,2$.

\item[(A3)] The function $e^{1/2}g( t) $ is of positive
type in the following sense:
\begin{equation*}
\int_0^{t}v( s) \int_0^{s}e^{(s-\tau )/2}
 g_{i}( s-\tau ) v( \tau ) d\tau ds\geq 0,\quad 
\forall v\in C^{1}[ 0,\infty ) \text{ and }\forall t>0.
\end{equation*}

\end{itemize}

To obtain the blow up result, we need the following lemma which repeats the
same one of \cite{LiHongLiu} with slight modification, we will omit it.

\begin{lemma} \label{lem1}
There exists positive constants $m_{i}$ and $s\geq 0$, $a>0$, $b\geq 0$, 
$\gamma >0$ such that
\begin{equation}
\frac{p+q+2}{2}\overline{M}( s) -\Big[ M( s) +\frac{
p+q+2}{2}\int_0^{\infty }g_{i}( \tau ) d\tau \Big] s\geq
m_{i}s,\quad \forall s\geq 0,  \label{305}
\end{equation}
where
\begin{equation*}
\overline{M}( s) =\int_0^{s}M( \tau ) d\tau .
\end{equation*}
\end{lemma}

\begin{lemma}[\cite{Munoz Rivera}] \label{lem2}
 For any $g\in C^{1}$ and $\phi \in H^{1}(0,T) $ we have
\begin{equation} \label{315}
\begin{aligned}
&\int_{\Omega }\int_0^{t}g( t-\tau )
\Delta \phi ( \tau ) \phi '( t) d\tau\,\,dx \\
&= -\frac{1}{2}( g'\circ \nabla \phi ) (t) +\frac{1}{2}g( t) \| \nabla \phi \| ^2
 +\frac{1}{2}\frac{d}{dt}[ ( g\circ \nabla \phi ) (
t) -\int_0^{t}g( \tau ) \| \nabla \phi \| ^2d\tau ] .
\end{aligned}
\end{equation}
\end{lemma}

\begin{lemma}[Sobolev-Poincar\'{e} inequality \cite{Adams}] \label{lem3}
Let $p$ be a number with $2\leq p<\infty $ $( n=1,2) $ or
 $2\leq p\leq 2n/( n-2) $ $( n\geq 3) $, then there is a constant 
$C_{\ast }=C_{\ast }( \Omega ,\text{ }p) $ such that
\begin{equation*}
\| u\| _p\leq C_{\ast }\| \nabla u\| ,\quad \forall u\in H_0^{1}( \Omega ) .
\end{equation*}
\end{lemma}

\begin{lemma}[\cite{Levine}] \label{lem4}
Suppose that $F( t) $ is a twice
continuously differentiable positive function satisfying
\begin{equation*}
F''( t) F( t) -( 1+\alpha )[ F'( t) ] ^2\geq 0,\quad  \forall t\geq 0
\end{equation*}
where $\alpha >0$. If $F( 0) >0$ and $F'( 0)>0$. Then there exists a
positive constant $T^{\ast }\leq \frac{F(0) }{\alpha F'( 0) }$ such that 
$\lim_{t\to T^{\ast }}F( t) =\infty $.
\end{lemma}

