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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 45, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/45\hfil Blow-up of solutions]
{Blow-up of solutions for an integro-differential equation with 
 a nonlinear source}
\author[S.-T. Wu\hfil EJDE-2006/45\hfilneg]
{Shun-Tang Wu}

\address{Shun-Tang Wu \newline
General Educational Center \\
China University of Technology \\
Taipei, 116, Taiwan}
\email{stwu@cute.edu.tw}
\date{}

\thanks{Submitted February 7, 2006. Published April 6, 2006.}
\subjclass[2000]{35L05, 35L15, 35L70, 37B25}
\keywords{Blow-up; life span; viscoelastic; integro-differential equation}

\begin{abstract}
 We study the nonlinear viscoelastic wave equation 
 \begin{equation*}
 u_{tt}-\Delta u+\int_0^t g(t-s)\Delta u(s)ds=|u|^p u,
 \end{equation*}
 in a bounded domain, with the initial and Dirichlet boundary conditions.
 By modifying the method in \cite{l6}, we prove that there are solutions, 
 under some conditions on the initial data, which blow up in finite time 
 with nonpositive initial energy as well as positive initial energy. 
 Estimates of the lifespan of solutions are also given.
\end{abstract}

\maketitle

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

\section{Introduction}

In this paper we consider the initial boundary value problem for the
nonlinear integro-differential equation 
\begin{gather}
u_{tt}-\Delta u+\int_0^t g(t-s)\Delta u(s)ds=|u|^p u,  \label{e1.1} \\
u(x,0)=u_0(x),\text{ }u_{t}(x,0)=u_{1}(x),\text{ }x\in \Omega ,  \label{e1.2}
\\
u(x,t)=0,\text{ }x\in \partial \Omega ,\text{ }t\geq 0,  \label{e1.3}
\end{gather}
where $\Delta =\sum_{j=1}^N\frac{\partial ^{2}}{\partial x_{j}^{2}}$ and 
$\Omega $ is a bounded domain in $R^N$, $N\geq 1$, with a smooth boundary 
$\partial \Omega $ so that the Divergence theorem can be applied. Here, $g$
is a positive function satisfying some conditions to be specified later 
and $p>0$.

When $g\equiv 0$, the equation \eqref{e1.1} becomes a nonlinear wave
equation. There is a large body of literature on nonexistence of global
solutions and blowup for solutions with negative initial energy 
\cite{b1,g2,k1}. Levine \cite{l1,l2,l3} considered the interaction between linear
or strong damping and the source terms and showed that solutions with
negative initial energy blow up in finite time. Georgiev and Todorova 
\cite{g1} extended Levine's result to wave equations with nonlinear damping
terms. Under some conditions, they proved that the solutions blow up in
finite time provided they have sufficiently negative initial energy. This
result was generalized by Levine and Serrin \cite{l4}, and then by Levine
and Park \cite{l5}. Vitillaro \cite{v1} combined the arguments in \cite{g1}
and \cite{l4} to extend these results to the case of positive initial energy.

On the contrary, when $g$ is not trivial on $R$, \eqref{e1.1} becomes a
semilinear viscoelastic equation. Cavalcanti et al. \cite{c1} treated 
\eqref{e1.1} with a localized damping mechanism acting on a part of the
domain. By assuming the kernel $g$ in the memory term decays exponentially,
they obtained an exponential decay rate of the energy function. Later,
Cavcalanti \cite{c3} and Berrimi and Messaoudi \cite{b2} improved this work
by using different methods. Also, Cavcalcanti et al. \cite{c2} established
an existence result and a decay result for problem \eqref{e1.1} with
nonlinear boundary damping. Regarding nonexistence, Messaoudi \cite{m2}
studied problem \eqref{e1.1} with an internal nonlinear damping term and
showed under some restrictions on the initial energy the solutions blow up
in finite time. Recently, Berrimi and Messaoudi \cite{b3} considered problem 
\eqref{e1.1} and proved, for suitable initial data, that the solution is
bounded and global and the damping caused by the integral term is enough to
obtain uniform decay of solutions. However, no blow up result is discussed
for this problem \eqref{e1.1}.

In this paper we shall deal with the blow up behavior of solutions for
problem \eqref{e1.1}-\eqref{e1.3}. We derive the blow-up properties of
solutions of problem \eqref{e1.1}-\eqref{e1.3} with nonpositive and positive
initial energy by modifying the method in \cite{l6}. The content of this
paper is organized as follows. In section 2, we give some lemmas and the
local existence theorem 2.4. In section 3, we define an energy function 
$E(t) $ and show that it is a non-increasing function of $t$. Then, we obtain
theorem 3.5, which gives the blow-up phenomena of solutions even for
positive initial energy. Estimates for the blow-up time $T^{\ast }$ are also
given.

