\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 129, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/129\hfil Decay of solutions] {Decay of solutions for a plate equation with $p$-Laplacian and memory term} \author[W. Liu, G. Li, L. Hong \hfil EJDE-2012/129\hfilneg] {Wenjun Liu, Gang Li, Linghui Hong} % in alphabetical order \address{Wenjun Liu\newline College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China} \email{wjliu@nuist.edu.cn} \address{Gang Li\newline College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China} \email{ligang@nuist.edu.cn} \address{Linghui Hong\newline College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China} \email{hlh3411006@163.com} \thanks{Submitted April 20, 2012. Published August 15, 2012.} \subjclass[2000]{35L75, 35B40} \keywords{Rate of decay; plate equation; $p$-Laplacian; memory term} \begin{abstract} In this note we show that the assumption on the memory term $g$ in Andrade \cite{a1} can be modified to be $g'(t)\leq -\xi(t)g(t)$, where $\xi(t)$ satisfies $$\xi'(t)\leq0,\quad \int_0^{+\infty}\xi(t){\rm d}t=\infty.$$ Then we show that rate of decay for the solution is similar to that of the memory term. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Consider a bounded domain $\Omega$ in $\mathbb{R}^N$ with smooth boundary $\Gamma=\partial \Omega$, and study the solutions to the problem \begin{gather} \label{eq1} u_{tt}+\Delta^2u-\Delta_pu+\int_0^tg(t-s)\Delta u(s) {\rm d}s-\Delta u_t+f(u)=0 \quad \text{in } \Omega\times \mathbb{R}^+ ,\\ \label{eq2} u=\Delta u= 0 \quad \text{on }\Gamma\times \mathbb{R}^+ , \\ \label{eq3} u(\cdot,0) = u_0,\quad u_t(\cdot,0)=u_1 \quad \text{in } \Omega, \end{gather} where $\Delta_pu=\mathrm{div }(|\nabla u|^{p-2}\nabla u )$ is the $p$-Laplacian operator. This problem without the memory term models elastoplastic flows. We refer to \cite{a1} for a motivation and references concerning the study of problem \eqref{eq1}-\eqref{eq3}. We will us the following assumptions: \begin{itemize} \item[(A1)] The memory kernel $g$ has typical properties $$\label{eq4} g(0)>0,\quad l=1-\mu_1\int_0^\infty g(s){\rm d}s>0,$$ where $\mu_1>0$ is the embedding constant for $\|\nabla u\|_2^2\leq \mu_1\|\Delta u\|_2^2$. There exists a constant $k_1>0$ such that $$\label{eq5} g'(t)\leq-k_1g(t),\quad \forall\ t\geq0.$$ \item[(A2)] The forcing term $f$ satisfies \begin{gather} \label{eq6} f(0)=0,\quad |f(u)-f(v)|\leq k_2(1+|u|^\rho+|v|^\rho)|u-v|,\quad \forall u, v \in \mathbb{R}, \\ \label{eq7} 0\leq \widehat{f}(u)\leq f(u)u,\quad \forall\ u\in \mathbb{R}, \end{gather} where $k_2$ is a positive constant, $\widehat{f}(z)=\int_0^z f(s){\rm d}s$, and $$0<\rho\leq\frac{4}{N-4} \text{ if } N\geq5 \quad \text{and} \quad \rho>0 \text{ if } 1\leq N\leq4.$$ \item[(A3)] The constant $p$ satisfies $$\label{eq9} 2\leq p\leq\frac{2N-2}{N-2} \text{ if } N\geq3 \quad \text{and} \quad p\geq2 \text{ if } N=1, 2.$$ \end{itemize} \begin{theorem}[{\cite[Theorem 2.1]{a1}}] \label{th1} Assume that {\rm (A1)--(A3)} hold. \begin{itemize} \item[(i)] If the initial data $(u_0,u_1)\in(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$, then problem \eqref{eq1}-\eqref{eq3} has a unique weak solution $$u\in C(\mathbb{R}^+; H^2(\Omega)\cap H_0^1(\Omega))\cap C^1(\mathbb{R}^+;L^2(\Omega)).