\documentclass[twoside]{article}
\input amssym.def     % used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil On a mixed problem for a coupled nonlinear system\hfil EJDE--1997/06}%
{EJDE--1997/06\hfil M.R. Clark. \& O.A. Lima\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1997}(1997), No.\ 06, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
 \vspace{\bigskipamount} \\
On a mixed problem for a coupled nonlinear system
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35M10.\newline\indent
{\em Key words and phrases:} Mixed problem, nonlinear system, 
weak solutions, uniqueness.
\newline\indent
\copyright 1997 Southwest Texas State University  and University of
North Texas.\newline\indent
Submitted November 26, 1996. Published March 6, 1997.} }
\date{}
\author{ M.R. Clark. \& O.A. Lima \\ \\
({\normalsize Dedicated to professor Luiz A. Medeiros for his 70th 
birthday)}}

\maketitle

\begin{abstract} 
In this article we prove the existence and uniqueness of solutions to
 the mixed problem associated with the nonlinear system
$$
u_{tt}-M(\int_\Omega |\nabla u|^2dx)\Delta u+|u|^\rho u+\theta =f $$
$$\theta _t -\Delta \theta +u_{t}=g 
$$ 
where $M$ is a positive real function, and $f$ and $g$ are known real 
functions.
\end{abstract}

\def\text#1{\mbox{ #1 }}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}

\section{Introduction}

Let $\Omega$ be an open and bounded subset of ${\Bbb R}^m$, with smooth
boundary $\Gamma$. Let $Q$ be the cylinder $Q=\Omega \times
]0,T[$ and $\sum$ its lateral boundary. Let us denote the usual norm in 
$H_0^m(\Omega )$ by $\|\cdot \|$ and the usual norm in $L^2(\Omega )$ 
by $|\cdot |$, where $H_0^m(\Omega )$ is the closure of 
$C_0^\infty(\Omega)$ in $H^m(\Omega )$, and $H^m(\Omega )$ is the
standard Sobolev space.

We shall consider the nonlinear system 
\begin{eqnarray}
&u_{tt}-M(\int_\Omega |\nabla u|^2dx)\Delta u+|u|^\rho u+\theta
=f\text{ in } Q & \label{1.1}\\
&\theta _t\ -\;\Delta \theta +u_{t\;}=g\;\text{in\ }Q & \label{1.2}\\
&\ u=\theta =0\;\text{on}\sum  & \label{1.3}\\
&u(0)=u_{0;\;\;}u'(0)=u_1;\;\theta (0)=\theta _0 & \label{1.4}
\end{eqnarray}

When $M(s)$ is a positive constant $\alpha $ and $\theta =0$, the dynamical
part of the above system   is a nonlinear perturbation of the 
linear wave equation $u_{tt}-\alpha \Delta u=f$, 
 (cf. Lions \cite{lions2}). 
When\ $M(s)=m_0+m_1s$, with $m_0$ and $m_1$ positive constants and
$\theta =0$, Equation (\ref{1.1}) is a nonlinear perturbation of the 
canonical
Kirchhoff-Carrier's model which describes small vibrations of a stretched
string when tension is assumed to have only a vertical component at each
point of the string (cf. Pohozhaev \cite{pohozhaev}, Arosio-Spagnolo \cite
{arosio-spagnolo}). For $\theta =0$, Hosoya-Yamada \cite{hosoya-yamada},
investigate the existence, uniqueness and regularity of solutions of (1.1).

In \cite{medeiros}, L. A. Medeiros studies the equation (\ref{1.1})
 when $\theta =0$ and the nonlinear perturbation is equal to $u^2$. 
Lastly, in  \cite{maciel-lima} Maciel-Lima, studied the existence of a 
local weak solution
of the mixed problem for the perturbed Kirchhoff-Carrier's equation
$$
u''-M(\int_\Omega |\nabla u|^2dx)\Delta u+\lambda |u|^\rho u=f\,, 
$$
when $\lambda =-1$, $M:[0,\infty )\rightarrow [0,\infty )$ is a
$C^1$ function such that $M(s)\geq m_0>0,\forall s\in {\Bbb R}$, where 
$\rho \in {\Bbb R}$
and satisfies $0<\rho \leq  2/(n-4)$ if $n\geq 5$  or $\rho \geq 0$
if $n=1,2,3$, or 4. For other perturbations of
Kirchhoff-Carrier's operator, among several works, we cite 
D'ancona-Spagnolo  \cite{d'ancona-spagnolo}, and Bisognin \cite{bisognin}.

In the present work we discuss the existence of a weak solution for the
coupled nonlinear system (\ref{1.1})--(\ref{1.3}) where we impose the appropriate
assumptions on $M$, $\rho$,$f$ and $g$. For the proof of existence, we 
employ the Galerkin's approximation method plus a compactness argument 
(see, e.g., Lions \cite{lions1}).


