\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 82, pp. 1--6.\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-2006/82\hfil Exponential decay]
{Exponential decay for the semilinear wave equation with source terms}
\author[J. Fan, H. Wu\hfil EJDE-2006/82\hfilneg]
{Jishan Fan, Hongwei Wu}

\address{Jishan Fan \newline
Department of Mathematics, Suzhou University, Suzhou 215006, China}
\curraddr{College of Information Science and technology,
Nanjing Forestry University, Nanjing 210037, China}
\email{fanjishan@njfu.edu.cn}

\address{Hongwei Wu \newline
Department of Mathematics,
Southeast University, Nanjing, 210096, China}
\email{hwwu@seu.edu.cn}

\date{}
\thanks{Submitted May 1, 2006. Published July 21, 2006.}
\thanks{Supported by grant 10101034 from NSFC}
\subjclass[2000]{35L05, 35L15, 35L20}
\keywords{Wave equation; source terms; exponential decay;
 multiplier method}

\begin{abstract}
In this paper, we prove that for a semilinear wave equation
with source terms, the energy decays exponentially as
time approaches infinity. For this end we use the
the multiplier method.
\end{abstract}

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

\section{Introduction}

\subsection*{Main results}

Let $\Omega$ be a bounded subset of $\mathbb{R}^n$  with smooth boundary
$\partial\Omega$. We are concerned with the  mixed problems
\begin{gather}
u_{tt}-\Delta u+\delta u_t=|u|^{p-1}u,\quad x\in\Omega,\quad t\ge 0,\label{1.1}\\
u(0,x)=u_0(x)\in H^1_0(\Omega),\quad
u_t(0,x)=u_1(x)\in L^2(\Omega),\quad x\in\Omega,\label{1.2}\\
u(t,x)|_{\partial\Omega}=0,\quad \mbox{for } t\ge 0.\label{1.3}
\end{gather}
Here $\delta>0$ and $1<p\le\frac{n}{n-2}$ ($n\ge 3$),
$1<p$ ($n=1,2$).
Set
\begin{gather}
I(u):=\int_{\Omega}(|\nabla{u}|^2-|u|^{p+1})dx,\label{1.4}\\
J(u):=\int_{\Omega}(\frac{1}{2}|\nabla{u}|^2-\frac{1}{p+1}|u|^{p+1})dx,\label{1.5}\\
E(t):=\frac{1}{2}\int_{\Omega}|u_t|^2dx+J(u)\quad .\label{1.6}
\end{gather}
Also let the Nehari manifold
\begin{equation}
N:=\{u\in{H_0^1(\Omega)}: I(u)=0,\; u\neq{0}\}; \label{1.7}
\end{equation}
and the potential depth
\begin{equation}
d:=\inf_{u\in{N}}J(u).\label{1.8}
\end{equation}
%
For problem (\ref{1.1})-(\ref{1.3}), Ikehata and Suzuki
\cite{Ikehata1} have shown the following results:
\begin{gather}
d>0; \label{1.9}\\
E(t)+\int_0^t\int_{\Omega}\delta|u_t|^2dx\,dt=E(0); \label{1.10}
\end{gather}
If $E(0)<d$ and $I(u(0,x))>0$ then we have
\begin{gather}
 E(t)<d\quad \mbox{and}\quad  I(u(t,x))>0,\quad \forall t\in[0,\infty); \label{1.11}\\
 \theta\int_{\Omega}|\nabla u|^2dx\ge\int_{\Omega}|u|^{p+1}dx,\quad
  \theta\in(0,1),\ \forall t\in[0,\infty); \label{1.12}\\
 \lim_{t\to +\infty}\int_{\Omega}(|u_t|^2+|\nabla u|^2)dx=0; \label{1.13}\\
 \int_0^t\int_{\Omega}|\nabla u|^2dx\,dt\le C.
\label{1.14}
\end{gather}
In this paper we will use the multiplier technique to prove the following
result.

\begin{theorem} \label{thm1}
If $E(0)<d$ and $I(u(0,x))>0$, then there exists positive constant
$\gamma$ and $C>1$ such that
\begin{equation}
E(t)\le Ce^{-\gamma t},\quad \forall t\in[0,\infty).\label{1.15}
\end{equation}
\end{theorem}

\subsection*{\label{sec:level2}Our results and their relationship to the literature}
The Problem
\begin{equation} \label{1.16}
\begin{gathered}
u_{tt}-\Delta u+a(x)|u_t|^{m-1}u_t+|u|^{p-1}u=0,\quad \text{in } \Omega, \\
u\big|_{\partial\Omega}=0,\quad
(u,u_t)\big|_{t=0}=(u_0,u_1)
\end{gathered}
\end{equation}
has been studied, among others, by Nakao \cite{Nakao1,Nakao2} and
 Zuazua \cite{Zuazua}. In \cite{Nakao1,Nakao2,Zuazua},
the authors assumed that $a(x)\ge 0$ in $\Omega$,
$\inf a(x)>0$ in $\Omega_0\subset\subset\Omega$ and $m=1$.
The case $m>1$ is still open \cite{Zuazua}.