\section{Preliminary results}

In this section, we shall give some lemmas which will be used throughout
this work.

\begin{lemma}[Sobolev-Poincar\'e inequality \cite{m1}] \label{lem2.1}
 If $2\leq p\leq \frac{2N}{N-2}$, then
\begin{equation*}
\| u\|_{p}\leq B\| \nabla u\|_{2},
\end{equation*}
for $u\in H_0^{1}(\Omega )$ holds with some constant
$B$,  where $\| \cdot \|_{p}$ denotes
the norm of $L^p (\Omega )$.
\end{lemma}

\begin{lemma}[\cite{l6}] \label{lem2.2}
Let $\delta >0$ and $B(t)\in C^{2}(0,\infty )$
 be a nonnegative function satisfying
\begin{equation}
B''(t)-4(\delta +1)B'(t)+4(\delta +1)B(t)\geq 0.
\label{e2.1}
\end{equation}
If
\begin{equation}
B'(0)>r_{2}B(0),  \label{e2.2}
\end{equation}
then $B'(t)>0$ for $t>0$,  where
$\ r_{2}=2(\delta +1)-2\sqrt{(\delta+1)\delta }$ is the smallest
root of the equation
\begin{equation*}
r^{2}-4(\delta +1)r+4(\delta +1)=0.
\end{equation*}
\end{lemma}

\begin{lemma}[\cite{l6}] \label{lem2.3}
If $J(t)$ is a non-increasing function on
$[t_0,\infty )$, $t_0\geq 0$ and satisfies the differential
inequality
\begin{equation}
J'(t)^{2}\geq a+bJ(t)^{2+\frac{1}{\delta }} \quad
\text{for }t_0\geq 0,  \label{e2.3}
\end{equation}
where $a>0$, $\delta >0$ and $b\in R$, then there exists a
finite time $T^{\ast }$ such that
\begin{equation*}
\lim_{t\to T^{\ast -}}J(t)=0
\end{equation*}
and the upper bound of $T^{\ast }$ is estimated
respectively by the following cases:

\noindent(i)  If $b<0$ and $J(t_0)<\min \{1,
\sqrt{\frac{a}{-b}}\}$,  then
\begin{equation*}
T^{\ast }\leq t_0+\frac{1}{\sqrt{-b}}\ln \frac{\sqrt{\frac{a}{-b}}}{\sqrt{
\frac{a}{-b}}-J(t_0)}.
\end{equation*}
(ii) If $b=0$, then
\begin{equation*}
T^{\ast }\leq t_0+\frac{J(t_0)}{\sqrt{a}}.
\end{equation*}
(iii) If $b>0$, then
\begin{equation*}
T^{\ast }\leq \frac{J(t_0)}{\sqrt{a}},
\end{equation*}
or
\begin{equation*}
T^{\ast }\leq t_0+2^{\frac{3\delta +1}{2\delta }}\frac{\delta c}{\sqrt{a}}
\{1-[1+cJ(t_0)]^{\frac{-1}{2\delta }}\},
\end{equation*}
where $c=(\frac{b}{a})^{\frac{\delta }{2+\delta }}$.
\end{lemma}

Now, we state the local existence theorem which is proved in \cite{m2}.

\begin{theorem}[Local existence] \label{thm2.4}
Let $u_0\in H_0^{1}(\Omega )$, $u_{1}\in L^{2}(\Omega )$
 and $0<p<p^{\ast }$,  here $p^{\ast }=\frac{2}{N-2}$,
if $N\geq 3(\infty $,  if $N\leq 2)$.
Let $g$ be a bounded $C^{1}$ function satisfying
\begin{equation}
g(0)>0,\quad g'(s)\leq 0,\quad 1-\int_0^{\infty
}g(s)ds=l>0.  \label{e2.4}
\end{equation}
Then problem \eqref{e1.1}-\eqref{e1.3} has a unique weak solution
$u$ in $C([0,T],H_0^{1}(\Omega ))$ with
$u_{t}$ in $C([0,T],L^{2}(\Omega ))$,
for some $T>0$.
\end{theorem}