\section{Blow up of solution}

In this section, we shall discuss the global nonexistence of the problem 
\eqref{300}. Let us first introduce the  functionals
\begin{equation} \label{320}
\begin{aligned}
J( t) &=\frac{1}{2}\int_0^{t}g_1( \tau) d\tau \| \nabla u\| ^2+\frac{1}{2}
\int_0^{t}g_2( \tau ) d\tau \| \nabla v\| ^2   \\
&\quad +\frac{1}{2}[ ( g_1\circ \nabla u) ( t)
+( g_2\circ \nabla v) ( t) ] -\int_{\Omega}| u| ^{p+1}| v| ^{q+1}\,dx,
\end{aligned}
\end{equation}
and
\begin{equation} \label{325}
\begin{aligned}
I( t) &= M( \| \nabla u( t) \|^2) \| \nabla u\| ^2+M( \| \nabla
v( t) \| ^2) \| \nabla v\| ^2 \\
&\quad -( p+q+2) \int_{\Omega }| u|^{p+1}| v| ^{q+1}\,dx.
\end{aligned}
\end{equation}
We also define the energy function
\begin{equation} \label{330}
\begin{aligned}
E( t) &=\frac{1}{2}( \| u_t\|^2+\| v_t\| ^2) +\frac{1}{2}[ \overline{M}
( \| \nabla u( t) \| _2^2) +\frac{
1}{2}\overline{M}( \| \nabla v( t) \|_2^2) ]   \\
&\quad -\frac{1}{2}\int_0^{t}g_1( \tau ) d\tau
\| \nabla u\| ^2-\frac{1}{2}\int_0^{t}g_2
( \tau ) d\tau \| \nabla v\| ^2    \\
&\quad +\frac{1}{2}[ ( g_1\circ \nabla u) ( t)
+( g_2\circ \nabla v) ( t) ] -\int_{\Omega}| u| ^{p+1}| v| ^{q+1}\,dx,
\end{aligned}
\end{equation}
where
\[
( \phi \circ \psi ) ( t)
=\int_0^{t}\phi ( t-\tau ) \int_{\Omega }| \psi (
t) -\psi ( \tau ) | ^2\,dx\,d\tau
= \int_0^{t}\phi ( t-\tau ) \| [ \psi( t) -\psi ( \tau ) ] \| ^2d\tau .
\]
Finally, we define
\begin{equation}
W=\big\{ ( u,v) :( u,v) \in H_0^{1}( \Omega
) \times H_0^{1}( \Omega ) ,\ I( u,v)
>0\} \cup \{ ( 0,0) \big\} .  \label{335}
\end{equation}
The next lemma shows that our energy functional \eqref{330} is a
nonincreasing function along the solution of the problem \eqref{300}.

\begin{lemma} \label{lem5}
$E( t) $ is a non-creasing function for $t\geq 0$, that is
\begin{equation}
E'( t) \leq -( \| u_t\|^2+\| v_t\| ^2) +\frac{1}{2}[ (
g_1'\circ \nabla u) ( t) +( g_2'\circ \nabla v) ( t) ]
\leq  0,  \label{340}
\end{equation}
and
\begin{equation}
E( t) \leq E( 0) -\int_0^{t}( \|u_{\tau }\| ^2+\| v_{\tau }\| ^2) d\tau . \label{345}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying the first equation of \eqref{300} by $u_t$ and the second
equation by $v_t$, integrating over $\Omega $, and using \eqref{315} and
assumption (A1)--(A2), we obtain \eqref{340}.
\end{proof}

\begin{lemma}[\cite{YWang}] \label{lem6}
Assume that $g_{i}$ satisfies  assumptions {\rm (A1), (A2)}  and $H( t) $ 
is a function that is twice continuously differentiable, satisfying
\begin{equation}
\begin{gathered}
\begin{aligned}
&H''( t) +H'( t)\\
&>2\int_0^{t}g( t-\tau ) \int_{\Omega }[ \nabla
u( \tau ,x) \nabla u( t,x) +\nabla
u( \tau ,x) \nabla u( t,x) ] \,dx\,d\tau
\end{aligned}\\
H( 0) >0,\quad H'( 0) >0
\end{gathered}  \label{395}
\end{equation}
for every $t\in [ 0,T_0) $ and $( u( x,t),v( x,t) ) $ is the solution 
of problem \eqref{300}.
Then the function $H( t) $ is strictly increasing on $[0,T_0)$.
\end{lemma}

\begin{lemma} \label{lem7}
Assume  $( u_0,v_0) \in ( H_0^{1}( \Omega
) \cap H^2( \Omega ) ) \times (
H_0^{1}( \Omega ) \cap H^2( \Omega ) ) $, 
$( u_1,v_1) \in H_0^{1}( \Omega ) \times
H_0^{1}( \Omega ) $ and satisfy
\begin{equation}
\int_{\Omega }( u_0u_1+v_0v_1) \,dx\geq 0.  \label{400}
\end{equation}
If the local solution $( u( t) ,v( t) ) $ of \eqref{300} satisfies
\begin{equation*}
I( u( t) ,v( t) ) <0,
\end{equation*}
then $H( t) =\| u( t,\cdot) \|_2^2+\| v( t) \| _2^2$ is strictly
increasing on $[ 0,T) $. 
\end{lemma}

