\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 113, 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}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/113\hfil Decay of solutions ]
{Decay of solutions for a system of nonlinear wave equations}

\author[X. Wei \hfil EJDE-2013/113\hfilneg]
{Xiao Wei} 

\address{Xiao Wei \newline
School of Science, Chang'An University, Xi'an 710064, China}
\email{xiaowei1802002@163.com}

\thanks{Submitted December 28, 2012. Published April 30. 2013.}
\subjclass[2000]{35B40, 35L10}
\keywords{Nonlinear wave equations; initial boundary value problem;
\hfill\break\indent  exponential decay}

\begin{abstract}
 This article concerns the decay of solutions
 of the semi-linear wave equation
 $$
  u_{tt}+\delta u_t-\phi(x)\triangle u=\lambda u|u|^{\beta-1}\quad
 x\in \mathbb{R}^N\; t\geq0
 $$
 Introducing an appropriate Lyaponuv function, we find
 exponential decay for certain initial data.
\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} \label{e1}
\begin{gathered}
 u_{tt}+\delta u_t-\phi(x)\triangle u=\lambda u|u|^{\beta-1}\quad
 x\in \mathbb{R}^N,\; t\geq0\\
 u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x)\quad x\in \mathbb{R}^N
\end{gathered}
\end{equation}
with initial conditions $u_0(x)$, $u_1(x)$ in appropriate function
spaces and $\delta>0$. Models of this type are of interest in
applications in various areas of mathematical physics
\cite{Antm,KarSta,ReeSim,Zaud},
as well as in geophysics and ocean acoustics, where, for example,
the coefficient $\phi(x)$ represents the speed of sound at the point
$x\in \mathbb{R}^N$ (see \cite{Klib}).
Throughout this article,  we assume that the
functions $\phi(x)$ and $g: \mathbb{R}^N\to R$ satisfy the
following conditions:
\begin{itemize}
\item[(G0)]  $\phi(x)>0$, for all $x\in \mathbb{R}^N$,
$(\phi(x))^{-1}=:g(x)$ is $\mathcal{C}^{0,\gamma}(\mathbb{R}^N)$-smooth, for some
$\gamma\in(0,1)$ and $g\in L^{N/2}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$,
\end{itemize}
Examples of functions $\phi$ of this type can be found in \cite[p. 632]{Zaud}.

The questions of global existence, nonexistence and
blow-up of solutions of the Cauchy problem for nonlinear wave
equations have been studied by many authors; see for example
\cite{Gril,Levi,PaySat}. 
In general, global existence happens, when the damping terms dominate
the source terms, while blow-up appears in the opposite
situation and under the assumption that the initial data is
sufficiently large (for example when the initial energy is assumed to be
sufficiently negative). 
In \cite{Ikeh} it is shown that for sufficiently
small initial data global existence can be obtained, even when the
influence of the source term is stronger than that of the damping
term. 
In the works 
\cite{AlvCav,AlCaDoVaRaTo,BoRaTo,BroSta,CaDoVaLa,GeoTod,Ikeh,MeLiFr,Ramm} 
the spatial domain is assumed to be bounded. On the other hand, 
in \cite{NakOno} the problem is considered in
the whole of $\mathbb{R}^N$ and the method of modified potential well 
is used to construct the global solutions. 
In \cite{GeoTod,Ikeh,NakOno} the
coefficient $\phi(x)=1$, which makes possible the treatment of the
equations in the classical Sobolev space setting. In 
\cite{Naka96,Naka97,Zuaz}  decay properties of solutions of wave equations,
involving weighted dissipative terms, are discussed. 
Cavalcanti \cite{CavDom} considers the nonlinear evolution equation 
with source and damping terms on a compact manifold.

The purpose of this article is to obtain  decay estimate of
solutions to the problem \eqref{e1}. More precisely we show that we can
always find initial data in the stable set for which the solution of
 \eqref{e1} decays exponentially. The key tool in the proof is an
idea of Zuazua \cite{HarZua,Zuaz}, which is based on the construction of a
suitable Lyapunov function.

Notation: For simplicity we use the symbols $L^p$ and $D^{1,2}$,
for the spaces $L^p(\mathbb{R}^N)$ and
$D^{1,2}(\mathbb{R}^N)$, respectively, with $1\leq p\leq\infty$.
We use  $\|\cdot\|_p$ for the norm
$\|\cdot\|_{L^p(\mathbb{R}^N)}$. Also differentiation with respect to
time is denoted by a dot over the function. The constants $C$ and
$c$ are considered in a generic sense.


\section{Asymptotic stability}

In this section we introduce and prove our main result. For this
purpose we use the definition of the solution of problem
\eqref{e1} given by Karachalios and Stavrakakis in \cite{KarSta}.
For later use, we briefly mention here some facts, notation and
results from paper \cite{KarSta}. 

