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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 113, pp. 1--7.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/113\hfil A blow-up result]
{A blow-up result for a viscoelastic system in $\mathbb{R}^n$}

\author[M. Kafini, S. A. Messaoudi \hfil EJDE-2007/113\hfilneg]
{Mohammad Kafini, Salim A. Messaoudi}  % in alphabetical order

\address{Mohammad Kafini \newline
Dammam Community College\\
Dammam, Saudi Arabia}
 \email{mkafini@kfupm.edu.sa}

\address{Salim A. Messaoudi \newline
 Mathematical Sciences Department\\
KFUPM, Dhahran 31261, Saudi Arabia}
\email{messaoud@kfupm.edu.sa}

\thanks{Submitted February 14, 2007. Published August 18, 2007.}
\thanks{Supported by Project SABIC-2006/01 from KFUPM}
\subjclass[2000]{35B05, 35L05, 35L15, 35L55}
\keywords{Blow up; coupled system; relaxation function; viscoelastic}

\begin{abstract}
 In this paper we consider a coupled system of nonlinear viscoelastic
 equations. Under suitable conditions on the initial data and
 the relaxation functions, we prove a finite-time blow-up result.
\end{abstract}

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

\section{Introduction}

In \cite{m1}, Messaudi considered the following initial-boundary value
problem
\begin{equation} \label{e1}
\begin{gathered}
u_{tt}-\Delta u+\int_{0}^{t}g(t-\tau )\Delta u(\tau )d\tau
+u_{t}|u_{t}|^{m-2}=u|u|^{p-2},\quad \text{in }\Omega \times (0,\infty ) \\
u(x,t)=0,\quad x\in \partial \Omega, \; t\geq 0 \\
u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_{1}(x),\quad x\in \Omega ,
\end{gathered}
\end{equation}
where $\Omega $ is a bounded domain of $\mathbb{R}^n $ ($n\geq 1$)
with a smooth boundary $\partial \Omega $, $p>2$, $m\geq 1$, and
$g: \mathbb{R}^{+}\to \mathbb{R}^{+}$ is a
positive non-increasing function. He showed, under suitable conditions on $g$,
that solutions with initial negative energy blow up in finite time if $p>m$
and continue to exist if $m\geq p$. This result has been later pushed, by
the same author \cite{m5}, to certain solutions with positive initial energy. A
similar result been also obtained by Wu \cite{w1} using a different method.

In the absence of the viscoelastic term ($g=0$), problem \eqref{e1} has been
extensively studied and many results concerning global existence and
nonexistence have been proved. For instance, for the equation
\begin{equation} \label{e2}
u_{tt}-\Delta u+au_{t}|u_{t}|^{m}=b|u|^{\gamma }u,\quad \text{in }\Omega
\times (0,\infty )
\end{equation}
$m,\gamma \geq 0$, it is well known that, for $a=0$, the source term
$bu|u|^{\gamma }$, ($\gamma >0$) causes finite time blow up of solutions with
negative initial energy (see \cite{b1}). The interaction between the damping and
the source terms was first considered by Levine \cite{l2}, and in \cite{l3}
the linear damping case ($m=0$).
He showed that solutions with negative initial energy
blow up in finite time. Georgiev and Todorova \cite{g1} extended Levine's result
to the nonlinear damping case ($m>0$). In their work, the authors introduced
a different method and showed that solutions with negative energy continue
to exist globally ``in time'' if $m\geq \gamma $ and blow up in finite time if
$\gamma >m$ and the initial energy is sufficiently negative. This last
blow-up result has been extended to solutions with negative initial energy
by Messaoudi \cite{m2} and others. For results of same nature,
we refer the reader
to Levine and Serrin \cite{l1}, and Vitillaro \cite{t1}, Messaoudi
and Said-Houari \cite{m4}.

For problem \eqref{e2} in $\mathbb{R}^n $, we mention, among others,
the work of Levine Serrin and Park \cite{l4}, Todorova \cite{t1,t2},
 Messaoudi \cite{m3}, and Zhou \cite{z1}.