The following problem,  with $m>1$ and $a(x)\ge a_0>0$ in $\bar{\Omega}$,
\begin{equation} \label{1.17}
\begin{gathered}
u_{tt}-\Delta{u}+a(x)|u_t|^{m-1}u_t=|u|^{p-1}u,\quad \text{in}\ \Omega, \\
u\big|_{\partial\Omega}=0,\quad (u,u_t)\big|_{t=0}=(u_0,u_1)
\end{gathered}
\end{equation}
has been studied by many authors, Ball \cite{Ball}, Ikehata \cite{Ikehata3},
Ikehata and Tanizawa\cite{Ikehata4},  Levine \cite{Levine1,Levine2},
Georgiev and Todorova \cite{Georgiev1}, Georgiev and Milani \cite{Georgiev2},
Todorova \cite{Todorova1}, Barbu, et al \cite{Barbu},
Todorova and Vitillaro \cite{Todorova2}, Messaoudi \cite{Messaoudi},
Serrin \cite{Serrin}, Kawashima, et al \cite{Kawashima}.
Ball \cite{Ball} proved the existence of a global attactor when
$m=1$. In \cite{Ikehata3,Ikehata4,Todorova2, Kawashima}, the authors
obtained a time-decay result when $\Omega=\mathbb{R}^N$.
In \cite{Levine1,Levine2,Georgiev1,Georgiev2,Todorova1,Barbu,Messaoudi,Serrin},
the authors mainly concerned the existence or nonexistence of
global weak (or strong) solutions.

By the multiplier method in \cite{Komornik}, Benaissa and
Mimouni \cite{Benaissa} studied very recently the
decay properties of the solutions to the wave equation of $p$-Laplacian
type with a weak nonlinear dissipative.

Here it should be noted that our main result Theorem \ref{thm1} is
also true for the locally damping case i.e., $\delta=\delta(x)\ge
0$ in $\Omega$ and $\delta(x)\ge \delta_0>0$ in
$\Omega_0\subset\subset\Omega$. We did not find references for the
case with boundary damping term.


\section{Proof of the Main Result}
Take $x_0\in R^n$ and set $m(x):=x-x_0$. Let $\nu$ denote the
outward normal vector to $\partial\Omega$.
Set
\begin{gather*}
\Gamma(x_0):=\{x\in\partial\Omega:  (x-x_0)\cdot\nu>0\},\\
\chi:=\int_{\Omega}\big(u_t(m\cdot\nabla
u)+\frac{n}{p+1}u(u_t+\frac{\delta}{2}u)\big)dx\big|_0^T.
\end{gather*}

\begin{lemma}\label{lem2.1}
There exists positive constant $C$ depending only on $n,p,\delta,\Omega$
such that
\begin{equation}
\int_0^TE(t)dt\le
C\big\{\int_0^T\int_{\Gamma(x_0)}(m\cdot\nu)|\frac{\partial
u}{\partial\nu}|^2d\Gamma
dt+\int_0^T\int_{\Omega}|u_t|^2dx\,dt+|\chi|\big\}.\label{2.1}
\end{equation}
\end{lemma}