\section{Notation and main result}

We make the following assumptions:
$$
M\in C^1[0,\infty )\;\text{and}\;M(s)\geq m_0>0\;\text{for\ }s\geq 0. 
\eqno{(A.1)}
$$
$$
0<\rho \leq \frac 2{n-2}\text{\ \ if\ }n\geq 5\;\text{and }0\leq \rho
<\infty \;\text{if\ }n=1,\;2,\;3\;\text{or}\;4  \eqno{(A.2)}
$$
$$
f,\;g\;\in \;C^0(0,T;H_0^1\left( \Omega \right) )  \eqno{(A.3)}
$$

The main result of the present work is given in the following theorem. 

\begin{theorem}
Assume (A.1)--(A.3). For 
$$
u_0\in H_0^1(\Omega )\cap H^2(\Omega ),\ u_1\in H_0^1(\Omega ),\text{ and }
\theta _0\in H_0^1\left( \Omega \right) 
$$
there exist  $T_0\in {\Bbb R}$, $0<T_0<T$
such that (\ref{1.1})--(\ref{1.4}) has a unique weak solution $\{u,\theta\}$ on 
$[0,T_0]$ satisfying (\ref{1.1}) and (\ref{1.2}) in the following sense:
\begin{eqnarray*}
&\frac d{dt}(u'(t),w)+M(\int_\Omega |\nabla
u(t)|dx)a(u(t),w)+(|u(t)|^\rho u(t),w)+
(\theta (t),w) 
=(f(t),w)&\\
&\frac d{dt}(\theta (t),w)+a(\theta (t),w)+(u'(t),w)=(g(t),w)& 
\end{eqnarray*}
for all $w\in H_0^1(\Omega )$ in the sense of $D'(0,T)$.
$$
u(0)=u_0,\ u'(0)=u_1,\ \theta '(0)=\theta _0 
$$
\end{theorem}

\paragraph{Proof of Theorem 1.}
Let $w_1,...,w_m$ be the eigenfunctions of the Laplacian on $\Omega $ and
let $V_m\;$be the space generated by the first$\;m$ eigenfunctions. Now let
us consider the approximated system 
\begin{eqnarray}
&(u_m^{\prime \prime }(t),w_k)-M(\|u_m(t)\|^2)(\Delta u_m(t),w_k) 
&\nonumber \\
&+(|u_m(t)|^\rho u_m(t),w_k)+ (\theta _m(t),w_k)=(f(t),w_k)& \label{2.6}\\
&(\theta _m'(t),w_k)-(\Delta \theta _m(t),w_k)+(u_m'(t),w_k)=(g(t),w_k)&
  \label{2.7}\\
&u_m(0)=u_{0m}\longrightarrow u_0\text{ strongly in }H_0^1(\Omega )\cap
H^2(\Omega )&  \label{2.8}\\
&u_m'\left( 0\right) =u_{1m}\longrightarrow u_1\text{ strongly in }
H_0^1(\Omega ) & \label{2.9}\\
&\theta _m(0)=\theta _{0m}\longrightarrow \theta _0 \text{ strongly in }
H_0^1(\Omega ) & \label{2.10} 
\end{eqnarray}
where $1\leq k\leq m$.
Then there exist functions $c_{km}$ and $d_{mk}$ such that
\[
u_m(t)=\sum_{k=1}^mc_{km}(t) w_k  \text{ and }  \theta
_m(t)=\sum_{k=1}^md_{km}(t)w_k
\]
are the unique local solutions of the above system on some
interval $[0,t_m[$, where $t_m\in [0,T[$.

The estimates that we obtain below will allow us to extend the solutions $%
\{u_m,\theta _m\}$ to the interval $[0,T[$.

\paragraph{Estimate (i).} Multiply (\ref{2.6}) by $c_{km}'(t)$ and multiply 
(\ref{2.7}) by $d_{km}(t)$, then sum over $k$ to obtain: 
\begin{eqnarray}
\lefteqn{\frac 12\frac d{dt}\{|u_m'(t)|^2+\hat M(\|u_m(t)\|^2)\}+\frac
1p\frac d{dt}\|u_m(t)\|_{L^p(\Omega )}^p} \\
&= &-(\theta _m(t),u_m'(t))+(f(t),u_m'( t) )
\label{2.11} 
\end{eqnarray}
where $p=\rho +2$.
\begin{equation}
\frac 12\frac d{dt}\{|\theta _m(t)|^2+\|\theta _m(t)\|^2\}
=-(u_m'(t),\theta _m(t))+(g(t),\theta _m(t))  \label{2.12}
\end{equation}
Define 
\[
E(u(t),\theta (t))=\frac 12\{|u'(t)|^2+|\theta (t)|^2+\hat
M(\|u(t)\|^2)+\|\theta (t)\|^2\}+\frac 1p\|u(t)\|_{L^p(\Omega )}^p 
\]
where $\hat M(\lambda )=\int_0^\lambda M(s)ds$.