\begin{proof}
Since
\begin{align*}
I( t) &=M( \| \nabla u( t) \|
^2) \| \nabla u\| ^2+M( \| \nabla
v( t) \| ^2) \| \nabla v\| ^2\\
&\quad -( p+q+2) \int_{\Omega }| u|
^{p+1}| v| ^{q+1}\,dx < 0,
\end{align*}
and $( u( t) ,v( t) ) $ is the local
solution of problem \eqref{300}, by a simple computation, we have
\begin{gather}
H( t) =\| u( t,\cdot) \|_2^2+\| v( t,\cdot) \| _2^2  
= \int_{\Omega }| u( t) | ^2\,dx+\int_{\Omega}| v( t) | ^2\,dx,  \label{405} \\
\frac{1}{2}\frac{d}{dt}H( t) 
=\int_{\Omega }uu_t\,dx+\int_{\Omega}vv_t\,dx  \label{410}
\end{gather}
\begin{align*}
&\frac{1}{2}\frac{d^2}{dt^2}H( t) \\
&= \int_{\Omega }|u_t| ^2\,dx+\int_{\Omega }uu_{tt}\,dx
 +\int_{\Omega }|v_t| ^2\,dx+\int_{\Omega }vv_{tt}\,dx    \\
&= \int_{\Omega }| u_t| ^2\,dx+\int_{\Omega }|
v_t| ^2\,dx+\int_{\Omega }uM( \| \nabla u\| ^2) \Delta u\,dx    \\
&\quad -\int_{\Omega }u\int_0^{t}g_1( t-\tau )
\Delta u( \tau ) d\tau \,dx-\int_{\Omega
}uu_t\,dx+\int_{\Omega }u( p+1) | v|
^{q+1}| u| ^{p-1}u\,dx    \\
&\quad +\int_{\Omega }vM( \| \nabla v\| ^2)
\Delta v\,dx-\int_{\Omega }v\int_0^{t}g_2( t-\tau
) \Delta v( \tau ) d\tau \,dx    \\
&\quad -\int_{\Omega }vv_t\,dx+\int_{\Omega }v( q+1) |
u| ^{p+1}| v| ^{q-1}v\,dx    \\
&\geq \int_{\Omega }M( \| \nabla u\|^2) u\Delta u\,dx-\int_{\Omega
}\int_0^{t}g_1( t-\tau ) u\Delta u(
\tau ) d\tau \,dx    \\
&\quad -\int_{\Omega }uu_t\,dx+\int_{\Omega }( p+1) |
v| ^{q+1}| u| ^{p-1}u\,dx    \\
&\quad+\int_{\Omega }M( \| \nabla v\| ^2)
v\Delta v\,dx-\int_{\Omega }\int_0^{t}g_2( t-\tau
) v\Delta v( \tau ) d\tau \,dx    \\
&\quad-\int_{\Omega }vv_t\,dx+\int_{\Omega }( q+1) |
u| ^{p+1}| v| ^{q-1}v\,dx    \\
&> -\int_{\Omega }( uu_t+vv_t)
\,dx+\int_0^{t}g_1( t-\tau ) \int_{\Omega
}\nabla u( \tau ) \nabla u( t) \,dx\,d\tau
  \\
&\quad +\int_0^{t}g_2( t-\tau ) \int_{\Omega
}\nabla v( \tau ) \nabla v( t) \,dx\,d\tau
  \\
&= -\frac{1}{2}\frac{dH}{dt}+\int_0^{t}g_1( t-\tau
) \int_{\Omega }\nabla u( \tau ) \nabla u( t) \,dx\,d\tau \\
&\quad +\int_0^{t}g_2( t-\tau ) \int_{\Omega}\nabla v( \tau ) \nabla v( t) \,dx\,d\tau
%\label{416}
\end{align*}
which yields
\begin{align*}
&\frac{1}{2}\frac{d^2H}{dt^2}+\frac{1}{2}\frac{dH}{dt} \\
&>\int_0^{t}g_1( t-\tau ) \int_{\Omega }\nabla
u( \tau ) \nabla u( t) \,dx\,d\tau
+\int_0^{t}g_2( t-\tau ) \int_{\Omega}\nabla v( \tau ) \nabla v( t) \,dx\,d\tau .
\end{align*}
Therefore, by \eqref{395}, the proof is complete.
\end{proof}