\section{Blow-up property}

In this section, we shall discuss the blow up phenomena of problem 
\eqref{e1.1}-\eqref{e1.3}. For this purpose, we make the following
assumption on $g$: 
\begin{equation}
\int_{0}^{\infty }g(s)ds<\frac{4\delta }{1+4\delta },  \label{e3.1}
\end{equation}
here $\delta =p/4$. First, we define the energy function for the solution $u$
of \eqref{e1.1}-\eqref{e1.3} by 
\begin{equation} \label{e3.2}
\begin{aligned} 
E(t) &=\frac{1}{2}\| u_{t}\|_{2}^{2}+\frac{1}{2}( 1-\int_0^t
g(s)ds)\| \nabla u(t)\|_{2}^{2}+\frac{ 1}{2}(g\diamond \nabla u)(t) \\ 
&\quad -\frac{1}{p+2}\| u\|_{p+2}^{p+2}, \end{aligned}
\end{equation}
for $t\geq 0$, where $(g\diamond \nabla u)(t)=\int_{0}^{t}g(s)\Vert \nabla
u(t)-\nabla u(s)\Vert _{2}^{2}ds$ \smallskip 

\noindent \textbf{Remark.} From \eqref{e3.2}, \eqref{e2.4} and Lemma
\ref{lem2.1}, we have
\begin{equation}
\begin{aligned}
E(t) &\geq \frac{1}{2}\Big( 1-\int_{0}^{t}g(s)ds\Big) \Vert \nabla
u(t)\Vert _{2}^{2}+\frac{1}{2}(g\diamond \nabla u)(t)-\frac{1}{p+2}
\Vert u\Vert _{p+2}^{p+2}   \\
&\geq \frac{1}{2}\big( l\Vert \nabla u(t)\Vert
_{2}^{2}+(g\diamond \nabla u)(t)\big) -\frac{B_{1}^{p+2}l^{\frac{p+2}{2}}}{%
p+2}\left\Vert \nabla u\right\Vert _{2}^{p+2}   \\
&\geq  G\big[ \big( l\Vert \nabla u(t)\Vert _{2}^{2}+(g\diamond
\nabla u)(t)\big) ^{1/2}\big] ,\quad t\geq 0,  
\end{aligned}\label{e3.3}
\end{equation}
where 
\begin{equation*}
G(\lambda )=\frac{1}{2}\lambda ^{2}-\frac{B_{1}^{p+2}}{p+2}\lambda
^{p+2},\quad B_{1}=\frac{B}{\sqrt{l}}.
\end{equation*}%
It is easy to verify that $G(\lambda )$ has a maximum at $\lambda
_{1}=B_{1}^{-\frac{p+2}{p}}$ and the maximum value is 
\begin{equation*}
E_{1}=\frac{p}{2(p+2)}B_{1}^{-\frac{p+2}{p}}.
\end{equation*}

Before proving our main result, we need the following lemmas.

\begin{lemma}[\cite{m2}] \label{lem3.1}
Assume that the conditions of theorem 2.4  hold and let
$u$ be a solution of \eqref{e1.1}-\eqref{e1.3}. Then
$E(t)$ is a non-increasing function on $[0,T]$ and
\begin{equation}
E'(t)=\frac{1}{2}(g'\diamond \nabla u)(t)-\frac{1}{2}
g(t)\| \nabla u(t)\|_{2}^{2},  \label{e3.5}
\end{equation}
for almost every $t\in [0,T]$.
\end{lemma}

\begin{proof}
Multiplying \eqref{e1.1} by $u_{t}$ and integrating it over $\Omega $, and
integrating by parts, we obtain \eqref{e3.5} for any regular solution. Then
by density arguments, we have the result.
\end{proof}

\begin{lemma} \label{lem3.2}
Suppose that the conditions of theorem 2.4 hold. Let
$u$ be a solution of \eqref{e1.1}-\eqref{e1.3}
with initial data satisfying $E(0)<E_{1}$ and
$l^{\frac{1}{2}}\| \nabla u_0\|_{2}>\lambda _{1}$, then there
exists $\lambda _{2}>\lambda _{1}$ such that
\begin{equation}
l\| \nabla u(t)\|_{2}^{2}+(g\diamond \nabla u)(t)\geq
\lambda _{2}^{2},\quad  \text{for } t>0.  \label{e3.6}
\end{equation}
\end{lemma}