\begin{proof}
Multiplying (\ref{1.1}) by $q(x)\cdot\nabla u$ and
integrating by parts gives, \cite{Zuazua,Lions},
\begin{equation}
\begin{aligned}
&\Big(\int_{\Omega}u_t(q\cdot\nabla u)dx\Big)\big|_0^T
+\frac{1}{2}\int_0^T\int_{\Omega}(\mbox{div}q)(|u_t|^2-|\nabla u|^2)dx\,dt
\\
&+\int_0^T\int_{\Omega}(\sum_{k,j=1}^n\frac{\partial q_k}{\partial
x_j}\frac{\partial u}{\partial x_k}\frac{\partial u}{\partial
x_j})dx\,dt
+\int_0^T\int_{\Omega}(\mbox{div}q)\frac{|u|^{p+1}}{p+1}dx\,dt  \\
&+\int_0^T\int_{\Omega}\delta u_t(q\cdot\nabla u)dx\,dt\\
&=\frac{1}{2}\int_0^T\int_{\partial\Omega}(q\cdot\nu)|\frac{\partial
u}{\partial\nu}|^2d\Gamma \,dt.
\end{aligned}\label{2.2}
\end{equation}
Here $q(x)\in W^{1,\infty}(\Omega)$.
Applying identity (\ref{2.2}) with $q(x)=m(x)$, we deduce
\begin{equation}
\begin{aligned}
&\Big(\int_{\Omega}u_t(m\cdot\nabla u)dx\Big)\big|_0^T
 +\frac{n}{2}\int_0^T\int_{\Omega}(|u_t|^2-|\nabla u|^2)dx\,dt
 +\int_0^T\int_{\Omega}|\nabla u|^2dx\,dt  \\
&+\frac{n}{p+1}\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt
+\int_0^T\int_{\Omega}\delta u_t(m\cdot\nabla u)dx\,dt \\
&=\frac{1}{2}\int_0^T\int_{\partial\Omega}(m\cdot\nu)|
  \frac{\partial u}{\partial\nu}|^2d\Gamma dt \\
&\le \frac{1}{2}\int_0^T\int_{\Gamma(x_0)}(m\cdot\nu)|
  \frac{\partial u}{\partial\nu}|^2d\Gamma dt\,.
\end{aligned}\label{2.3}
\end{equation}
We now multiply (\ref{1.1}) by $u$ and integrate by parts, then we
have
\begin{equation}
\Big(\int_{\Omega}u(u_t+\frac{\delta}{2}u)dx\Big)\big|_0^T
=\int_0^T\int_{\Omega}(|u_t|^2-|\nabla
u|^2)dx\,dt+\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt.\label{2.4}
\end{equation}
Combining (\ref{2.3}) and (\ref{2.4}) we obtain
\begin{equation}
\begin{aligned}
&\chi+(\frac{n}{2}-\frac{n}{p+1})\int_0^T\int_{\Omega}|u_t|^2dx\,dt
+(1+\frac{n}{p+1}-\frac{n}{2})\int_0^T\int_{\Omega}|\nabla u|^2dx\,dt \\
&+\int_0^T\int_{\Omega}\delta u_t(m\cdot\nabla u)dx\,dt\\
& \le \frac{1}{2}\int_0^T\int_{\Gamma(x_0)}(m\cdot\nu)|\frac{\partial
u}{\partial\nu}|^2d\Gamma dt.
\end{aligned}\label{2.5}
\end{equation}
On the other hand, for any given $\varepsilon>0$,
\begin{equation}
\begin{aligned}
&\big|\int_0^T\int_{\Omega}\delta u_t(m\cdot\nabla u)dx\,dt\big|  \\
&\le\varepsilon\|m\|_{L^{\infty}(\Omega)}^2\int_0^T\int_{\Omega}|\nabla
u|^2dx\,dt+\frac{\delta^2}{2\varepsilon}\int_0^T\int_{\Omega}|u_t|^2dx\,dt.
\end{aligned}\label{2.6}
\end{equation}
Taking $\varepsilon$ sufficiently small in (\ref{2.6}), then
substituting (\ref{2.6}) into (\ref{2.5}) we obtain
(\ref{2.1}).
\end{proof}

\begin{lemma}\label{lem2.2}
With the above notation,
\begin{equation}
E(t)\le C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt.\label{2.7}
\end{equation}
\end{lemma}