The space setting for the initial conditions
and the solutions of the problem \eqref{e1} is the product space
$\mathcal{X}_0=D^{1,2}(\mathbb{R}^N)\times L^2_g(\mathbb{R}^N)$. The space
$D^{1,2}(\mathbb{R}^N)$ is defined as the closure of 
$C^\infty_0 (\mathbb{R}^N)$
functions with respect to the energy norm 
$\|u\|_{D^{1,2}}=:\int_{\mathbb{R}^N}|\nabla u|^2\,dx$. It is well known that
$$
D^{1,2}(\mathbb{R}^N)=\big\{u\in L^{\frac{2N}{N-2}}(\mathbb{R}^N):\nabla u
\in(L^2(\mathbb{R}^N))^N\big\}
$$
and that $D^{1,2}$ is embedded continuously
in $L^{\frac{2N}{N-2}}$.i.e., there exists $k>0$ such that
\begin{equation}
\|u\|_{\frac{2N}{N-2}}\leq k\|u\|_{D^{1,2}}
\end{equation}
We shall frequently use the following generalized version of
Poincar\'{e} inequality
\begin{equation}
\int_{\mathbb{R}^N}|\nabla u|^2\,dx\geq \alpha \int_{\mathbb{R}^N}gu^2\,dx
\end{equation}
for all $u\in C^\infty_0$ and $g\in L^{N/2}$, where 
$\alpha =: k^{-2}\|g\|^{-1}_{N/2}$ (see \cite[Lemma 2.1]{BroSta}). 
It has been shown that $D^{1,2}(\mathbb{R}^N)$ is a separable Hilbert space. 
The space $L^2_g(\mathbb{R}^N)$ is defined to be the closure of 
$u\in C^\infty_0(\mathbb{R}^N)$
functions with respect to the inner product
\begin{equation}
(u,v)_{L^2_g(\mathbb{R}^N)}=:\int_{\mathbb{R}^N}guv\,dx
\end{equation}
Clearly, $L^2_g(\mathbb{R}^N)$ is a separable Hilbert space.

We consider the potential well
\begin{equation*}
\mathcal{W}=:\big\{u\in D^{1,2}(\mathbb{R}^N): K(u)=:\|u\|^2_{D^{1,2}}
-\lambda \|u\|^{\beta+1}_{L^{\beta+1}_g}>0 \big\}
\end{equation*}
Also consider the functional
\begin{equation} \label{e5}
\mathcal{J}(u)=:\frac{1}{2}\|u\|^2_{D^{1,2}}-\frac{\lambda}{\beta+1}
\int_{\mathbb{R}^N}g(x)|u(t)|^{\beta+1}\,dx.
\end{equation}
The energy of the problem is defined as
\begin{equation} \label{e6}
\mathcal{E}^*(u(t),u_t(t))=\mathcal{E}^*(t)
:=\frac{1}{2}\|u_t(t)\|^2_{L^2_g}+\mathcal{J}(u).
\end{equation}
From equation \eqref{e1}, we  have
\begin{equation}
\mathcal{E}^*(t)=\mathcal{E}^*(0)-\delta\int^t_0\|u_t(\tau)\|^2_{L^2_g}d\tau
\end{equation}

Under certain assumptions on the initial data, solutions exist
globally in the energy space $\mathcal{X}_0$. In addition to the
principal condition (G0) in the introduction, we shall use
the following additional hypotheses for
the function $g$ and the nonlinearity exponent $\beta$. 
\begin{itemize}
\item[(G1)] $g\in L^1(\mathbb{R}^N)$ and $1<\beta\leq\frac{N}{N-2}$,
for all $N\geq3$.

\item[(G2)] $N\geq3$ and
$\frac{N+2}{N}\leq\beta\leq\frac{N}{N-2}$.

\item[(G3)] $N=3,4$ and
$\frac{N+4}{N}\leq\beta\leq\frac{N}{N-2}$.

\end{itemize}
Let us note that since 
$g\in L^{N/2}(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$ by
hypothesis (G0), then any $g$ satisfying hypothesis
(G1) belongs to all spaces $L^p(\mathbb{R}^N)$, for
$p\in[1,+\infty)$.