\begin{proof}
 From the definition of $G(\lambda )$, we see that $G(\lambda )$ is
increasing in $(0,\lambda _{1})$ and decreasing in $(\lambda _{1},\infty )$,
and $G(\lambda )\rightarrow -\infty $, as $\lambda \rightarrow \infty $.
Since $E(0)<E_{1}$, there exist $\lambda _{2}'$ and $\lambda _{2}$
such that $\lambda _{2}'<\lambda _{1}<\lambda _{2}$ and $G(\lambda
_{2}')=G(\lambda _{2})=E(0)$. When $l^{1/2}\Vert \nabla u_{0}\Vert
_{2}>\lambda _{1}$, by $\eqref{e3.3}$, we have 
\begin{equation*}
G(l^{1/2}\Vert \nabla u_{0}\Vert _{2})\leq E(0)=G(\lambda _{2}).
\end{equation*}
This implies$\ l^{1/2}\Vert \nabla u_{0}\Vert _{2}>\lambda _{2}$. To
establish \eqref{e3.6}, we suppose by contradiction that 
\begin{equation*}
l\Vert \nabla u(t)\Vert _{2}^{2}+(g\diamond \nabla u)(t)<\lambda _{2}^{2},
\end{equation*}
for some $t_{0}>0$.

\noindent Case 1: If $\lambda _{2}'<(l\Vert \nabla u(t_{0})\Vert
_{2}^{2}+(g\diamond \nabla u)(t_{0}))^{\frac{1}{2}}<\lambda _{2}$, then 
\begin{equation*}
G((l\Vert \nabla u(t_{0})\Vert _{2}^{2}+(g\diamond \nabla
u)(t_{0}))^{1/2})>E(0)\geq E(t_{0}),
\end{equation*}%
which contradicts \eqref{e3.3}.

\noindent Case 2: If $(l\| \nabla u(t_0)\| _{2}^{2}+(g\diamond \nabla
u)(t_0))^{1/2}<\lambda _{2}'$, then by the continuity of $(l\|
\nabla u(t)\|_{2}^{2}+(g\diamond \nabla u)(t))^{1/2}$, there exists 
$0<t_{1}<t_0$ such that 
\begin{equation*}
\lambda _{2}'<(l\| \nabla u(t_{1})\| _{2}^{2}+(g\diamond \nabla
u)(t_{1}))^{1/2}<\lambda _{2}.
\end{equation*}
Then 
\begin{equation*}
G((l\| \nabla u(t_{1})\|_{2}^{2}+(g\diamond \nabla
u)(t_{1}))^{1/2})>E(0)\geq E(t_{1}).
\end{equation*}
This is a contradiction.
\end{proof}

\noindent\textbf{Definition.} A solution $u$ of \eqref{e1.1}-\eqref{e1.3} is
said to blowup if there exists a finite time $T^{\ast }$ such that 
\begin{equation}
\lim_{t\to T^{\ast -}} \Big(\int_{\Omega }u^{2}dx\Big)^{-1}=0.  \label{e3.7}
\end{equation}
For short notation, we define 
\begin{equation}
a(t)=\int_{\Omega }u^{2}dx,\quad t\geq 0.  \label{e3.8}
\end{equation}

\begin{lemma} \label{lem3.3}
Assume that the conditions of theorems 2.4
 and \eqref{e3.1}  hold and let $u$ be a solution of
\eqref{e1.1}-\eqref{e1.3}, then we have
\begin{equation}
a''(t)-4(\delta +1)\int_{\Omega }u_{t}^{2}dx\geq Q_{1}(t),  \label{e3.9}
\end{equation}
where $Q_{1}(t)=(-4-8\delta )E(0)+m(l\| \nabla u\|_{2}^{2}
+(g\diamond \nabla u)(t))$,
$m=(1+4\delta )-1/l>0$.
\end{lemma}