Sum (\ref{2.11}) and (\ref{2.12}). Using the inequality 
$ab\leq \frac12\left( a^2+b^2\right) $ and the Poincar\'e inequality we
integrate from $0$ to $t\leq t_m$ to obtain 
\begin{eqnarray*} 
\lefteqn{\frac 12\{|u_m'(t)|^2+|\theta _m(t)|^2+m_0\|u_m(t)\|^2+\|\theta
_m(t)\|^2\}+\frac 1p \|u_m(t)\|_{L^p(\Omega )}^p } \\
&\leq & E(u_m(t),\theta _m(t)) \\
&\leq & E(u_{0m},\theta _{0m})+\frac 12\int_0^T\|f(s)\|^2
+\frac 12\int_0^T\|g(s)\|^2ds \\ 
&&+\frac 32\int_0^t\|u_m'(s)\|^2ds+\int_0^t|\theta _m(s)|^2ds 
\end{eqnarray*}


From (\ref{2.8})--(\ref{2.10}) and  hypotheses (A.3), it follows from Gronwall's
inequality that 
\begin{eqnarray*}
|u_m'(t)|^2+|\theta _m(t)|^2+m_0\|u_m(t)\|^2+\|\theta
_m(t)\|^2+\frac 12\|u_m(t)\|_{L^p}^p \\ 
\leq \{2E(u_0,\theta _0)+\int_0^T\|f(s)\|^2ds+\int_0^T\|g(s)\|^2ds\}e^T
\end{eqnarray*}

Then we extend the approximate solution $\{u_m(t),\theta _m(t)\}\;$to
the interval\ $[0,T[$ and\ we have the estimates 
\begin{equation}
|u_m'(t)|\leq C_1,\quad \|u_m(t)\|\leq C_2,\text{ and }
\|\theta _m(t)\|\leq C_1
\label{2.15}
\end{equation}
where $C_1=\{2E(u_0,\theta_0)+\int_0^T\|f(s)\|^2ds+\int_0^T\|g(s)\|^2ds\}
e^T$ and $C_2=C_1m_0^{-1}$.

From now on we denote by $C$ various positive constants
independent of $m$ and $t$ in $[0,T[$.

\paragraph{Estimate (ii).} Observe that the system (\ref{2.6}), (\ref{2.7}) is 
equivalent to
\begin{eqnarray}
&(u_m''(t),w)-M(\|u_m(t)\|^2)(\Delta u_m(t),w)+
(|u_m(t)|^\rho u_m(t),w)+ (\theta _m(t),w) &\nonumber  \\ 
&=(f(t),w)& \label{2.16}\\
&(\theta _m'(t),w)-(\Delta \theta _m(t),w)+(u_m'(t),w)
=(g(t),w)& \label{2.17}
\end{eqnarray}
for all $w\in V_m$. Putting $w=-\Delta u_m'(t)\in V_m$ in (\ref{2.16}) and 
$w=-\Delta\theta_m(t)\in V_m$ in (\ref{2.17}) we have 
\begin{eqnarray}
\lefteqn{\frac 12\frac d{dt}\{\|u_m'(t)\|^2+M(\|u_m(t)\|^2)|\Delta
u_m(t)|^2\} }&& \label{2.18} \\ 
&=& -(\nabla (|u_m(t)|^\rho u_m(t),\nabla u_m'\left( t\right) ) 
 +M'(\|u_m'(t)\|^2)(\nabla u_m(t),\nabla u_m'(t))|\Delta u_m(t)|^2
  \nonumber \\
&& -(\nabla u_m'\left( t\right) ,\nabla \theta _m(t))+(\nabla
f(t),\nabla u_m'( t) ) \nonumber
\end{eqnarray}
\begin{equation}
\frac 12\frac d{dt}\|\theta _m(t)\|^2+|\Delta \theta _m(t)|^2=-(\nabla
u_m'\left( t\right) ,\nabla \theta _m(t))+ 
(\nabla g(t),\nabla \theta _m(t))
\label{2.19}
\end{equation}
Adding equations (\ref{2.18}) and (\ref{2.19}) we have:
\begin{eqnarray}
\lefteqn{\frac 12\frac d{dt}\{\|u_m'(t)\|^2+\|\theta
_m(t)\|^2+M(\|u_m(t)\|^2)|\Delta u_m(t)|^2\}+  
|\Delta \theta _m(t)|^2 }&& \label{2.20} \\
&=&-(\nabla (|u_m(t)|^\rho u_m(t)),\nabla u_m'(t) )+
M'(|u_m'(t)|^2)(\nabla u_m(t),\nabla u_m^{\prime
}(t))|\Delta u_m(t)|^2\nonumber \\
&&-2(\nabla u_m'\left( t\right) ,\nabla \theta _m(t))+(\nabla
f(t),\nabla u_m'\left( t\right) )+(\nabla g(t),\nabla \theta_m(t))
\nonumber \end{eqnarray}
We have that
\begin{eqnarray*}
&|\left( \nabla (|u_m(t)|^\rho u_m(t)),\nabla u_m'\left( t\right)
\right) | \leq \\
&(\rho +1)\int_\Omega |u(t)|^\rho |\nabla u_m(t)\|\nabla
u_m'\left( t\right) |dx 
(\rho +1)|u(t)|_{L^{\rho q}}^\rho \cdot |\nabla u_m(t)|_{L^r}\cdot
\|u_m'\left( t\right) \|&
\end{eqnarray*}
with $1/q+1/r=1/2$.