\begin{proof} First, we construct a function $h(x)\in W^{1,\infty}(\Omega)$
such that $h(x)=\nu$ on $\Gamma(x_0)$; $h(x)\cdot\nu>0$ a.e in
$\partial\Omega$;  see\cite{Zuazua}.
Applying (\ref{2.2}) with $q(x)=h(x)$, we have
\begin{equation}
\begin{aligned}
\int_0^T\int_{\Gamma(x_0)}|\frac{\partial u}{\partial\nu}|^2d\Gamma dt
&\le\int_0^T\int_{\partial\Omega}(h\cdot\nu)|\frac{\partial u}{\partial\nu}|^2d\Gamma dt \\
&\le C\int_0^T\int_{\Omega}(|u_t|^2+|\nabla
u|^2)dx\,dt+2\Big(\int_{\Omega}u_t(h\cdot\nabla
u)dx\Big)\big|_0^T .
\end{aligned} \label{2.8}
\end{equation}
 From (\ref{2.4}), we see that
\begin{equation}
\int_0^T\int_{\Gamma(x_0)}|\frac{\partial
u}{\partial\nu}|^2d\Gamma dt\le
C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt+Y,\label{2.9}
\end{equation}
where
$$
Y=\Big(\int_{\Omega}u(u_t+\frac{\delta}{2}u)dx\Big)\big|_0^T
+2\Big(\int_{\Omega}u_t(h\cdot\nabla u)dx\Big)\big|_0^T.
$$
Combining (\ref{2.1}), (\ref{2.9}) and (\ref{1.10}) we obtain
\begin{equation}
\begin{aligned}
TE(T)&\le\int_0^TE(t)dt \\
& \le C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt+|\chi|+|Y| \\
& \le C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt+C(E(0)+E(T)) \\
& \le C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt
 +C\Big(2E(T)+\delta
\int_0^T\int_{\Omega}|u_t|^2dx\,dt\Big).
\end{aligned} \label{2.10}
\end{equation}
Taking $T$ sufficiently large we get (\ref{2.7}).
\end{proof}

\begin{lemma}\label{lem2.3}
\begin{equation}
\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt\le
C\int_0^T\int_{\Omega}|u_t|^2dx\,dt.\label{2.11}
\end{equation}
\end{lemma}

\begin{proof}
We argue by contradiction. If
(\ref{2.11}) is not satisfied for some $C>0$, then there exists a
sequence of solutions $\{u_n\}$ of (\ref{1.1})-(\ref{1.3})
with
\begin{equation}
\lim_{n\to\infty}\frac{\int_0^T\int_{\Omega}|u_n|^{p+1}dx\,dt}
{\int_0^T\int_{\Omega}|u_{nt}|^2dx\,dt}=\infty.\label{2.12}
\end{equation}
 From (\ref{1.12}) and (\ref{1.14}) we have
\begin{equation}
\int_0^T\int_{\Omega}|u_n|^{p+1}dx\,dt\le\theta\int_0^T\int_{\Omega}|\nabla
u_n|^2dx\,dt\le C.\label{2.13}
\end{equation}
Thus we get
\begin{equation}
\lim_{n\to\infty}\int_0^T\int_{\Omega}|u_{nt}|^2dx\,dt=0.\label{2.14}
\end{equation}
We extract a subsequence (still denote by $\{u_n\}$) such that
\begin{gather}
 u_n\rightharpoonup u \quad \mbox{weakly in }
  H^1(\Omega\times(0,T)),\label{2.15}\\
 u_n\to u\quad \mbox{strongly in } L^2(\Omega\times(0,T)),\label{2.16}\\
 u_n\to u\quad  \mbox{a.e. in } \Omega\times(0,T),\label{2.17}\\
 |u_n|^{p-1}u_n\to |u|^{p-1}u \quad \mbox{strongly in }
L^{\infty}(0,T;L^r(\Omega))\label{2.18}
\end{gather}
where $r\in [1,\frac{2n}{p(n-2)})$ if $n\ge 3$ and
$r\in[1,\infty)$ if $n=1,2$.
 From (\ref{2.14}) we know that
\begin{equation}
u_t=0,\quad \mbox{a.e. in } \Omega\times(0,T)\label{2.19}
\end{equation}
and so we have
\begin{gather}
-\Delta u = |u|^{p-1}u,\quad \mbox{in } \Omega\times(0,T)\label{2.20}\\
u = 0,\quad \mbox{on } \partial\Omega\times(0,T).\label{2.21}
\end{gather}
 From (\ref{2.13}) we get
\begin{equation}
\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt\le\theta\int_0^T\int_{\Omega}|\nabla
u|^2dx\,dt<\int_0^T\int_{\Omega}|\nabla u|^2dx\,dt \label{2.22}
\end{equation}
which contradicts (\ref{2.20}) and (\ref{2.21}). This proves
(\ref{2.11}).
\end{proof}

 By Lemmas \ref{lem2.2} and \ref{lem2.3}, we obtain
\begin{equation}
E(T)\le C\int_0^T\int_{\Omega}|u_t|^2dx\,dt. \label{2.23}
\end{equation}
This inequality, (\ref{1.10}), and semigroup properties
complete the proof of Theorem \ref{thm1}. For properties of
semigroups, we refer the reader to \cite{Rauch}.


\subsection*{Acknowledgments}
The authors are indebted to the referee who has given many
valuable suggestions for improving the presentation of this article.

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\end{document}