\begin{theorem} \label{thm8}
Under  {\rm (A1)--(A3)} hold, and the initial data
\begin{gather*}
( u_0,v_0) \in ( H_0^{1}( \Omega ) \cap
H^2( \Omega ) ) \times ( H_0^{1}( \Omega ) \cap H^2( \Omega ) ), \\
( u_1,v_1) \in H_0^{1}( \Omega ) \times H_0^{1}( \Omega )
\end{gather*}
satisfy
\begin{gather}
E( 0) >0,  \label{420} \\
I( u_0,v_0) <0,  \label{421} \\
\int_{\Omega }( u_0u_1+v_0v_1) \,dx\geq 0,  \label{422}\\
\| u_0\| ^2+\| v_0\| ^2\geq \frac{
( p+q+2) \eta }{\min \{ m_1,m_2\} }E(0) .  \label{423}
\end{gather}
Then the solution of problem \eqref{300} blows up in finite $T<\infty $.
\end{theorem}

\begin{lemma} \label{lem9}
If $( u_0,v_0) \in ( H_0^{1}( \Omega ) \cap
H^2( \Omega ) ) \times ( H_0^{1}( \Omega) \cap H^2( \Omega ) ) $ and 
$(u_1,v_1) \in H_0^{1}( \Omega ) \times H_0^{1}(\Omega ) $ satisfy the 
assumptions in Theorem \ref{thm8}, then the solution 
$( u,v) $ of the problem \eqref{300} satisfies
\begin{gather}
I( u( t,x) ,v( t,x) ) <0,  \label{425} \\
\| u( t) \| ^2+\| v( t)
\| ^2\geq \frac{( p+q+2) \eta }{\min \{
m_1,m_2\} }E( 0) ,  \label{430}
\end{gather}
for every $t\in [ 0,T) $.
\end{lemma}

\begin{proof}
We will prove this lemma by a contradiction argument. First we assume that 
\eqref{425} is not true over $[ 0,T) $, so, that there exists a
time $t_1>0$ such that
\begin{equation}
t_1=\min \{ t\in ( 0,T) :I( u,v) =0\} .  \label{435}
\end{equation}
Since $I( u,v) <0$ on $[ 0,t_1) $, by Lemma \ref{lem7}, we
see that $H( t) =\| u( t,\cdot) \|_2^2+\| v( t,\cdot) \| _2^2$ is strictly
increasing over $[ 0,t_1) $, which implies
\[
H( t) =\| u( t,\cdot) \|_2^2+\| v( t,\cdot) \| _2^2
> \| u_0\| ^2+\| v_0\| ^2 
> \frac{( p+q+2) \eta }{\min \{ m_1,m_2\} }E( 0) .
\]
It is obvious that $H( t) =\| u( t,\cdot)
\| _2^2+\| v( t,\cdot) \| _2^2$ is
continuous on $[ 0,t_1) $. Thus we obtain the inequality
\begin{equation}
H( t_1) =\| u( t_1,.) \|
_2^2+\| v( t_1,.) \| _2^2\geq \frac{
( p+q+2) \eta }{\min \{ m_1,m_2\} }E( 0)\,.
\label{440}
\end{equation}
 On the other hand, by \eqref{435} we have
\begin{align*}
E( 0) &\geq E( t_1) +\int_0^{t}[ \|
u_{\tau }\| ^2+\| v_{\tau }\| ^2] d\tau   \\