\begin{proof}
Form \eqref{e3.8}, we have 
\begin{equation}
a'(t)=2\int_{\Omega }uu_{t}dx  \label{e3.10}
\end{equation}%
and by \eqref{e1.1} and the Divergence theorem, we get 
\begin{equation}
a''(t)=2\Vert u_{t}\Vert _{2}^{2}-2\Vert \nabla u\Vert
_{2}^{2}+2\Vert u\Vert _{p+2}^{p+2}+2\int_{0}^{t}\int_{\Omega }g(t-s)\nabla
u(s)\cdot \nabla u(t)dxds.  \label{e3.11}
\end{equation}%
Then, using $\eqref{e3.5}$, we obtain 
\begin{equation}
\begin{aligned} &a''(t)-4(\delta +1)\| u_{t}\|_{2}^{2} \\ 
&\geq (-4-8\delta
)E(0)+2\big[1-\frac{2+4\delta }{p+2}\big]\| u\|_{p+2}^{p+2}+4\delta \|
\nabla u(t)\|_{2}^{2} \\ &\quad -(2+4\delta )\int_0^t g(s)ds\| \nabla u(t)\|
_{2}^{2}+2\int_0^t \int_{\Omega }g(t-s)\nabla u(s)\cdot \nabla u(t)dxds \\
&\quad -(2+4\delta )\int_0^t (g'\diamond \nabla u)(t)dt+(2+4\delta
)(g\diamond \nabla u)(t). \end{aligned}  \label{e3.12}
\end{equation}
It follows from H\"{o}lder's inequality and Young's inequality that 
\begin{equation*}
\begin{aligned} &\int_{\Omega }\int_0^t g(t-s)\nabla u(s)\cdot \nabla
u(t)dsdx \notag \\ &=\int_{\Omega }\int_0^t g(t-s)\nabla u(t)\cdot (\nabla
u(s)-\nabla u(t))dsdx+\int_0^t g(t-s)ds\| \nabla u(t)\|_{2}^{2} \notag \\
&\geq -[\frac{1}{2}(g\diamond \nabla u)(t)+\frac{1}{2} \int_0^t g(s)ds\|
\nabla u(t)\|_{2}^{2}] +\int_0^t g(s)ds\| \nabla u(t)\|_{2}^{2}.
\end{aligned}
\end{equation*}%
Hence, \eqref{e3.12} becomes 
\begin{align*}
& a''(t)-4(\delta +1)\Vert u_{t}\Vert _{2}^{2} \\
& \geq (-4-8\delta )E(0)+\Big(4\delta -(1+4\delta )\int_{0}^{t}g(s)ds\Big)
\Vert \nabla u(t)\Vert _{2}^{2} \\
& \quad +(1+4\delta )(g\diamond \nabla u)(t)-(2+4\delta
)\int_{0}^{t}(g'\diamond \nabla u)(t)dt.
\end{align*}%
Therefore, by \eqref{e2.4} and \eqref{e3.1}, we obtain 
\begin{align*}
& a''(t)-4(\delta +1)\Vert u_{t}\Vert _{2}^{2} \\
& \geq (-4-8\delta )E(0)+(4\delta -(1+4\delta )(1-l))\Vert \nabla u(t)\Vert
_{2}^{2}+(1+4\delta )(g\diamond \nabla u)(t) \\
& \geq (-4-8\delta )E(0)+m(l\Vert \nabla u(t)\Vert _{2}^{2}+(g\diamond
\nabla u)(t)),
\end{align*}%
where $m=(1+4\delta )-\frac{1}{l}>0$.

Now, we consider different cases on the sign of the initial energy $E(0)$.

\noindent Case 1: If $E(0)<0$, then from $\eqref{e3.9}$, we have 
\begin{equation*}
a'(t)\geq a'(0)-4(1+2\delta )E(0)t,\quad t\geq 0.
\end{equation*}%
Thus we get $a'(t)>0$ for $t>t^{\ast }$, where 
\begin{equation}
t^{\ast }=\max \big\{\frac{a'(0)}{4(1+2\delta )E(0)},0\big\}.
\label{e3.14}
\end{equation}%
\noindent Case 2: If $E(0)=0$, then $a''(t)\geq 0$ for $t\geq
0$. Furthermore, if $a'(0)>0$, then $a'(t)>0$, $t\geq 0$.

\noindent Case 3: If $0<E(0)<\frac{m}{p}E_{1}$ and $l^{1/2}\Vert \nabla
u_{0}\Vert _{2}>\lambda _{1}$, then using lemma 3.2, we see that 
\begin{align*}
Q_{1}(t)& =(-4-8\delta )E(0)+m(l\Vert \nabla u\Vert _{2}^{2}+(g\diamond
\nabla u)(t)) \\
& \geq (-4-8\delta )E(0)+m\lambda _{2}^{2} \\
& >(-4-8\delta )E(0)+m\frac{2(p+2)}{p}E_{1} \\
& =(4+8\delta )(-E(0)+\frac{m}{4+8\delta }\frac{2(p+2)}{p}E_{1}) \\
& =(4+8\delta )(-E(0)+\frac{m}{p}E_{1}).
\end{align*}%
Thus, from \eqref{e3.9}, we have 
\begin{equation}
a''(t)\geq Q_{1}(t)>C_{1}>0,\quad t>0,  \label{e3.15}
\end{equation}%
where $C_{1}=(4+8\delta )(-E(0)+\frac{m}{p}E_{1})$. Hence, we get 
$a'(t)>0$ for $t>t_{1}^{\ast }$, where 
\begin{equation}
t_{1}^{\ast }=\max \{\frac{-a'(0)}{C_{1}},0\}.  \label{e3.16}
\end{equation}%
\noindent Case 4: If $E(0)\geq \frac{m}{p}E_{1}$, using H\"{o}lder's
inequality and Young's inequality, we get 
\begin{equation*}
a'(t)\leq \Vert u\Vert _{2}^{2}+\Vert u_{t}\Vert _{2}^{2}.
\end{equation*}
Hence, from \eqref{e3.9}, we have 
\begin{equation*}
a''(t)-4(\delta +1)a'(t)+4(\delta +1)a(t)+(4+8\delta
)E(0)\geq 0.
\end{equation*}%
Let 
\begin{equation*}
b(t)=a(t)+\frac{(1+2\delta )E(0)}{1+\delta },\quad t>0.
\end{equation*}%
Then $b(t)$ satisfies \eqref{e2.1}. By \eqref{e2.2}, we see that if 
\begin{equation}
a'(0)>r_{2}\big[a(0)+\frac{(1+2\delta )E(0)}{1+\delta }\big],
\label{e3.17}
\end{equation}%
then $\ a'(t)>0$, $t>0$.
\end{proof}