 A weak solution of \eqref{e1} is a function
$u(x,t)$ such that
\begin{itemize}
\item[(i)] $u\in L^2[0,T;D^{1,2}(\mathbb{R}^N)]$, 
$u_t\in L^2[0,T; L^2_g(\mathbb{R}^N)]$,
$u_{tt}\in L^2[0,T;D^{-1,2}(\mathbb{R}^N)]$,

\item[(ii)] for all $v\in C^\infty_0([0,T]\times \mathbb{R}^N)$, $u$ 
satisfies the generalized formula
\begin{equation}
\begin{aligned}
&\int^T_0(u_{tt}(\tau),
v(\tau))_{L^2_g}d\tau+\delta\int^T_0(u_{t}(\tau),
v(\tau))_{L^2_g}d\tau\\
&+\int^T_0\int_{\mathbb{R}^N}\nabla u(\tau)\nabla
v(\tau)\,dxd\tau-\lambda\int^T_0(f(u(\tau)), v(\tau))_{L^2_g}d\tau=0,
\end{aligned}
\end{equation}
where $f(s)=|s|^{\beta-1}s$,  and \\
(iii) $u$ satisfies the initial conditions
$$
u(x,0)=u_0(x)\in D^{1,2}(\mathbb{R}^N),\quad
 u_t(x,0)=u_1(x)\in L^2_g(\mathbb{R}^N).
$$
\end{itemize}

The following two lemmas come from \cite{KarSta}.

\begin{lemma}[{\cite[Proposition 3.1]{KarSta}}]
 Let $g$, $\beta$, $N$ satisfy conditions {\rm (G0) or (G2)}.
Suppose that the constants $\delta>0$, $\lambda<\infty$ and the initial
conditions
\begin{equation} \label{e9}
u_0\in D^{1,2}(\mathbb{R}^N) \quad {\rm{and}} \quad u_1\in L^2_g(\mathbb{R}^N).
\end{equation}
are given. Then for sufficiently small $T>0$ the problem \eqref{e1} admits
a unique (weak) solution such that
\begin{equation}
u\in C[0,T;D^{1,2}(\mathbb{R}^N)], \quad
u_t\in C[0,T;L^2_g(\mathbb{R}^N)].
\end{equation}
\end{lemma}

\begin{lemma}[{\cite[Theorem 3.2]{KarSta}]}] \label{lem2}
 Let condition {\rm (G3)} be satisfied and $u_0\in\mathcal{W}$. 
Assume that the initial data satisfy \eqref{e9} and they are sufficiently 
small in the sense
\begin{equation}
\mathcal{E}^*(0)<\Big(\frac{1}{C_0\lambda\mu_0^{p_1}}\Big)^{1/p_2}
\end{equation}
where 
\[
p_1=\frac{2(\beta+1)-N(\beta-1)}{2},\quad
p_2=\frac{N\beta-N-4}{4}.
\]
Then the (weak) solution of \eqref{e1} is such that
\begin{equation}
u\in C[0,\infty;D^{1,2}(\mathbb{R}^N)], \quad 
u_t\in C[0,\infty;L^2_g(\mathbb{R}^N)].
\end{equation}
\end{lemma}

\begin{theorem} \label{thm1}  
If the assumption of Lemma \ref{lem2} is satisfied,
then there exist two positive constants $\widehat{C}$ and $\xi$,
independent of $t$ such that
\begin{equation}
0<\mathcal{E}^*(t)\leq \widehat{C}e^{-\xi t}\quad \text{for all }t\geq0
\end{equation}
\end{theorem}

\begin{proof}
 From \eqref{e5}, we have
\begin{equation*}
\mathcal{E}^*(t)=\frac{1}{2}\|u_t(t)\|^2_{L^2_g}
+\frac{\beta-1}{2(\beta+1)}\|u\|^2_{D^{1,2}}+K(u).
\end{equation*}
Theorem 3.2 in \cite{KarSta} shows that for all $t\geq0$,
$u(t)\in\mathcal{W}$, so we have
\begin{equation}
0<\mathcal{E}^*(t) \quad {\rm{for\quad all}}\quad t\geq0.
\end{equation}
The proof of the other inequality relies on the construction of a
Lyapunov function by performing a suitable modification of the
energy. To this end, for $\varepsilon>0$, to be chosen later, we
define
\begin{equation} \label{e15}
L(t)=\mathcal{E}^*(t)+\varepsilon\int guu_t\,dx
\end{equation}
It is straightforward to see that $L(t)$ and $\mathcal{E}^*(t)$ are
equivalent in the sense that there exist two positive constants
$\beta_1$ and $\beta_2$ depending on $\varepsilon$ such that for
$t\geq0$,
\begin{equation} \label{e16}
\beta_1\mathcal{E}^*(t)\leq L(t)\leq \beta_2\mathcal{E}^*(t)
\end{equation}
By taking the time derivative of the function $L$ defined above in
equation \eqref{e15}, using problem \eqref{e1}, and performing several 
integration by parts, we obtain
\begin{equation}
\begin{aligned}
&\frac{dL(t)}{dt}\\
&=-\delta\|u_t\|_{L_g^2}^2+\varepsilon\int
gu^2_t\,dx+\varepsilon\int guu_{tt}\,dx\\
&=-\delta\|u_t\|_{L_g^2}^2+\varepsilon\|u_t\|_{L_g^2}^2+\varepsilon\int
gu\lambda u|u|^{\beta-1}\,dx-\varepsilon\int gu\delta
u_t\,dx+\varepsilon\int gu\phi\Delta u\,dx\\
&=-\delta\|u_t\|_{L_g^2}^2+\varepsilon\|u_t\|_{L_g^2}^2+\varepsilon\lambda\|u\|_{L_g^{\beta+1}}^{\beta+1}
-\varepsilon\|\nabla u\|_2^2-\varepsilon\delta\int guu_t\,dx
\end{aligned} \label{e17}
\end{equation}
Using Young inequality and  Sobolev inequality,  for
any $\gamma>0$, we obtain
\begin{equation}
\int guu_t\,dx\leq\frac{1}{4\gamma}\int gu_t^2\,dx+\gamma\int gu^2\,dx
\leq \frac{1}{4\gamma}\int gu_t^2\,dx+\frac{\gamma}{\alpha}\|\nabla
u\|_2^2 \label{e18}
\end{equation}
where $\alpha$ is the Sobolev constant.