&=\frac{1}{2}\Big( \| u_t\| ^2+\|
v_t\| ^2\Big) +\frac{1}{2}[ \overline{M}(
\| \nabla u\| ^2) +\overline{M}(
\| \nabla v\| ^2) ]    \\
&\quad -\frac{1}{2}\Big(\int_0^{t}g_1( \tau ) \|\nabla u( t_1) \| ^2d\tau
+\int_0^{t}g_2( \tau ) \| \nabla v(t_1) \| ^2d\tau \Big)    \\
&\quad +\frac{1}{2}( ( g_1\circ \nabla u) (
t_1) +( g_2\circ \nabla v) (t_1) )    \\
&\quad -\int_{\Omega }| u| ^{p+1}| v|
^{q+1}\,dx+\int_0^{t}[ \| u_{\tau }\| ^2+\|
v_{\tau }\| ^2] d\tau    \\
&\geq \frac{1}{2}[ \overline{M}( \| \nabla
u\| ^2) +\overline{M}( \| \nabla v\| ^2) ] \\
&\quad -\frac{1}{2}\Big( \int_0^{t}g_1(
\tau ) \| \nabla u( t_1) \|
^2d\tau +\int_0^{t}g_2( \tau ) \| \nabla
v( t_1) \| ^2d\tau \Big) 
-\int_{\Omega }| u| ^{p+1}| v|^{q+1}\,dx.  %\label{441}
\end{align*}
Combining this inequality and \eqref{440}, we have
\begin{align*}
&( p+q+2) E( 0) \\
&\geq \frac{p+q+2}{2}\overline{M}
( \| \nabla u( t_1) \| ^2)
+\frac{p+q+2}{2}\overline{M}( \| \nabla v(
t_1) \| ^2) \\
&\quad -\frac{p+q+2}{2}( \int_0^{t}g_1( \tau ) \|
\nabla u( t_1) \| ^2d\tau
+\int_0^{t}g_2( \tau ) \| \nabla v(
t_1) \| ^2d\tau ) \\
&\quad -M( \| \nabla u( t_1) \| ^2)
\| \nabla u( t_1) \| ^2-M( \|
\nabla v( t_1) \| ^2) \| \nabla v( t_1) \| ^2
\end{align*}
By \eqref{305}, we get
\begin{align*}
&( p+q+2) E( 0) \\
&\geq \frac{p+q+2}{2}\overline{M}( \| \nabla u( t_1) \| ^2)
\\
&\quad -[ M( \| \nabla u( t_1) \|
^2) +\frac{p+q+2}{2}\int_0^{t}g_1( \tau ) d\tau
] \| \nabla u( t_1) \| ^2 \\
&\quad +\frac{p+q+2}{2}\overline{M}( \| \nabla v(
t_1) \| ^2) \\
&\quad -[ M( \| \nabla v( t_1) \|
^2) +\frac{p+q+2}{2}\int_0^{t}g_2( \tau ) d\tau
] \| \nabla v( t_1) \| ^2 \\
&\geq m_1\| \nabla u( t_1) \|^2+m_2\| \nabla v( t_1) \| ^2
\\
&\geq \min \{ m_1,m_2\} [ \| \nabla u( t_1) \| ^2+\| \nabla v(t_1) \| ^2] .
\end{align*}
Thus, by the Poincar\'{e} inequality, we have
\begin{gather*}
( p+q+2) E( 0) \geq \min \{ m_1,m_2\} \frac{1}{\eta }[ \| u( t_1) \|
^2+\| v( t_1) \| ^2] , \\
H( t_1) =\| u( t_1) \|
^2+\| v( t_1) \| ^2\leq \frac{(p+q+2) \eta }{\min \{ m_1,m_2\} }E( 0)
\end{gather*}
for every $t\in [ 0,T) $. The proof  is complete.
\end{proof}