In this work, we are concerned with the  Cauchy problem
\begin{equation} \label{e3}
\begin{gathered}
u_{tt}-\Delta u+\int_{0}^{t}g(t-s)\Delta u(x,s)ds=f_{1}(u,v),
\quad \text{in }\mathbb{R}^n \times (0,\infty ) \\
v_{tt}-\Delta v+\int_{0}^{t}h(t-s)\Delta v(x,s)ds=f_{2}(u,v),
\quad \text{in }\mathbb{R}^n \times (0,\infty ) \\
u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_{1}(x),\quad x\in \mathbb{R}^{n} \\
v(x,0)=v_{0}(x),\quad v_{t}(x,0)=v_{1}(x),\quad x\in \mathbb{R}^{n}
\end{gathered}
\end{equation}
where $g,h$, $u_{0}$, $u_{1,}$ $v_{0}$, $v_{1}$ are functions to be
specified later. This type of problems arises in viscoelasticity and in
systems governing the longitudinal motion of a viscoelastic configuration
obeying a nonlinear Boltzmann's model. Our aim is to extend the result of
\cite{z1}, established for the wave equation, to our problem. To achieve this
goal some conditions have to be imposed on the relaxation functions $g$ and
$h$.

\section{Preliminaries}

In this section we present some material needed in the proof of our main
result. So, we make the following assumption
\begin{itemize}
\item[(G1)] $g,h:\mathbb{R}_{+}\to \mathbb{R}_{+}$ are nonincreasing
differentiable functions satisfying
\begin{gather*}
1-\int_{0}^{\infty }g(s)ds = l>0,\quad g'(t)\leq 0,\quad t\geq 0. \\
1-\int_{0}^{\infty }h(s)ds = k>0,\quad h'(t)\leq 0,\quad t\geq 0.
\end{gather*}
\item[(G2)] There exists a function $I(u,v)\geq 0$ such that
\[
\frac{\partial I}{\partial u}=f_{1}(u,v),\quad
\frac{\partial I}{\partial v}=f_{2}(u,v).
\]
\item[(G3)] There exists a constant $\rho >2$ such that
\[
\int_{\mathbb{R}^n }[uf_{1}(u,v)+vf_{2}(u,v)-\rho I(u,v)]dx\geq
0.
\]
\item[(G4)] There exists a constant $d>0$ such that
\begin{gather*}
| f_{1}(\xi ,\varsigma )| \leq d(| \xi | ^{\beta
_{1}}+| \varsigma | ^{\beta _{2}}),\quad \forall (\xi
,\varsigma )\in \mathbb{R}^{2}, \\
| f_{2}(\xi ,\varsigma )| \leq d(| \xi | ^{\beta
_{3}}+| \varsigma | ^{\beta _{4}}),\quad \forall (\xi
,\varsigma )\in \mathbb{R}^{2},
\end{gather*}
where
\[
\beta _{i}\geq 1,\quad (n-2)\beta _{i}\leq n,\quad i=1,2,3,4.
\]
\end{itemize}
Note that (G1) is necessary to guarantee the hyperbolicity of the
system \eqref{e3}.
As an example of functions satisfying (G2)-(G4), we have
\[
I(u,v)=\frac{a}{\rho }| u-v| ^{\rho },\quad
\rho >2,\quad (n-2)\rho \leq 2(n-1).
\]
Condition (G4) is necessary for the existence of a
local solution to \eqref{e3}.

We introduce the ``modified'' energy functional
\begin{equation} \label{e4}
\begin{aligned}
E(t) &:=\frac{1}{2}\| u_{t}\| _{2}^{2}+\frac{1}{2}\|
v_{t}\| _{2}^{2}+\frac{1}{2}(1-\int_{0}^{t}g(s)ds)\|
\nabla u\| _{2}^{2}+\frac{1}{2}(1-\int_{0}^{t}h(s)ds)
\| \nabla v\| _{2}^{2}   \\
&\quad +\frac{1}{2}(g\circ \nabla u)+\frac{1}{2}(h\circ \nabla
v)-\int_{\mathbb{R}^n }I(u,v)dx,
\end{aligned}
\end{equation}
where
\begin{equation} \label{e5}
\begin{gathered}
(g\circ \nabla u)(t) =\int_{0}^{t}g(t-\tau
)||u(t)-\nabla u(\tau )||_{2}^{2}d\tau .   \\
(h\circ \nabla v)(t) =\int_{0}^{t}h(t-\tau )||\nabla
v(t)-\nabla v(\tau )||_{2}^{2}d\tau .
\end{gathered}
\end{equation}

\section{Blow up results}

In this section we state and prove our main result.