Consequently, we have the following result.

\begin{lemma} \label{lem3.4} Assume that the conditions of theorem 2.4
 and \eqref{e3.1} hold, and that either one of the following four
conditions is satisfied:
\begin{itemize}
\item[(i)] $E(0)<0,$
\item[(ii)] $E(0)=0$ and $a'(0)>0$,
\item[(iii)] $0<E(0)<\frac{m}{p}E_{1}$ and
      $l^{1/2}\| \nabla u_0\|_{2}>\lambda _{1}$
\item[(iv)] $\frac{m}{p}E_{1}\leq E(0)$ and
 \eqref{e3.17} holds.
\end{itemize}
Then $a'(t)>0$ for $t>t_0$, where $t_0=t^{\ast }$ is given by
\eqref{e3.14} in case 1, $t_0=0$ in cases 2 and 4,
and $t_0=t_{1}^{\ast }$ is given by \eqref{e3.16} in
case 3.
\end{lemma}

Hereafter, we will find an estimate for the life span of $a(t)$. Let 
\begin{equation}
J(t)=a(t)^{-\delta },\quad \text{for }t\geq 0.  \label{e3.18}
\end{equation}
Then we have 
\begin{equation}
\begin{gathered} J'(t)=-\delta J(t)^{1+\frac{1}{\delta }}a'(t), \\
J''(t)=-\delta J(t)^{1+\frac{2}{\delta }}V(t), \end{gathered}  \label{e3.19}
\end{equation}
where 
\begin{equation}
V(t)=a^{\prime\prime}(t)a(t)-(1+\delta )(a'(t))^{2}.  \label{e3.20}
\end{equation}
Using \eqref{e3.9} and exploiting H\"{o}lder's inequality on $a(t)$, we
deduce that 
\begin{align*}
V(t)&\geq [Q_{1}(t)+4(1+\delta )\|u_{t}\|_{2}^{2}]a(t)-4(1+\delta ) \|
u\|_{2}^{2}\| u_{t}\|_{2}^{2} \\
&=Q_{1}(t)J(t)^{-\frac{1}{\delta }}.
\end{align*}
Therefore, \eqref{e3.19} yields 
\begin{equation}
J^{\prime\prime}(t)\leq -\delta Q_{1}(t)J(t)^{1+ \frac{1}{\delta }},\quad
t\geq t_0.  \label{e3.21}
\end{equation}