\section{Proof of Theorem \ref{thm8}}

To prove our main result, we adopt the concavity method introduced by Levine
and define the  auxiliary function
\begin{equation}
\begin{aligned}
F( t) &=\| u( t) \| ^2+\|
v( t) \| ^2+\int_0^{t}( \| u( \tau
) \| ^2+\| v( \tau ) \|^2) d\tau    \\
&\quad +( t_2-t) ( \| u_0\| ^2+\|
v_0\| ^2) +\beta ( t_{3}+t) ^2
\end{aligned}  \label{445}
\end{equation}
where $t_2$, $t_{3}$ and $\beta $ are positive constants, which will be
determined later.

By direct computations, we obtain
\begin{equation}
\begin{aligned}
F'( t) &= 2\int_{\Omega }( uu_t+vv_t)
d\tau +2\int_0^{t}\int_{\Omega }( uu_{\tau }+vv_{\tau })
\,dx\,d\tau -\| u_0\| ^2-\| v_0\| ^2
  \\
&\quad -( \| u_0\| ^2+\| v_0\|
^2) +2\beta ( t_{3}+t)    \\
&= 2\int_{\Omega }( uu_t+vv_t) \,dx+2\int_0^{t}\int_{\Omega
}( uu_{\tau }+vv_{\tau }) \,dx\,d\tau +2\beta ( t_{3}+t)
\end{aligned} \label{450}
\end{equation}
and
\begin{align*}
& F''( t) \\
&= 2\int_{\Omega }(u_t^2+v_t^2) \,dx+2\int_{\Omega }( uu_{tt}+vv_{tt})
\,dx+2\int_{\Omega }( uu_t+vv_t) \,dx+2\beta    \\
&= 2\| u_t\| ^2+2\| v_t\|
^2+2\int_{\Omega }M( \| \nabla u\| ^2)
u\Delta u\,dx-2\int_0^{t}g_1( t-\tau ) \int_{\Omega
}u\Delta u( \tau ) \,dx\,d\tau    \\
&\quad -2\int_{\Omega }( uu_t+vv_t) \,dx+2( p+1)
\int_{\Omega }| v| ^{q+1}| u|
^{p+1}\,dx+2\int_{\Omega }M( \| \nabla v\|
^2) v\Delta v\,dx    \\
&\quad -2\int_0^{t}g_2( t-\tau ) \int_{\Omega
}v\Delta v( \tau ) \,dx\,d\tau +2( q+1)
\int_{\Omega }| u| ^{p+1}| v| ^{q+1}\,dx
  \\
&\quad +2\int_{\Omega }( uu_t+vv_t) \,dx+2\beta 
\end{align*} %\label{455}
By Young and Poincare inequalities, \eqref{345}, \eqref{423}, 
Lemma \ref{lem7}, we obtain
\begin{align*}
&F''( t) \\
&\geq ( p+q+4) (\| u_t\| ^2+\| v_t\| ^2)
+2\min \{ m_1,m_2\} [ \| \nabla u( t)
\| ^2+\| \nabla v( t) \| ^2]
  \\
&\quad +2( p+q+2) \Big( -E( 0) +\int_0^{t}[
\| u_{\tau }\| ^2+\| v_{\tau }\| ^2
] d\tau \Big) +2\beta    \\
&=( p+q+4) ( \| u_t\| ^2+\|
v_t\| ^2) +2\min \{ m_1,m_2\} [
\| \nabla u( t) \| ^2+\| \nabla v( t) \| ^2]    \\
&\quad -2( p+q+2) E( 0) +2( p+q+2) \int_0^{t}
[ \| u_{\tau }\| ^2+\| v_{\tau }\|
^2] d\tau +2\beta    \\
&\geq ( p+q+4) ( \| u_t\|
^2+\| v_t\| ^2) +2\min \{
m_1,m_2\} \frac{1}{\eta }( \| \nabla u( t)
\| ^2+\| \nabla v( t) \| ^2)
  \\
&\quad -2( p+q+2) E( 0) +2( p+q+2) \int_0^{t}
[ \| u_{\tau }\| ^2+\| v_{\tau }\|^2] d\tau +2\beta    \\
&\geq ( p+q+4) ( \| u_t\|^2+\| v_t\| ^2) 
+2\min \{ m_1,m_2\} \frac{1}{\eta }( \| \nabla
u( t) \| ^2+\| \nabla v( t)\| ^2) \\
&\quad -2( p+q+2) E( 0)    
+2( p+q+2) \int_0^{t}[ \| u_{\tau }\|^2
 +\| v_{\tau }\| ^2] d\tau +2\beta   
\geq 0  %\label{465}
\end{align*}
which means that $F''( t) >0$ for every 
$t\in( 0,T) $. Since $F'( t) \geq 0$ and 
$F(t) \geq 0$, thus we obtain that $F'( t) $ and 
$F( t) $ are strictly increasing on $[ 0,T) $.