From hypotheses (A.2) we can take $q$ and $r$ such that 
\[
\frac 1q\geq \frac{\rho(n-4)}{2n}  \text{ and }\frac 1r\geq
\frac{n-2}{2n}\,.
\]
 Sobolev's inequality gives 
\[
|\nabla u_m(t)|_{L^r}\leq C|u_m(t)|_{H^2} \text{ and } |u_m(t)|_{L^{\rho
q}}\leq C|u_m(t)|_{H^2}
\]
and the regularity theory for elliptic equations ensures that 
\[
|u_m(t)|_{H^2}\leq C|\Delta u_m(t)| 
\]
(see, e. g., Friedman\thinspace  \cite{friedman}).

Therefore, 
\begin{equation}
|\left( \nabla (|u_m(t)|^\rho u_m(t)),\nabla u_m'\left( t\right)
\right) |\leq C|\Delta u_m(t)|^{\rho +1}\|u_m'(t)\|  \label{2.22}
\end{equation}
The second, third, fourth, and fifth terms of the right side in
(\ref{2.20}) are bounded as follows
$$
\left| M'(|u_m'(t)|^2)\left( \nabla u_m(t),\nabla
u_m'(t)\right) |\Delta u_m(t)|^2\right| \leq M_1C_2\|u_m^{\prime
}(t)\|\cdot |\Delta u_m(t)|^2 
$$
where $M_1=\max \{|M'(s)|;0\leq s\leq C_2\}$,
\begin{eqnarray}
2|(\nabla u_m'\left( t\right) ,\nabla \theta _m(t))|
&\leq& \|u_m'(t)\|^2+\|\theta _m(t)\|^2  \label{2.24}\\
|(\nabla f(t),\nabla u_m'\left( t\right) )|
&\leq & \frac12\|f(t)\|^2+\frac 12\|u_m'(t)\|^2  \label{2.25}\\
|(\nabla g(t),\nabla \theta _m(t))|
&\leq& \frac 12\|g(t)\|^2+\frac12\|\theta _m(t)\|^2  \label{2.26}
\end{eqnarray}

Let us define the functional 
\[
F(u(t),\theta (t))=\|u_m'(t)\|^2+\|\theta
_m(t)\|^2+M(\|u(t)\|^2)|\Delta u(t)|^2+|\Delta \theta (t)|^2\,. 
\]
Then by (\ref{2.15}) we have 
\begin{eqnarray}
\lefteqn{\|u_m'(t)\|^2+\|\theta _m(t)\|^2+m_0|\Delta u_m(t)|^2+|\Delta
\theta _m(t)|^2}&&  \nonumber\\ 
&\leq& F(u_m(t),\theta _m(t)) \label{2.27} \\
&\leq& \|u_m'(t)\|^2+\|\theta _m(t)\|^2+ M_2|\Delta u_m(t)|^2+
|\Delta \theta _m(t)|^2 \nonumber
\end{eqnarray}
where $M_2=\max \{M(s);0\leq s\leq C_2^2\}$.
Making use of inequalities (\ref{2.22})--(\ref{2.27}) in (\ref{2.20}) it follows that 
\begin{eqnarray*}
\lefteqn{ \frac d{dt}F(u_m(t),\theta _m(t))} \\
&\leq& 2C|\Delta u_m(t)|^{\rho +1}\cdot
\|u_m'(t)\|+ 2M_1C_2\|u_m'(t))\|\cdot |\Delta u_m(t)|^2\\ 
&&+\|f(t)\|^2+\|g(t)\|^2+
3\|u_m'(t)\|^2+3\|\theta _m(t)\|^2
\end{eqnarray*}
By (\ref{2.27}) we have,
\begin{eqnarray*}
\lefteqn{ \frac d{dt}F(u_m(t),\theta _m(t))}\\
&\leq& C\left\{ F(u_m(t),\theta _m(t))^{%
\frac{\rho +2}2}+F(u_m(t),\theta _m(t))^{\frac 32}+F(u_m(t),\theta
_m(t))\right\} \\ 
&&+\|f(t)\|^2+\|g(t)\|^2
\end{eqnarray*}
A simple computation shows that\ 
\[
\frac d{dt}F(u_m(t),\theta _m(t))\leq C\{F(u_m(t),\theta _m(t))^\gamma
+\|f(t)\|^2+\|g(t)\|^2\}, 
\]
with $\gamma =\max \{(\rho +2)/2,\;3/2\}$. Here we
need the following lemma which will be proved later.