\begin{theorem} \label{thm3.5}
Assume the conditions of theorem 2.4
 and \eqref{e3.1} hold, and that either one of the following four
conditions is satisfied:
\begin{itemize}
\item[(i)] $E(0)<0$,
\item[(ii)] $E(0)=0$  and $a'(0)>0$,
\item[(iii)] $0<E(0)<\frac{m}{p}E_{1}$ and
$l^{\frac{1}{2}}\| \nabla u_0\|_{2}>\lambda _{1}$,
\item[(iv)] $\frac{m}{p}E_{1}\leq E(0)<\frac{a'(t_0)^{2}}{8a(t_0)}$
 and \eqref{e3.17}  holds.
\end{itemize}
Then the solution $u$ blows up at finite time $T^{\ast }$
 in the sense of \eqref{e3.7}. Moreover, the upper bounds
for $T^{\ast }$ can be estimated according to the sign of $E(0)$:
In case (1),
\begin{equation*}
T^{\ast }\leq t_0-\frac{J(t_0)}{J'(t_0)}.
\end{equation*}
Furthermore, if $J(t_0)<\min \{1,\sqrt{\frac{\alpha }{-\beta }}\}$,
then
\begin{equation*}
T^{\ast }\leq t_0+\frac{1}{\sqrt{-\beta }}\ln \frac{\sqrt{\frac{\alpha }{
-\beta }}}{\sqrt{\frac{\alpha }{-\beta }}-J(t_0)}.
\end{equation*}
In case (2),
\begin{equation*}
T^{\ast }\leq t_0-\frac{J(t_0)}{J'(t_0)}
\quad{or}\quad
T^{\ast }\leq t_0+\frac{J(t_0)}{\sqrt{\alpha }}.
\end{equation*}
In case (3),
\begin{equation*}
T^{\ast }\leq t_0-\frac{J(t_0)}{J'(t_0)}.
\end{equation*}
Furthermore, if $J(t_0)<\min \{1,\sqrt{\frac{\alpha _{1}}{-\beta _{1}}}\}$,
then
\begin{equation*}
T^{\ast }\leq t_0+\frac{1}{\sqrt{-\beta _{1}}}\ln \frac{\sqrt{\frac{\alpha
_{1}}{-\beta _{1}}}}{\sqrt{\frac{\alpha _{1}}{-\beta _{1}}}-J(
t_0)}.
\end{equation*}
In case (4),
\begin{equation*}
T^{\ast }\leq \frac{J(t_0)}{\sqrt{\alpha }}
\quad\text{or}\quad
T^{\ast }\leq t_0+2^{\frac{3\delta +1}{2\delta }}\frac{\delta c}{\sqrt{
\alpha }}\{1-[1+cJ(t_0)]^{\frac{-1}{
2\delta }}\},
\end{equation*}
where $c=(\frac{\beta }{\alpha })^{\frac{\delta }{
2+\delta }}$. Here $\alpha $, $\beta$, $\alpha_{1}$ and
$\beta _{1}$ are given in \eqref{e3.23}-\eqref{e3.26},
respectively. Note that in case 1,
$t_0=t^{\ast }$ is given in \eqref{e3.14} and
$t_0=0$  in cases 2  and 4,
and in case 3, $t_0=t_{1}^{\ast }$
 is given in \eqref{e3.16}.
 \end{theorem}

\begin{proof}
(1) For $E(0)\leq 0$, from \eqref{e3.21}, 
\begin{equation}
J^{\prime\prime}(t)\leq \delta (4+8\delta ) E(0)J(t)^{1+\frac{1}{\delta }}.
\label{e3.22}
\end{equation}
Note that by lemma 3.4, $J'(t)<0$ for $t>t_0$. Multiplying 
\eqref{e3.22} by $J'(t)$ and integrating it from $t_0$ to $t$, we
have 
\begin{equation*}
J'(t)^{2}\geq \alpha +\beta J(t)^{2+\frac{ 1}{\delta }}\quad \text{%
for }t\geq t_0,
\end{equation*}
where 
\begin{equation}
\alpha =\delta ^{2}J(t_0)^{2+\frac{2}{\delta }}\big[ a^{%
\prime}(t_0)^{2}-8E(0)J(t_0)^{\frac{-1}{ \delta }}\big]>0.  \label{e3.23}
\end{equation}
and 
\begin{equation}
\beta =8\delta ^{2}E(0).  \label{e3.24}
\end{equation}
Then by lemma 2.3, there exists a finite time $T^{\ast }$ such that 
$\lim_{t\to T^{\ast -}}J(t)=0$ and this will imply that 
$\lim_{t\to T^{\ast -}}(\int_{\Omega}u^{2}dx)^{-1}=0$.

\noindent (2) For the case $0<E(0)<\frac{m}{p}E_{1}$, from \eqref{e3.21} and 
\eqref{e3.15}, we have 
\begin{equation*}
J^{\prime\prime}(t)\leq -\delta C_{1}J(t)^{1+ \frac{1}{\delta }}\quad 
\text{for }t\geq t_0.
\end{equation*}
Then using the same arguments as in (1), we have 
\begin{equation*}
J'(t)^{2}\geq \alpha _{1}+\beta _{1}J(t) ^{2+\frac{1}{\delta }}\text{
for }t\geq t_0,
\end{equation*}
where 
\begin{equation}
\alpha _{1} =\delta ^{2}J(t_0)^{2+\frac{2}{\delta }}[ a'(t_0)^{2}+
\frac{2C_{1}}{1+2\delta }J(t_0)^{\frac{ -1}{\delta }}] >0.  \label{e3.25}
\end{equation}
and 
\begin{equation}
\beta _{1}=-\frac{2C_{1}\delta ^{2}}{1+2\delta }.  \label{e3.26}
\end{equation}
Thus, by lemma 2.3, there exists a finite time $T^{\ast }$ such that 
$\lim_{t\to T^{\ast -}}(\int_{\Omega }u^{2}dx)^{-1}=0$.