$$ \item[(ii)] If the initial data $(u_0,u_1)\in H_\Gamma^3(\Omega)\times H_0^1(\Omega)$, where $$H_\Gamma^3(\Omega)=\{u\in H^3(\Omega)|u=\Delta u=0 \text{on } \Gamma \},$$ then problem \eqref{eq1}-\eqref{eq3} has a unique strong solution satisfying $$u\in L^\infty(\mathbb{R}^+;H_\Gamma^3(\Omega)), \quad u_t\in L^\infty(\mathbb{R}^+;H_0^1(\Omega)),\quad u_{tt}\in L^2(0,T; H^{-1}(\Omega)).$$ \item[(iii)] In both cases, the energy $E(t)$ of problem \eqref{eq1}-\eqref{eq3} satisfies the decay rate $$E(t)\leq CE(0)e^{-\gamma t},\quad t\geq0,$$ for some $C, \gamma>0$, where $$\label{eq8} E(t)=\frac{1}{2}\|u_t(t)\|_2^2+\frac{1}{2}\|\Delta u(t)\|_2^2+\frac{1}{p}\|\nabla u(t)\|_p^p+\int_\Omega\widehat{f}(u(t)){\rm d}x.$$ \end{itemize} \end{theorem} In this note, we shall extend the above exponential rate of decay to the general case, which is similar to that of $g$. We use the following assumption which is weaker than \eqref{eq5}. \begin{itemize} \item[(A4)] There exists a positive differentiable function $\xi(t)$ such that $$g'(t)\leq -\xi(t)g(t),\quad \forall t\geq0,$$ and $\xi(t)$ satisfies $$\xi'(t)\leq0,\ \forall\ t>0,\ \int_0^{+\infty}\xi(t){\rm d}t=\infty.$$ \end{itemize} Then, we can prove the following main result. \begin{theorem}\label{th2} Assume that {\rm (A2)--(A4)} and \eqref{eq4} hold. If the initial data $(u_0,u_1)\in(H^2(\Omega)\cap H_0^1(\Omega))\times L^2(\Omega)$ or $(u_0,u_1)\in H_\Gamma^3(\Omega)\times H_0^1(\Omega)$, then the energy $E(t)$ of problem \eqref{eq1}-\eqref{eq3} satisfies the inequality $$\label{eq10'} E(t)\leq KE(0)e^{-k\int_{0}^t\xi(s){\rm d}s}, \quad t\geq 0,$$ for some $K, k>0$. \end{theorem} \begin{remark} \rm We note that a similar decay rate was given in \cite[Theorem 3.5]{m1}. However, unlike \cite[(G2)]{m1} and \cite[(A1)]{w1}, we do not use the condition of $|\frac{\xi'(t)}{\xi(t)}|\le k$ here. \end{remark} \begin{remark} \rm For $\xi(t)\equiv k_1$, \eqref{eq10'} recaptures the exponential decay rate in \cite[Theorem 2.1]{a1}. For $\xi(t)=a (1+t)^{-1}$, we can get polynomial decay rate, which is nt addressed in \cite{a1}. \end{remark} \section{Proof of Theorem \ref{th2}} Let us first prove the decay property for the strong solution $u$ of problem \eqref{eq1}-\eqref{eq3}. We modify the perturbed energy method in \cite{a1} by using the idea of \cite{l1, m1}. Assume that condition (A4) holds and define the modified energy, as in \cite{a1}, \begin{align*} F(t)&=\frac{1}{2}\|u_t(t)\|_2^2+\frac{1}{2}\|\Delta u(t)\|_2^2+\frac{1}{p}\|\nabla u(t)\|_p^p+\int_\Omega\widehat{f}(u(t)){\rm d}x\\ &\quad -\frac{1}{2}\Big(\int_{0}^t g(s){\rm d}s\Big)\|\nabla u(t)\|_2^2+\frac{1}{2}(g\circ \nabla u)(t), \end{align*} where $$(g \circ \nabla u)(t) =\int_{0}^{t}g (t-s)\|\nabla u(t)-\nabla u(s)\|_{2}^{2}{\rm d}s.$$ Then we obtain $$E(t)\leq \frac{1}{l}F(t),$$ and $F(t)$ is decreasing because \label{eq10} \begin{aligned} F'(t)&=-\|\nabla u_t(t)\|_2^2+\frac{1}{2}(g'\circ \nabla u)(t)-\frac{1}{2}g(t)\|\nabla u(t)\|_2^2 \\ &\le -\|\nabla u_t(t)\|_2^2-\frac{1}{2}\xi(t)(g\circ \nabla u)(t)\le 0. \end{aligned} Let $$\Psi(t)=\int_\Omega u_t(t)u(t){\rm d}x$$ and $$F_\varepsilon(t)=F(t)+\varepsilon \Psi(t),\quad \forall\ \varepsilon>0.