\begin{lemma}
\ Let $\mu$ a positive and differentiable function such that 
\begin{equation}
\mu '(t)\leq \theta (t)+\alpha \mu (t)+\beta \mu ^\gamma (t) 
\label{*}
\end{equation}
where $\theta (t)$ is a positive function, $\theta \in L^1(0,T)$, $\alpha
$, $\beta $, and $\gamma $ are positive constants, with $\gamma >1$.
 Then there exists $T_0\in {\Bbb R}$, where $0<T_0<T$, such that 
$\mu $ is bounded on $[0,T_0]$.
\end{lemma}

By Lemma 1, there exist $T_0>0$ such that 
\[
F(u_m(t),\theta _m(t))\leq C \text{ for }  0\leq t\leq T_0
\]
Hence, we have 
\begin{eqnarray}
\|u_m'(t)\|    &\leq & C  \label{2.29}\\
|\Delta u_m(t)|&\leq & C  \label{2.30}\\
|\Delta \theta _m(t)| &\leq & C  \label{2.31}\\
\|\theta _m(t)\|  &\leq & C  \label{2.32}
\end{eqnarray}
for $0\leq t\leq T_0$.
Putting $w=\theta _m'(t)$ in (\ref{2.17}) we have
\begin{eqnarray*}
|\theta _m'(t)|^2 &\leq& (|g(t)|+|\Delta\theta_m(t)|+
|u_m'(t)|)\,|\theta_m'(t)| \\
|\theta _m'(t)| &\leq& |g(t)|+|\Delta \theta _m(t)|+|u_m'(t)| 
\end{eqnarray*}
Now, using  the Sobolev embedding $H_0^1(\Omega )\hookrightarrow
L^2(\Omega )$, it follows from (\ref{2.29}) and (\ref{2.31}) that
\[
|\theta _m'(t)|\leq C+|g(t)|\text{ or }|\theta _m^{\prime
}(t)|^2\leq C+2|g(t)|^2\,.
\]
Integrating from $0$ to $T_0$, we have
\begin{equation}
\int_0^{T_0}|\theta _m'(t)|^2dt\leq C  \label{2.33}
\end{equation}

\paragraph{Estimate (iii).} Putting $w=u_m''(t)$ in (\ref{2.16}) we have 
\begin{eqnarray*}
|u_m''(t)|^2&=&M(\|u_m(t)\|^2)(\Delta u_m(t),u_m''(t))
-(|u_m(t)|^\rho u_m(t),u_m^{\prime \prime }(t)) \\ 
&&-(\theta _m(t),u_m^{\prime \prime }(t))+(f(t),u_m^{\prime \prime }(t))
\end{eqnarray*}
Then estimating we obtain
\begin{eqnarray*}
|u_m''(t)|^2&\leq& M_2|\Delta u_m(t)|\;|u_m''(t)|
+|u_m(t)|_{L^{2(\rho +1)}}^{\rho +1}|u_m''(t)|\\
&&+|\theta_m(t)|\;|u_m''(t)|+|f(t)|\,|u_m''(t)|\\
|u_m''(t)|&\leq& M_2|\Delta u_m(t)|\;+|u_m(t)|_{L^{2(\rho
+1)}}^{\rho +1}+|\theta _m(t)|+|f(t)|\,  \label{2.34}
\end{eqnarray*}


By (A.3), it follows that $H_0^1(\Omega )\hookrightarrow L^{2(\rho +1)}$.
Using (\ref{2.15}), (\ref{2.29}) and Sobolev's embedding theorem, from 
(\ref{2.30}) we get 
$$
|u_m^{\prime \prime }(t)|\leq C\,.
$$