\begin{theorem} \label{thm1}
Assume that {\rm (G1)--(G4)} hold  and that
\begin{equation} \label{e6}
\max \Big\{ \int_{0}^{+\infty }g(s)ds,\int_{0}^{+\infty
}h(s)ds\Big\} \leq \frac{\rho (\rho -2)}{1+\rho (\rho -2)}.
\end{equation}
Then for  initial data $(u_{0},v_{0}),(u_{1},v_{1})\in H^{1}(
\mathbb{R}^{N})\times L^{2}(\mathbb{R}^{N})$,
with compact support, satisfying
\begin{equation} \label{e7}
E(0)=\frac{1}{2}\| u_{1}\| _{2}^{2}+\frac{1}{2}\|
\nabla u_{0}\| _{2}^{2}+\frac{1}{2}\| v_{1}\|
_{2}^{2}+\frac{1}{2}\| \nabla v_{0}\| _{2}^{2}
-\int_{\mathbb{R}^n }I(u_{0},v_{0})dx<0,
\end{equation}
the corresponding solution (of (1.3))blows up in finite time.
\end{theorem}

\begin{proof}
Multiplying  \eqref{e3} by $u_{t}$ and $v_{t}$
respectively, integrating over $\mathbb{R}^n $, using
integration by parts, and repeating the same computations as in
\cite{m1}, we obtain
\begin{equation} \label{e8}
E'(t)=\frac{1}{2}(g'\circ \nabla u)+\frac{1}{2}
(h'\circ \nabla v)-\frac{1}{2}g(s)\| \nabla
u\| _{2}^{2}-\frac{1}{2}h(s)\| \nabla v\|
_{2}^{2}\leq 0.
\end{equation}
Hence,
\begin{equation} \label{e9}
E(t)\leq E(0)<0.
\end{equation}
We then define
\begin{equation} \label{e10}
F(t)=\frac{1}{2}\int_{\mathbb{R}^n }[| u(x,t)|
^{2}+| v(x,t)| ^{2}]dx+\frac{1}{2}\beta (t+t_{0})^{2},
\end{equation}
for $t_{0}\smallskip >0$ and $\beta >0$ to be chosen later.
By differentiating $F$ twice we get
\begin{gather}
F'(t)=\int_{\mathbb{R}^n }(u_{t}u+v_{t}v)dx+\beta
(t+t_{0}), \label{e11}\\
F''(t)=\int_{\mathbb{R}^n }(u_{tt}u+v_{tt}v)dx+
\int_{\mathbb{R}^n }(| u_{t}| ^{2}+|
v_{t}| ^{2})dx+\beta . \label{e12}
\end{gather}
To estimate the term $\int_{\mathbb{R}^n }(u_{tt}u+v_{tt}v)dx$
in (3.7), we multiply the equations in \eqref{e3} by $u$
and $v$ respectively and integrate by parts over $\mathbb{R}^n $ to get
\begin{align*}
\int_{\mathbb{R}^n }(uu_{tt}+vv_{tt})dx
&=-\int_{\mathbb{R}^{n}}(| \nabla u| ^{2}+| \nabla v|^{2})dx
+\int_{\mathbb{R}^n }[uf_{1}(u,v)+vf_{2}(u,v)]dx \\
&\quad +\int_{0}^{t}g(t-s)\int_{\mathbb{R}^n }\nabla
u(x,t).\nabla u(x,s)\,dx\,ds \\
&\quad +\int_{0}^{t}h(t-s)\int_{\mathbb{R}^n }\nabla
v(x,t).\nabla v(x,s)\,dx\,ds.
\end{align*}
Using Young's inequality and (G3) we arrive at
\begin{equation} \label{e13}
\begin{aligned}
\int_{\mathbb{R}^n }(uu_{tt}+vv_{tt})dx
&\geq \Big[ -1-\delta +\int_{0}^{t}g(s)ds\Big] \|
 \nabla u\|_{2}^{2}+\rho \int_{\mathbb{R}^n }I(u,v)dx \\
&\quad -\frac{1}{4\delta }\Big(\int_{0}^{t}g(s)ds\Big)(g\circ \nabla
u)+\Big[ -1-\delta +\int_{0}^{t}h(s)ds\Big] \| \nabla v\| _{2}^{2}   \\
&\quad -\frac{1}{4\delta }\Big(\int_{0}^{t}h(s)ds\Big)(h\circ \nabla v)