Thus, we can choose $\beta $ to satisfy
\begin{equation}
\min \{ m_1,m_2\} ( \| u_0\|^2+\| v_0\| ^2) -( p+q+2) \eta
E( 0) >\beta ( p+q+2)  \label{480}
\end{equation}
consequently,
\begin{equation}
\begin{aligned}
F''( t) 
&\geq ( p+q+4) (\| u_t\| ^2+\| v_t\| ^2) 
 +2( p+q+2) \int_0^{t}[ \| u_{\tau }\|^2+\| v_{\tau }\| ^2] d\tau \\
&\quad +( p+q+4) \beta .
\end{aligned}  \label{485}
\end{equation}
As far as $\beta $ is fixed, we select $t_{3}$ large enough satisfying
\begin{equation}
\frac{p+q}{2}\Big( \int_{\Omega }( u_0u_1+v_0v_1)
\,dx+\beta t_{3}\Big) >\| u_0\| ^2+\|v_0\| ^2.  \label{490}
\end{equation}
From \eqref{445}, \eqref{450} and \eqref{490}, we now choose
\begin{equation*}
t_2>\frac{\| u_0\| ^2+\| v_0\| ^2}{\frac{p+q}{2}( \int_{\Omega }( u_0u_1+v_0v_1)
\,dx+\beta t_{3}) },
\end{equation*}
which ensures that
\begin{equation}
t_2 >\frac{\| u_0\| ^2+\| v_0\|
^2}{\frac{p+q}{2}( \int_{\Omega }( u_0u_1+v_0v_1)
\,dx+\beta t_{3}) }   
=\frac{4}{p+q}\frac{F( 0) }{F'( 0) }.
\label{495}
\end{equation}
Now let
\begin{gather*}
A = \| u( t) \| ^2+\| v(
t) \| ^2+\int_0^{t}[ \| u( \tau )
\| ^2+\| v( \tau ) \| ^2]\, d\tau +\beta ( t_{3}+t) ^2, \\
B = \frac{1}{2}F'( t) =\int_{\Omega }(
uu_t+vv_t) \,dx+\int_0^{t}\int_{\Omega }( uu_{\tau }+vv_{\tau
}) \,dx\,d\tau +\beta ( t_{3}+t) , \\
C = \| u_t( t) \| ^2+\| v_t(
t) \| ^2+\int_0^{t}[ \| u_{\tau }( \tau
) \| ^2+\| v_{\tau }( \tau ) \|^2] d\tau +\beta .
\end{gather*}
By \eqref{450} and a simple computation, for all $s\in R$, we have
\begin{align*}
As^2-2Bs+C
 &= [ \| u( t) \|
^2+\| v( t) \| ^2+\int_0^{t}[
\| u( \tau ) \| ^2+\| v( \tau
) \| ^2] d\tau +\beta ( t_{3}+t) ^2
] s^2 \\
&\quad -2[ \int_{\Omega }( uu_t+vv_t)
\,dx+\int_0^{t}\int_{\Omega }( uu_{\tau }+vv_{\tau }) \,dx\,d\tau
+\beta ( t_{3}+t) ] s \\
&\quad +\| u_t( t) \| ^2+\| v_t(
t) \| ^2+\int_0^{t}[ \| u_{\tau }( \tau
) \| ^2+\| v_{\tau }( \tau ) \|
^2] d\tau +\beta \\
&= \int_{\Omega }( su( t) -u_t( t) )
^2\,dx+\int_{\Omega }( sv( t) -v_t( t) )
^2\,dx \\
&\quad +\int_0^{t}\int_{\Omega }( su( \tau ) -u_{\tau }(
\tau ) ) ^2\,dx\,d\tau +\int_0^{t}\int_{\Omega }( sv(
\tau ) -v_{\tau }( \tau ) ) ^2\,dx\,d\tau \\
&\quad +\beta ( s( t_{3}+t) -1) ^2 
\geq 0
\end{align*}
which implies 
$B^2-AC\leq 0$.
Since we assume that the solution $( u,v) $ to  problem \eqref{300} exists 
for every $t\in [ 0,T) $, we have
\begin{equation*}
F( t) F''( t) -\frac{(
p+q+4) }{4}( F'( t) ) ^2\geq 0.
\end{equation*}
Let $\alpha =\frac{p+q}{2}>0$. As $\frac{p+q+4}{4}>1$, we have
\begin{equation*}
F( t) F''( t) -( 1+\alpha )( F'( t) ) ^2\geq 0.
\end{equation*}
We see that
\begin{equation*}
( F^{-\alpha }( t) ) '=-\alpha F^{-\alpha-1}F'<0,
\end{equation*}
\begin{equation}
\begin{aligned}
( F^{-\alpha }( t) ) '' 
&= -\alpha ( -\alpha -1) F^{-\alpha -2}F'F'-\alpha
F^{-\alpha -1}F''    \\
&= \alpha ( \alpha +1) F^{-\alpha -2}( F')
^2-\alpha F^{-\alpha -1}F''    \\
&= -\alpha F^{-\alpha -2}[ F''F-( 1+\alpha )
( F') ^2]  
\end{aligned} \label{500}
\end{equation}
for every $t\in [ 0,T) $, which means that the function 
$F^{-\alpha }$ is concave. Obviously $F( 0) >0$, then from 
\eqref{500} it follows that 
\begin{equation*}
F^{-\alpha }\to 0, \quad\text{as } 
t\to T<\frac{4}{p+q}\frac{F( 0) }{F'(0) }.
\end{equation*}
Therefore, we see that there exist a finite time $T>0$ such that
\begin{equation*}
\lim_{t\to T^{-}}\Big[ \| u\| ^2+\|
v\| ^2+\int_0^{t}( \| u_{\tau }( \tau
,x) \| ^2+\| v_{\tau }( \tau ,x)
\| ^2) d\tau \Big] =\infty .
\end{equation*}
The proof  is complete.

\begin{thebibliography}{99}

\bibitem{Adams} R. A. Adams, J. J. F. Fournier;
\emph{Sobolev Spaces}, Academic Press, 2003.