\section*{Passage to the limit}

From estimates (\ref{2.15}) and (\ref{2.29}) we have that $(u_m)$ and 
$(\theta _m)$ are bounded in $L^\infty (0,T_0;H_0^1(\Omega )\cap H^2(\Omega
))$ and $L^\infty (0,T_0;H_0^1(\Omega ))$, respectively. 
From (\ref{2.29})
the sequence $(u_m')$ is bounded in
 $L^\infty (0,T_0;H_0^1(\Omega ))$, and, by (2.35), the sequence 
$(u_m'')$ is bounded in $L^\infty (0,T_0;L^2(\Omega))$.
Because the embedding from $H_0^1(\Omega )\cap H^2(\Omega )$ into 
$H_0^1(\Omega )$ is compact we can extract a subsequence, again denoted 
by $(u_m)$, such that:
\[
u_m\longrightarrow u  \text{ strongly in } L^2(0,T_0;H_0^1(\Omega ))
\]

Analogously, from (\ref{2.32}), (\ref{2.33}), the compact embedding 
$H_0^1(\Omega )$\ into $L^2(\Omega )$, and the Aubin-Lions
lemma (see, e.g.,  \cite{lions1}) it follows that 
\[
\theta_m\longrightarrow \theta\text{ strongly in }L^2(0,T_0;L^2(\Omega))\,.
\]
Then taking the limit in equations (\ref{2.6})--(\ref{2.7}), when $m\longrightarrow
\infty$, we have that $\{u\,,\theta \}$ is a weak solution of the
system (\ref{1.1})--(\ref{1.4}).

\paragraph{Proof of the Lemma 1.} Multiply (\ref*) by $e^{-\alpha t}$
to obtain 
\begin{equation}
(\mu (t)e^{-\alpha t})'\leq \theta (t)+\beta \mu ^\gamma (t) 
\label{2.36}
\end{equation}
(Note that $e^{-\alpha t}\leq 1$). Integrating (\ref{2.36}) in 
$[0,t[\subset [0,T[$ we obtain
\[
\mu (t)\leq \left[ \mu (0)+\int_0^T\theta (s)ds+\beta \int_0^t\mu ^\gamma
(s)ds\right] e^{ \alpha T} 
\]
Letting
\[
K_1=\left[ \mu (0)+\int_0^T\theta (s)ds\right] e^{\alpha T}\,\,\,\,\text{%
and\thinspace \thinspace \thinspace \thinspace \thinspace }K_2=\beta
e^{\alpha T}
\]
it follows that 
\begin{equation}
\mu (t)\leq K_1+K_2\int_0^t\mu ^\gamma (s)\,,ds\,.  \label{2.37}
\end{equation}

If we denote by $z(t)$ the function$\;z(t)=\int_0^t\mu ^\gamma (s)ds,$ it
follows that $z(0)=0\,$ and $z'(t)=\mu ^\gamma (t).$ Then,$\;$
$$
\frac{z'(t)}{(K_1+K_2z(t))^\gamma }\leq 1
$$
Choosing $T_0$ such that 
\[
K_1+K_2z(t)\leq K_{3}\,, 
\]
where
\[
K_3=\left\{ [\frac{K_1^{1-\gamma }}{K_2(\gamma-1)}-T_0]^{1/(\gamma-1)}
\cdot [K_2(\gamma -1)]^{1/(\gamma -1)}\right\} ^{-1} 
\]

Thus, from (\ref{2.37}), we obtain $\mu (t)\leq K_3$, if 
$0\leq t\leq T_0$. This concludes the proof of this Lemma.