+\int_{\mathbb{R}^n }\big(| u_{t}| ^{2}+|v_{t}| ^{2}\big)dx,
\end{aligned}
\end{equation}
we then insert \eqref{e13} in \eqref{e12} to obtain
\begin{equation} \label{e14}
\begin{aligned}
F''(t) &\geq (-1-\delta +\int_{0}^{t}g(s)ds)\|
\nabla u\| _{2}^{2}-\frac{1}{4\delta }(\int
_{0}^{t}g(s)ds)(g\circ \nabla u)   \\
&\quad+(-1-\delta +\int_{0}^{t}h(s)ds)\| \nabla v\|
_{2}^{2}-\frac{1}{4\delta }(\int_{0}^{t}h(s)ds)(h\circ \nabla
v) \\
&\quad +\rho \int_{\mathbb{R}^n }I(u,v)dx+2\int_{\mathbb{R}
^{n}}(| u_{t}| ^{2}+| v_{t}| ^{2})dx+\beta .
\end{aligned}
\end{equation}
Now, we exploit \eqref{e4} to substitute for $\int_{\mathbb{R}
^{n}}I(u,v)dx$ , thus \eqref{e14} takes the form
\begin{equation} \label{e15}
\begin{aligned}
F''(t) &\geq -\rho E(t)+\beta +[(-1-\delta
+\int_{0}^{t}g(s)ds)+\frac{\rho }{2}\Big(1-\int_{0}^{t}g(s)ds\Big)]
\| \nabla u\| _{2}^{2}   \\
&\quad +[(-1-\delta +\int_{0}^{t}h(s)ds)+\frac{\rho }{2}
(1-\int_{0}^{t}h(s)ds)]\| \nabla v\| _{2}^{2}
 \\
&\quad +\Big[\frac{\rho }{2}-\frac{1}{4\delta }\big(\int_{0}^{t}g(s)ds\big)\Big](g\circ
\nabla u)   \\
&\quad +\Big[\frac{\rho }{2}-\frac{1}{4\delta }\big(\int_{0}^{t}h(s)ds\big)\Big]
(h\circ\nabla v)+(\frac{\rho }{2}+2)[\| u_{t}\| _{2}^{2}+\|
v_{t}\| _{2}^{2}].
\end{aligned}
\end{equation}
At this point, we introduce
\[
G(t):=F^{-\gamma }(t),
\]
for $\gamma >0$ to be chosen properly. By differentiating $G$ twice we
arrive at
\[
G'(t)=-\gamma F^{-(\gamma +1)}(t)F'(t),\quad G''(t)=-\gamma F^{-(\gamma +2)}(t)Q(t),
\]
where
\begin{equation} \label{e16}
\begin{aligned}
Q(t) &= F(t)F''(t)-(\gamma +1)(F')^ 2(t)   \\
&\geq F(t)\Big\{-\rho E(t)+\beta +\Big[ (-1-\delta +\frac{\rho }{2})-(
\frac{\rho }{2}-1)\int_{0}^{t}g(s)ds)\Big] \| \nabla
u\| _{2}^{2}   \\
&\quad +\Big[ (-1-\delta +\frac{\rho }{2})-(\frac{\rho }{2}-1)\int
_{0}^{t}h(s)ds)\Big] \| \nabla v\| _{2}^{2} \\
&\quad +\big[\frac{\rho }{2}-\frac{1}{4\delta }(\int_{0}^{t}g(s)ds)\big]
(g\circ \nabla u)+\big[\frac{\rho }{2}-\frac{1}{4\delta }(\int
_{0}^{t}h(s)ds)\big](h\circ \nabla v)   \\
&\quad +(\frac{\rho }{2}+2)[\| u_{t}\| _{2}^{2}+\| v_{t}\|
_{2}^{2}]\Big\}
-(\gamma +1)\Big[\int_{\mathbb{R} ^{n}}(u_{t}u+v_{t}v)dx+\beta (t+t_{0})\Big]^{2}.
\end{aligned}
\end{equation}
Using Young's and Cauchy-Schwartz inequalities, we estimate the last term
in (3.11) as follows:
\begin{align*}
&\Big[\int_{\mathbb{R}^n }(u_{t}u+v_{t}v)dx+\beta (t+t_{0})\Big]^{2} \\
&\leq \Big(\int_{\mathbb{R}^n }(u_{t}u+v_{t}v)dx\Big)^{2}
+2\Big[\frac{ \varepsilon }{2}\Big(\int_{\mathbb{R}^n }(u_{t}u+v_{t}v)dx\Big)^{2} \\
&\quad +\frac{1}{2\varepsilon }\beta ^{2}(t+t_{0})^{2}\Big]