\noindent (3) For the case $\frac{m}{p}E_{1}\leq E(0)$. Applying the same
discussion as in part (1), we also have the equalities \eqref{e3.23} and 
\eqref{e3.24}. In this way, we observe that 
\begin{equation*}
\alpha >0\quad \text{if and only if}\quad E(0)<\frac{a'(t_0)^{2}}{%
8a(t_0)}.
\end{equation*}
Hence, by lemma 2.3, there exists a finite time $T^{\ast }$ such that 
$\lim_{t\to T^{\ast -}}(\int_{\Omega }u^{2}dx)^{-1}=0$.
\end{proof}

\subsection*{Acknowledgements}

The author would like to express sincere gratitude to Professor Tsai Long-Yi
for his enthusiastic guidance and constant encouragement. Thanks are also
due to the anonymous referee for his/her constructive suggestions.

\begin{thebibliography}{99}
\bibitem{b1} Ball, J., Remarks on blow up and nonexistence theorems for
nonlinear evolution equations, Quart. J. Math. Oxford, 28(1977), 473-486.

\bibitem{b2} Berrimi, S. and Messaoudi, S. A., Exponential decay of
solutions to a viscoelastic equation with nonlinear localized damping,
Electronic J. Diff. Eqns., 2004(2004), No. 88, 1-10.

\bibitem{b3} Berrimi, S. and Messaoudi, S. A., Existence and decay of
solutions of a viscoelastic equation with a nonlinear source, To appear in
Nonlinear Anal. TMA.

\bibitem{c1} Cavalcanti, M. M., Domingos Cavalcanti, V. N. and Soriano, J.
A., Exponential decay for the solution of semilinear viscoelastic wave
equation with localized damping, Electronic J. Diff. Eqns., 2002(2002), No.
44, 1-14.

\bibitem{c2} Cavcalcanti, M. M., Domingos Cavcalcanti, V. N., Prates Filho,
J. S. and Soriano, J. A., Existence and uniform decay rates for viscoelastic
problems with nonlinear boundary damping, Differential Integral Equations,
14(2001), 85-116.

\bibitem{c3} Cavcalcanti, M. M. and Oquendo, H. P.., Frictional versus
viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim.,
42(2003), 1310-1324.

\bibitem{g1} Georgiev, V., and Todorova, D., Existence of solutions of the
wave equations with nonlinear damping and source terms, J. Diff. Eqns.
109(1994), 295-308.

\bibitem{g2} Glassey, R. T., Blow-up theorems for nonlinear wave equations,
Math. Z., 132(1973), 183-203.

\bibitem{k1} Keller, J. B., On solutions of nonlinear wave equations, Comm.
Pure. Appl. Math., 10(1957), 523-530.

\bibitem{l1} Levine, H. A., Instability and nonexistence of global solutions
of nonlinear wave equation of the form $Du_{tt}=Au+F(u)$, Trans. Amer. Math.
Soc., 192(1974), 1-21.

\bibitem{l2} Levine, H. A., Some additional remarks on the nonexistence of
global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5(1974),
138-146.

\bibitem{l3} Levine, H. A., Nonexistence of global weak solutions to some
properly and improperly posed problems of mathematical physics : The method
of unbound Fourier coefficients, Math. Ann., 214(1975), 205-220.

\bibitem{l4} Levine, H. A., and Serrin, J., Global nonexistence theorems for
quasilinear evolution equations with dissipation, Arch. Rational Mech.
Anal., 137(1997), 341-361.

\bibitem{l5} Levine, H. A., and Ro Park, S., Global existence and global
nonexistence of solutions of the cauchy problems for a nonlinearly damped
wave equation, J. Math. Anal. Appl., 228(1998), 181-205.

\bibitem{l6} Li, M. R. and Tsai, L. Y., Existence and nonexistence of global
solutions of some systems of semilinear wave equations, Nonlinear Anal.
TMA., 54(2003), 1397-1415.

\bibitem{m1} Matsutama, T. and Ikehata, R., On global solutions and energy
decay for the wave equations of Kirchhoff type with nonlinear damping, J.
Math. Anal. Appl., 204(1996). 729-753.

\bibitem{m2} Messaoudi, S. A. and Said-Houari, S., Blow up and global
existence in a nonlinear viscoelastic equation, Math. Nachr., 260(2003),
58-66.

\bibitem{v1} Vitillaro, E., Global nonexistence theorems for a class of
evolution equations with dissipation, Arch. Rational Mech. Anal., 149(1999),
155-182.
\end{thebibliography}

\end{document}