\section{Uniqueness}

Let $[u,\theta ]$ and $[\hat u,\hat \theta ]$ be solutions of
(\ref{1.1})--(\ref{1.4})
under the conditions of Theorem~1. Let $w=u-\hat u$ and $v=\theta -\hat
\theta$. Then $[w,v]$ satisfies
\begin{eqnarray}
\lefteqn{ \frac d{dt}(w',z)+M(\int_\Omega |\nabla u|^2dx)(\nabla w,\nabla
z)+(|u|^\rho u-|\hat u|^\rho \hat u,z)+(v,z)] }\nonumber\\ 
&=& M(\int_\Omega |\nabla \hat u|^2dx)(\nabla \hat u,\nabla z)
-M(\int_\Omega|\nabla u|^2dx)(\nabla \hat u,\nabla z) \label{3.1}\\
&&\frac d{dt}(v,z)+(\nabla v,\nabla z)+(w',z)=0  \label{3.2}\\
&&w(0)=0,\quad w'(0)=0\,\text{ and } v(0)=0  \label{3.3}
\end{eqnarray}
Taking $z=w'$ in (\ref{3.1}) and $z=v$ in (\ref{3.2}), we obtain
\begin{eqnarray}
\lefteqn{ \frac d{dt}|w'|^2+M(\int_\Omega |\nabla u|^2dx)
\frac d{dt}\|w\|^2+\int_\Omega (|u|^\rho u-|\hat u|^\rho \hat u)w'dx
+(v,w') } \nonumber \\
&=& M(\int_\Omega |\nabla \hat u|^2dx)(\nabla \hat u,\nabla
w')-M(\int_\Omega |\nabla u|^2dx)(\nabla \hat u,\nabla w')\label{3.4}\\
&&\frac d{dt}|v|^2+\|v\|^2+(w',v)=0  \label{3.5}
\end{eqnarray}
in the $D'(0,T)$ sense. Adding (\ref{3.4}) to (\ref{3.5}) we have
\begin{eqnarray*}
\lefteqn{\frac d{dt}|w'|^2+M(\int_\Omega |\nabla u|^2dx)\frac
d{dt}\|w\|^2+\frac d{dt}|v|^2+\|v\|^2} \\ 
&=&\int_\Omega (|\hat u|^\rho \hat u-|u|^\rho u)w'dx-2(v,w^{\prime
})+M(\int_\Omega |\nabla \hat u|^2dx)(\nabla \hat u,\nabla w')\\
&&- M(\int_\Omega |\nabla u|^2dx)(\nabla \hat u,\nabla w') \\
&\leq& \left|
\int_\Omega (|\hat u|^\rho \hat u-|u|^\rho u)w'dx\right|
+2|(v,w')|\\  
&&+\left| M(\int_\Omega |\nabla \hat u|^2dx)-M(\int_\Omega |\nabla
u|^2dx)\right| |(\nabla \hat u,\nabla w')|
\end{eqnarray*}

On the other hand, by Holder's inequality with $\frac 1q+\frac 1n+\frac
12=1$, we have
\begin{eqnarray*}
\left| \int_\Omega (|\hat{u}|^\rho \hat{u}-|u|^\rho u)w'dx\right|
&\leq& (\rho +1)\int_\Omega \sup (|u|^\rho ,|\hat{u}|^\rho )|w|\,|w^{\prime
}|dx\\ 
&\leq& C\left( \|\,|u|^\rho \|_{L^n(\Omega )}+\|\,|\hat{u}|^\rho \|_{_{L^n(\Omega
)}}\right) \,\|w\|_{L^q(\Omega )}|w'|_{L^2(\Omega )}\;
\end{eqnarray*}

By condition (A.2), we have $\rho n\leq q$ and from the immersion 
$H_0^1(\Omega )\hookrightarrow L^q(\Omega )$ with 
$1/q= 1/2-1/n$, we have
\[
\left| \int_\Omega (|\hat{u}|^\rho \hat{u}-|u|^\rho u)w'dx\right|
\leq C(\|u\|^\rho +\|\hat{u}\|^\rho )\,\|w\|\,|w'| 
\]
and since $u,\,\hat{u}\in L^\infty (0,T;H_0^1(\Omega ))$, we have
\begin{eqnarray}
\left| \int_\Omega (|\hat u|^\rho \hat u-|u|^\rho u)w'dx\right|
&\leq & C\|w\|\,|w'|  \label{3.6}\\
2|(v,w')|&\leq& 2|v|\,|w'|  \label{3.7}
\end{eqnarray}

Observe that
\begin{eqnarray*}
\lefteqn{ \left| M(\int_\Omega |\nabla \hat u|^2dx)-M(\int_\Omega |\nabla
u|^2dx)\right| |(\nabla \hat u,\nabla w')|} \\  
&\leq& |M'(\xi )|\,\left| |\nabla \hat u|^2-|\nabla u|^2\right| |(-\Delta
)\hat u|\;|w'|
\end{eqnarray*}
where $\xi$ is between $|\nabla \hat u|^2$ and $|\nabla u|^2$. Then we have
\begin{eqnarray}
\lefteqn{ \left| M(\int_\Omega |\nabla \hat u|^2dx)-M(\int_\Omega |\nabla
u|^2dx)\right| |(\nabla \hat u,\nabla w')| }\nonumber \\ 
&\leq & C\left| |\nabla \hat u|+|\nabla u|\right| \;\left| |\nabla \hat u|-|\nabla
u|\right| |(-\Delta )\hat u|\;|w'| \label{3.8}\\ 
&\leq & C\|\hat u-u\|\;|(-\Delta )\hat u|\;|w'| \nonumber \\ 
&\leq & C\|w\|\;|w'| \nonumber
\end{eqnarray}