+\beta ^{2}(t+t_{0})^{2} \\
&\leq (1+\varepsilon )(\int_{\mathbb{R}
^{n}}(u_{t}u+v_{t}v)dx)^{2}+(1+\frac{1}{\varepsilon })\beta ^{2}(t+t_{0})^{2}
\\
&\leq (1+\varepsilon )\Big[ \int_{\mathbb{R}^n }u^{2}dx+\int_{
\mathbb{R}^n }v^{2}dx\Big] \Big[ \int_{\mathbb{R}
^{n}}u_{t}^{2}dx+\int_{\mathbb{R}^n }v_{t}^{2}dx\Big] \\
&\quad +(1+\frac{1}{\varepsilon })\beta ^{2}(t+t_{0})^{2} \\
&\leq 2F(x)\Big[ (1+\varepsilon )\big(\int_{\mathbb{R}
^{n}}u_{t}^{2}dx+\int_{\mathbb{R}^n }v_{t}^{2}dx\big)+(1+\frac{1}{
\varepsilon })\beta \Big] .
\end{align*}
Hence, \eqref{e16} becomes
\begin{equation} \label{e17}
\begin{aligned}
Q(t) &\geq  F(t)\Big\{ \Big[ (-1-\delta +\frac{\rho }{2})-(\frac{\rho }{2}
-1)\int_{0}^{t}g(s)ds)\Big] \| \nabla u\|_{2}^{2}   \\
&\quad+\Big[ (-1-\delta +\frac{\rho }{2})-(\frac{\rho }{2}-1)\int
_{0}^{t}h(s)ds)\Big] \| \nabla v\| _{2}^{2}
 \\
&\quad +[\frac{\rho }{2}-\frac{1}{4\delta }\int_{0}^{t}g(s)ds](g\circ \nabla u)
+[\frac{\rho }{2}-\frac{1}{4\delta }\int_{0}^{t}h(s)ds](h\circ
\nabla v)   \\
&\quad+\big[ \frac{\rho }{2}+2-2(\gamma +1)(1+\varepsilon \big]
\big[ \| u_{t}\| _{2}^{2}+\| v_{t}\| _{2}^{2}\big]
 \\
&\quad-\rho E_{0}-2(\gamma +1)(1+\frac{1}{\varepsilon })\beta \Big\},\quad
\forall \varepsilon >0.
\end{aligned}
\end{equation}
We choose $\varepsilon <\rho/4$,
$0<\gamma <(\rho -4\varepsilon)/(4(1+\varepsilon ))$, and
$\beta $ small so that
\[
-\rho E_{0}-[2+\frac{2}{\varepsilon }+\gamma (2+\frac{2}{\varepsilon }
)]\beta \geq 0.
\]
Next, we choose $\delta >0$ so that
\[
-1-\delta +\int_{0}^{t}g(s)ds+\frac{\rho }{2}(1-\int
_{0}^{t}g(s)ds)\geq 0,\quad \frac{\rho }{2}-\frac{1}{4\delta }
\int_{0}^{t}g(s)ds\geq 0,
\]
and
\[
-1-\delta +\int_{0}^{t}h(s)ds +\frac{\rho }{2}
\big(1-\int _{0}^{t}h(s)ds\big) \geq 0,\quad
\frac{\rho }{2}-\frac{1}{4\delta }\int_{0}^{t}h(s)ds\geq 0.
\]
This is, of course, possible by \eqref{e6}, we then conclude,
from \eqref{e17}, that
$Q(t)\geq 0$, for all $t\geq 0$. Therefore
$G''(t)\leq 0$ for all $t\geq 0;$ which implies that $G'$ is decreasing. By
choosing $t_{0}$ large enough we get
\[
F'(0)=\int_{\mathbb{R}^n }(u_{0}u_{1}+v_{0}v_{1})dx+
\beta t_{0}>0,
\]
hence $G'(0)<0$. Finally Taylor expansion of $G$ yields
\[
G(t)\leq G(0)+tG'(0),\quad \forall t\geq 0,
\]
which shows that $G(t)$ vanishes at a time $t_{m}\leq -\frac{G(0)}{G'(0)}$.
Consequently $F(t)$ must become infinite at time $t_{m}$.
\end{proof}

\subsection*{Acknowledgments}
The authors would like to express their sincere
thanks to King Fahd University of Petroleum and Minerals for its support. 


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\end{document}