$$ To obtain the decay result, we use the following lemmas which are of crucial importance in the proof. \begin{lemma}[{\cite[Lemma 4.1]{a1}}]\label{le1} There exists $C_1>0$ such that $$|F_\varepsilon(t)-F(t)|\leq \varepsilon C_1F(t),\quad \forall t\geq0,\ \forall\ \varepsilon>0.$$ \end{lemma} \begin{lemma}[{\cite[(27) in Lemma 4.2]{a1}}] \label{le2} There exist positive constants $C_2, C_3$ such that $$\label{eq11} \Psi'(t)\leq-F(t)+C_2\|\nabla u_t(t)\|_2^2+C_3(g\circ \nabla u)(t).$$ \end{lemma} Now, we conclude the proof of the decay property. Let $$\varepsilon_0=\min\big\{\frac{1}{2C_1},\frac{1}{C_2}\big\}.$$ It follows from Lemma \ref{le1} that, for $\varepsilon<\varepsilon_0$, $$\label{eq12} \frac{1}{2}F(t)\leq F_\varepsilon(t)\leq\frac{3}{2}F(t),\quad t\geq0.$$ By the definition of $F_\varepsilon(t)$, \eqref{eq10} and \eqref{eq11}, we obtain \label{eq13} \begin{aligned} \xi(t)F'_\varepsilon(t) &=\xi(t)F'(t)+\varepsilon \xi(t)\Psi'(t) \\ &\leq -\xi(t)\|\nabla u_t(t)\|_2^2-\frac{\xi^2(t)}{2}(g\circ \nabla u)(t)-\varepsilon\xi(t)F(t) \\ &\quad +\varepsilon C_2\xi(t)\|\nabla u_t(t)\|_2^2+\varepsilon C_3\xi(t)(g\circ \nabla u)(t) \\ &\leq -(1-\varepsilon C_2)\xi(t)\|\nabla u_t(t)\|_2^2-\varepsilon\xi(t)F(t)+\varepsilon C_3\xi(t)(g\circ \nabla u)(t) \\ &\leq-\varepsilon\xi(t)F(t)+\varepsilon C_3\xi(t)(g\circ \nabla u)(t) \\ &\leq-\varepsilon\xi(t)F(t)-2\varepsilon C_3F'(t). \end{aligned} We set $$L(t)=\xi(t)F_\varepsilon(t)+2\varepsilon C_3F(t).$$ Then, $L(t)$ is equivalent to $F(t)$. In fact, we have $$L(t)\le \xi(0)F_\varepsilon(t)+2\varepsilon C_3F(t)\le \Big(\frac{3}{2}\xi(0)+2\varepsilon C_3\Big)F(t)$$ and $$L(t)\ge \frac{1}{2}\xi(t)F(t)+ 2\varepsilon C_3F(t)\ge 2\varepsilon C_3F(t).$$ Since $F(t)\ge l E(t)\ge 0$ and $\xi'(t)\le 0$, from \eqref{eq12} and \eqref{eq13} we obtain \label{eq14} \begin{aligned} L'(t)&=\xi'(t)F_\varepsilon(t)+\xi(t)F'_\varepsilon(t)+2\varepsilon C_3F'(t) \\ &\leq \xi(t)F'_\varepsilon(t)+2\varepsilon C_3F'(t) \\ &\leq -\varepsilon\xi(t)F(t)\leq-\varepsilon k\xi(t)L(t), \end{aligned} where we have used \eqref{eq13} and $k$ is a positive constant. A simple integration of \eqref{eq14} leads to \begin{align}\label{eq15} L(t)\leq L(0)e^{-k\int_{0}^t\xi(s){\rm d}s},\quad \forall\ t\geq 0. \end{align} This proves the decay property for strong solutions in $H_\Gamma^3(\Omega)$. The result can be extended to weak solutions by standard density arguments, as in Cavalcanti et al. \cite{c1, c2}. \subsection*{Acknowledgements} This work was partly supported by the Tianyuan Fund of Mathematics (Grant No. 11026211) and the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No. 09KJB110005). \begin{thebibliography}{0} \bibitem{a1} D. Andrade, M. A. Jorge Silva, T. F. Ma; Exponential stability for a plate equation with $p$-Laplacian and memory terms, {\it Math. Methods Appl. Sci.} {\bf 35} (2012), no. 4, 417--426. \bibitem{c1} M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma; Exponential decay of the viscoelastic Euler-Bernoulli equation with a nonlocal dissipation in general domains, {\it Differential Integral Equations} {\bf 17} (2004), no.~5-6, 495--510. \bibitem{c2} M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. 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