\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 $T_0\in {\Bbb R}$, $01$. Then there exists $T_0\in {\Bbb R}$, where $00$ 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 $$\int_0^{T_0}|\theta _m'(t)|^2dt\leq C \label{2.33}$$ \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 $$(\mu (t)e^{-\alpha t})'\leq \theta (t)+\beta \mu ^\gamma (t) \label{2.36}$$ (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 $$\mu (t)\leq K_1+K_2\int_0^t\mu ^\gamma (s)\,,ds\,. \label{2.37}$$ 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}