\bibitem{HanWang}  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., 7 (2009), 5427--5450.

\bibitem{Said-Houari1} B. S. Houari;
\emph{Global nonexistence of positive
initial-energy solutions of a system of nonlinear wave equations with
damping and source terms}, Diff. Integral Eqns., 23 (2010), 79--92.

\bibitem{HouariMesGu} B. S. Houari, S. A. Messaoudi, A. Guesmia;
\emph{General decay of solutions of a nonlinear system of viscoelastic 
wave equations}, NoDEA- Nonlinear Diff., 18 (2011), 659--684.

\bibitem{JieFei} L. Jie, L. Fei;
\emph{Blow-up of solution for an
integro-differential equation with arbitrary positive initial energy},
Boundary Value Problems,  2015 (2015) 96.

\bibitem{Kafini1} M. Kafini, S. A. Messaoudi;
\emph{A blow-up result in a system of
nonlinear viscoelastic wave equations with arbitrary positive initial
energy}, Indagationes Mathematicae, 24 (2013), 602--612.

\bibitem{Kafini2} M. Kafini, S. A. Messaoudi;
\emph{A blow-up result for a viscoelastic system in $\mathbb{R}^N$},
 Electron. J. Differential Equations, 2007 (2007), no. 113,  1--7.

\bibitem{Levine} H. A. Levine;
\emph{Instability and nonexistence of global
solutions to nonlinear wave equations of the form $Pu_{tt}=Au+F(u)$}, Trans.
Amer. Math. Soc., 192 (1974), 1--21.

\bibitem{LiHongLiu} Gand Li, Linghui Hong, Wenjun Liu; 
\emph{Global nonexistence of solutions for viscoelastic wave equations 
of Kirchhoff type with high energy}, J. Func. Spaces and Appl., 
2012 Art. ID 530861, 15 pp.

\bibitem{LuFei} Y. Lu, L. Fei, G. Zhenhua;
\emph{Lower bounds for blow up time of a nonlinear viscoelstic wave equation}, 
Boundary Value Problems, 219 (2015), 1-6.

\bibitem{MaMuZeng} J. Ma, C. Mu, R. Zeng;
\emph{A blow up result for viscoelastic aquations with arbitrary positive 
initial energy}, Boundary Value Problems, 2011 (2011) :6.

\bibitem{Messaoudi1} S. A. Messaoudi;
\emph{Blow up and global existence in a
nonlinear viscoelastic wave equation}, Math. Nachr., 260 (2003), 58--66.

\bibitem{Messaoudi2} S. A. Messaoudi;
\emph{Blow up of solutions with positive
initial energy in a nonlinear viscoelastic wave equations}, J. Math. Anal.
Appl., 320 (2006), 902--915.

\bibitem{MessaoudiHouari} S. A. Messaoudi, B. S. 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{Munoz Rivera} J. E. Munoz Rivera, M. Naso, E. Vuk;
\emph{Asymptotic behavior of the energy for electromagnetic system with memory}, 
Math. Methods Appl. Sci.,  25 (2004), 819-841.

\bibitem{Piskin1} E. Pi\c{s}kin;
\emph{Global nonexistence of solutions for a
system of viscoelastic wave equations with weak damping terms}, Malaya J.
Mat., 3(2), 168-174 (2015).

\bibitem{Piskin2} E. Pi\c{s}kin;
\emph{A lower bound for the blow up time of a
system of viscoelastic wave equations with nonlinear damping and source
terms}, J. Nonlinear Funct. Anal., 2017 (2017), 1-9.

\bibitem{YWang} Y. Wang;
\emph{A global nonexistence theorem for viscoelastic
equations with arbitrary positive initial energy}, Appl. Math. Lett., 
22 (2009), 1394--1400.

\end{thebibliography}

\end{document}