\documentclass[reqno]{amsart} \usepackage{hyperref} \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 $10; \label{1.9}\\ E(t)+\int_0^t\int_{\Omega}\delta|u_t|^2dx\,dt=E(0); \label{1.10} \end{gather} If$E(0)0$then we have \begin{gather} E(t)0,\quad \forall t\in[0,\infty); \label{1.11}\\ \theta\int_{\Omega}|\nabla u|^2dx\ge\int_{\Omega}|u|^{p+1}dx,\quad \theta\in(0,1),\ \forall t\in[0,\infty); \label{1.12}\\ \lim_{t\to +\infty}\int_{\Omega}(|u_t|^2+|\nabla u|^2)dx=0; \label{1.13}\\ \int_0^t\int_{\Omega}|\nabla u|^2dx\,dt\le C. \label{1.14} \end{gather} In this paper we will use the multiplier technique to prove the following result. \begin{theorem} \label{thm1} If$E(0)0$, then there exists positive constant$\gamma$and$C>1$such that $$E(t)\le Ce^{-\gamma t},\quad \forall t\in[0,\infty).\label{1.15}$$ \end{theorem} \subsection*{\label{sec:level2}Our results and their relationship to the literature} The Problem $$\label{1.16} \begin{gathered} u_{tt}-\Delta u+a(x)|u_t|^{m-1}u_t+|u|^{p-1}u=0,\quad \text{in } \Omega, \\ u\big|_{\partial\Omega}=0,\quad (u,u_t)\big|_{t=0}=(u_0,u_1) \end{gathered}$$ 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$\bar{\Omega}$, $$\label{1.17} \begin{gathered} u_{tt}-\Delta{u}+a(x)|u_t|^{m-1}u_t=|u|^{p-1}u,\quad \text{in}\ \Omega, \\ u\big|_{\partial\Omega}=0,\quad (u,u_t)\big|_{t=0}=(u_0,u_1) \end{gathered}$$ 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 $$\int_0^TE(t)dt\le C\big\{\int_0^T\int_{\Gamma(x_0)}(m\cdot\nu)|\frac{\partial u}{\partial\nu}|^2d\Gamma dt+\int_0^T\int_{\Omega}|u_t|^2dx\,dt+|\chi|\big\}.\label{2.1}$$ \end{lemma} \begin{proof} Multiplying (\ref{1.1}) by$q(x)\cdot\nabla uand integrating by parts gives, \cite{Zuazua,Lions}, \begin{aligned} &\Big(\int_{\Omega}u_t(q\cdot\nabla u)dx\Big)\big|_0^T +\frac{1}{2}\int_0^T\int_{\Omega}(\mbox{div}q)(|u_t|^2-|\nabla u|^2)dx\,dt \\ &+\int_0^T\int_{\Omega}(\sum_{k,j=1}^n\frac{\partial q_k}{\partial x_j}\frac{\partial u}{\partial x_k}\frac{\partial u}{\partial x_j})dx\,dt +\int_0^T\int_{\Omega}(\mbox{div}q)\frac{|u|^{p+1}}{p+1}dx\,dt \\ &+\int_0^T\int_{\Omega}\delta u_t(q\cdot\nabla u)dx\,dt\\ &=\frac{1}{2}\int_0^T\int_{\partial\Omega}(q\cdot\nu)|\frac{\partial u}{\partial\nu}|^2d\Gamma \,dt. \end{aligned}\label{2.2} Hereq(x)\in W^{1,\infty}(\Omega)$. Applying identity (\ref{2.2}) with$q(x)=m(x), we deduce \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} We now multiply (\ref{1.1}) byuand integrate by parts, then we have $$\Big(\int_{\Omega}u(u_t+\frac{\delta}{2}u)dx\Big)\big|_0^T =\int_0^T\int_{\Omega}(|u_t|^2-|\nabla u|^2)dx\,dt+\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt.\label{2.4}$$ Combining (\ref{2.3}) and (\ref{2.4}) we obtain \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} On the other hand, for any given\varepsilon>0, \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} 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, $$E(t)\le C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt.\label{2.7}$$ \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{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} From (\ref{2.4}), we see that $$\int_0^T\int_{\Gamma(x_0)}|\frac{\partial u}{\partial\nu}|^2d\Gamma dt\le C\int_0^T\int_{\Omega}(|u_t|^2+|u|^{p+1})dx\,dt+Y,\label{2.9}$$ 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{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} TakingT$sufficiently large we get (\ref{2.7}). \end{proof} \begin{lemma}\label{lem2.3} $$\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt\le C\int_0^T\int_{\Omega}|u_t|^2dx\,dt.\label{2.11}$$ \end{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 $$\lim_{n\to\infty}\frac{\int_0^T\int_{\Omega}|u_n|^{p+1}dx\,dt} {\int_0^T\int_{\Omega}|u_{nt}|^2dx\,dt}=\infty.\label{2.12}$$ From (\ref{1.12}) and (\ref{1.14}) we have $$\int_0^T\int_{\Omega}|u_n|^{p+1}dx\,dt\le\theta\int_0^T\int_{\Omega}|\nabla u_n|^2dx\,dt\le C.\label{2.13}$$ Thus we get $$\lim_{n\to\infty}\int_0^T\int_{\Omega}|u_{nt}|^2dx\,dt=0.\label{2.14}$$ 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 $$u_t=0,\quad \mbox{a.e. in } \Omega\times(0,T)\label{2.19}$$ 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 $$\int_0^T\int_{\Omega}|u|^{p+1}dx\,dt\le\theta\int_0^T\int_{\Omega}|\nabla u|^2dx\,dt<\int_0^T\int_{\Omega}|\nabla u|^2dx\,dt \label{2.22}$$ 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 $$E(T)\le C\int_0^T\int_{\Omega}|u_t|^2dx\,dt. \label{2.23}$$ This inequality, (\ref{1.10}), and semigroup properties complete the proof of Theorem \ref{thm1}. 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