Substituting (\ref{3.6})--(\ref{3.8}) in (\ref{3.4}) and noting that
\begin{eqnarray*}
\lefteqn{ M(\int_\Omega |\nabla u|^2dx)\frac d{dt}|\nabla w|^2 }\\
&=& \frac d{dt}\left(
M(\int_\Omega |\nabla u|^2dx)|\nabla w|^2\right) -\left[ \frac
d{dt}M(\int_\Omega |\nabla u|^2dx)\right] |\nabla w|^2 
\end{eqnarray*}
we obtain:
\begin{eqnarray}
\lefteqn{\frac d{dt}\left\{ |w'|^2+|v|^2+M(\int_\Omega |\nabla
u|^2dx)|\nabla w|^2\right\} +\|v\|^2} \nonumber \\ 
&\leq & |v|^2+C|w'|^2+C\|w\|^2+\left| \frac d{dt}M(\int_\Omega |\nabla
u|^2dx)\right| |\nabla w|^2 \label{3.9}\\ 
&\leq& C\left\{ |v|^2+|w'|^2+\|w\|^2\right\} \nonumber 
\end{eqnarray}
Integrating (\ref{3.9}) from $0$ to $t\leq T_0,\;$we have
\begin{eqnarray*}
\lefteqn{ |w'(t)|^2+|v(t)|^2+m_0\|w(t)\|^2+\int_0^T\|v(s)\|^2ds}\\ 
&\leq& C\int_0^t\left\{ |v(s)|^2+|w'(s)|^2+\|w(s)\|^2\right\} ds
\end{eqnarray*}
By Gronwall's Lemma it follows that 
\[
|v(s)|^2+|w'(s)|^2+\|w(s)\|^2\leq 0\,. 
\]
This implies that $v(t)=w(t)=0\;\forall t\in [0,T]$. Or
$u(t)=\hat{u}(t)$ and $\theta (t)=\hat{\theta}(t)\;\forall t\in [0,T]$. 
This concludes the proof of uniqueness.

\paragraph{Acknowledgment.} We would like to
express our sincere thanks to Professor Aldo Maciel for our useful
conversations about this work.

\begin{thebibliography}{99}
\bibitem{arosio-spagnolo}  Arosio, A. - Spagnolo, S., \textit{Global
solution of the Cauchy problem for a nonlinear hyperbolic, nonlinear 
partial differential equation and their applications,} 
Coll\`ege de France Seminar vol. 6 
(ed. by H. Brezis and J. L. Lions), Pitman, London, 1984.

\bibitem{bisognin}  Bisignin, E., \textit{Perturbation of
Kirchhoff-Carrier's operator by Lipschitz functions, } 
Proceedings of XXXI Bras. Sem. of Analysis, Rio de Janeiro, 1992.

\bibitem{d'ancona-spagnolo}  D'ancona, P. \& Spagnolo, S., 
\textit{Nonlinear perturbation of the Kirchhoff-Carrier equations,}
Univ. Pisa Lectures Notes, 1992.

\bibitem{friedman}  Friedman, A., \textit{Partial differential equations,}
Krieger Publishing Co., Florida, 1989.

\bibitem{lions1}  Lions, J. L., \textit{Quelques methods de r\'esolution
des probl\`emes aux limites nonlineares,} Dunod, Paris, 1969.

\bibitem{lions2}  Lions, J. L., \textit{On some questions in boundary value
problem of mathematical-physics, in contemporary} \textit{development in
continuum mechanics and PDE, } Ed. by G. M. da la Penha and L. A. Medeiros,
North Holland, Amsterdam, 1978.

\bibitem{medeiros}  Medeiros, L. A., \textit{On some nonlinear
perturbation of Kirchhoff-Carrier's operator,} Comp. Appl. Math. 13(3),
1994, 225-233.

\bibitem{maciel-lima}  Maciel, A. \& Lima, O., \textit{Nonlinear
perturbation of Kirchhoff-Carrier's equations,} Proceedings of XLII Bras.
Sem. of Analysis, Maring\'a Brasil, 1995.

\bibitem{hosoya-yamada} Hosoya, M. \& Yamada, Y., \textit{On some
nonlinear wave equation I- local existence and} \textit{regularity of
solutions}, Journal Fac. Sci. Tokyo, Sec IA, Math. 38(1991), 225-238.

\bibitem{pohozhaev}  Pohozhaev, S. I., \textit{On a class of quasilinear
hyperbolic equations,} Mat. Sbornic 96 (138)(1)(1975), 152-166 (Mat.
Sbornic 25(1)(1975), 145-158, english translation).

\end{thebibliography}
\bigskip

{\sc M. R. Clark\newline 
Universidade Federal da Para\'{\i}ba - PB - Brasil}\newline
E-mail address: mclark@dme.ufpb.br

\bigskip

{\sc O. A. Lima\newline 
Universidade Estadual da Para\'\i ba - DM - Brasil}\newline 
E-mail address: olima@dme.ufpb.br
\end{document}