Using the result \cite[(3.20) in Theorem 3.2]{KarSta}, we have
\begin{equation*}
\|u(t)\|_{L_g^{\beta+1}}^{\beta+1}\leq
C_0\mu_0^{p_1}\mathcal{E}^*(0)^{p_2}\|\nabla u\|_2^2
\end{equation*}
Consequently, inserting \eqref{e18} into \eqref{e17}, we have
\begin{equation}
\frac{dL(t)}{dt}
\leq(\varepsilon+\frac{\varepsilon\delta}{4\gamma}-\delta)\|u_t\|_{L_g^2}^2
+(\varepsilon\lambda
C_0\mu_0^{p_1}\mathcal{E}^*(0)^{p_2}
-\varepsilon+\frac{\varepsilon\delta\gamma}{\alpha})\|\nabla u\|_2^2
\end{equation}
By the condition $\mathcal{E}^*(0)^{p_2}C_0\lambda\mu_0^{p_1}<1$,
let us choose $\gamma$ small enough such that
\begin{equation}
\varepsilon (\lambda
C_0\mu_0^{p_1}\mathcal{E}^*(0)^{p_2}-1+\frac{\delta\gamma}{\alpha})<0
\end{equation}
From this inequality we may find $\eta>0$, which depends only on $\gamma$ such
that
\begin{equation}
\frac{dL(t)}{dt}
\leq\Big(\varepsilon(\frac{\delta}{4\gamma}+1)-\delta\Big)\|u_t\|_{L_g^2}^2
-\varepsilon\eta\|\nabla u\|_2^2
\end{equation}
Consequently, using the definition of the energy \eqref{e6}, for any
positive constant $M$ (we will choose the suitable $M$),  we always
obtain
\begin{equation} \label{e22}
\frac{dL(t)}{dt}
\leq-M\varepsilon\mathcal{E}^*(t)
+\Big(\varepsilon(\frac{\delta}{4\gamma}+1+\frac{M}{2})-\delta\Big)
\|u_t\|_{L_g^2}^2
+\varepsilon(\frac{M}{2}-\eta)\|\nabla u\|_2^2
\end{equation}
Choose $M<2\eta$, and $\varepsilon$ small enough such that
\begin{equation}
\varepsilon(\frac{\delta}{4\gamma}+1+\frac{M}{2})-\delta<0
\end{equation}
inequality \eqref{e22} becomes
\begin{equation}
\frac{dL(t)}{dt}\leq-M\varepsilon\mathcal{E}^*(t)\quad {\rm{for\
all}}\quad t\geq0
\end{equation}
On the other hand, by  \eqref{e16}, setting
$\xi=\frac{M\varepsilon}{\beta_2}$, the last inequality
becomes
\begin{equation}
\frac{dL(t)}{dt}\leq-\xi L(t)\quad {\rm{for\ all}}\quad t\geq0
\end{equation}
Integrating this differential inequality  between $0$
and $t$ gives the following estimate for the function $L$
\begin{equation}
L(t)\leq Ce^{-\xi t}\quad \text{for all } t\geq0
\end{equation}
Consequently, by using \eqref{e16} once again, we conclude
\begin{equation}
E(t)\leq \widehat{C}e^{-\xi t}\quad \text{all }d t\geq0
\end{equation}
This completes the proof.
\end{proof}

\subsection*{Acknowledgements}
 The authors want to thank the anonymous referees  for their
valuable comments and suggestions.

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\end{document}