\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil A blowup result \hfil EJDE--2001/30} {EJDE--2001/30\hfil Salim A. Messaoudi \hfil} \begin{document} \title{\vspace{-1in} \parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No.~30, pp. 1--9. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % A blowup result in a multidimensional semilinear thermoelastic system % \thanks{\emph{Mathematics Subject Classifications:} 35K22, 58D25, 73B30. \hfil\protect\break\indent \emph{Key words:} Thermoelasticity, weak solutions, negative energy, blowup, finite time. \hfil\protect\break\indent \copyright 2001 Southwest Texas State University. \hfil\protect\break \indent Submitted February 20, 2001. Published May 7, 2001.} } \date{} \author{Salim A. Messaoudi} \maketitle \begin{abstract} In this work, we consider a multidimensional semilinear system of thermoelasticity and show that the energy of any weak solution blows up in finite time if the initial energy is negative. This work generalizes earlier results in [5] and [8]. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 % \@addtoreset{equation}{section} \catcode`@=12 \section{Introduction} In [8], we considered the one-dimensional Cauchy problem $$\displaylines{ u_{tt}(x,t) =au_{xx}(x,t)+b\theta _{x}(x,t)+|u(x,t)|^{\alpha -1}u(x,t) \cr c\theta _t(x,t) = k\theta _{xx}(x,t)+bu_{xt}(x,t),\quad x\in \mathbb{R} ,t\geq 0\cr u(x,0) = u_0(x),\quad u_t(x,0)=u_1(x),\quad \theta (x,0)=\theta (x),\quad x\in \mathbb{R}, }$$ where $a,$ $c,$ $k$ are strictly positive constants, $b$ is a nonzero constant, and $\alpha $ $\geq \sqrt{1+b^2/\left( ac\right) }$. We showed that any weak solution with negative initial energy blows up in finite time if $u_0$ and $u_1$ are cooperative ($\int u_0u_1>0$). This result was improved by Kirane and Tatar [5], where the authors studied a more general system by allowing gradient terms in both equations. To overcome the difficulty caused by these extra terms, they defined a functional which satisfies the conditions of a theorem by Kalantarov and Ladyzhenskaya [4]. Their result, when applied to the system in [8], omits the condition of cooperative initial data, however the condition on $\alpha $ remained (see relation 13 of [5]). In [17], Racke and Wang discussed the propagation of singularities for systems of homogeneous thermoelasticity in one spatial dimension. They considered some linear and semilinear Cauchy problems and described the propagation of singularities, as well as, the distribution of regular domains if the initial data have different regularity in different parts of the real line. Concerning global existence and asymptotic behavior of weak solutions, it is worth noting the work of Aassila [1], where a purely linear multidimensional system of inhomogeneous and anisotropic thermoelasticity, associated with nonlinear boundary conditions, has been studied. Under suitable requirements on the nonlinear terms at the boundary, the author proved a decay result. As he mentioned, his result extends the one in [14] to the nonlinear case. For results regarding the matter of existence, regularity, controllability, and long-time behavior of systems of thermoelasticity, we refer the reader to articles [2], [3], [12], [13], [15], [16], and [18]. In this paper we are concerned with the initial boundary value problem \begin{eqnarray} & u_{tt}(x,t)=\mathop{\rm div}(A(x)\nabla u(x,t))+b(x)\cdot \nabla \theta (x,t)+D(x)\cdot \nabla u(x,t) & \nonumber \\ & \hspace{18mm}-m(x)u_t(x,t)+e^{\beta t}u(x,t)|u(x,t)|^{p-2},\quad x\in \Omega , t>0 & \nonumber \\ & c(x)\theta_t(x,t)=\mathop{\rm div}[K(x)\nabla \theta (x,t)+b(x)u_t(x,t)] +R(x)\cdot \nabla u(x,t) & \\ & u(x,t)=0,\quad \theta (x,0)=0,\quad x\in \partial \Omega ,t\geq 0 & \nonumber \\ & u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x),\quad \theta (x,0)=\theta (x),\quad x\in \Omega , & \nonumber \end{eqnarray} where $b,D,R$ are ``function entry" $n$-component real vectors; $c,m$ are functions; $A$, $K$ are $n\times n$ ``function entry" matrices such that $A$ is symmetric; $b\neq 0$, $p>2,\beta >0$; and $\Omega $ is a bounded domain of $\mathbb{R}^n$ ($n\geq 1$), with a smooth boundary $\partial \Omega $. We will show that any weak solution, with negative ``enough" initial energy blows up in finite time. This work generalizes the result of Kirane and Tatar [5] to the multidimensional setting and includes an earlier result by the author [9] (see comments below). To establish our result, we impose the following \begin{description} \item[\textbf{H1)}] $c,m$ $\in L^{\infty }(\Omega )$, $b,D,R\in [L^{\infty }(\Omega )]^n$, and for $a,c$ $,k>0$ the functions $A$, $K$ $\in [L^{\infty }(\Omega )]^{n\times n}$ satisfy \[ \displaylines{c(x)\geq c,\quad m(x)\geq m_0\geq 0,\quad \forall x\in \Omega .\cr A(x)\xi \cdot \xi \geq a|\xi |^2,\quad K(x)\xi \cdot \xi \geq k|\xi |^2,\quad \forall x\in \Omega ,\ \forall \xi \in \mathbb{R}^n\,.} \] \item[\textbf{H2)}] $(u_0,u_1,\theta _0)\in H_0^1(\Omega )\times L^2(\Omega )\times H_0^1(\Omega )$ \item[\textbf{H3)}] $p\leq 2(n-1)/(n-2)$ if $n\geq 3$. \end{description} \paragraph{Definition} By a weak solution of (1.1), we mean a pair $(u,\theta )$ such that \begin{eqnarray} &u\in C\left([0, T); H_0^1(\Omega )\right) \cap C^1\left([0, T); L^2(\Omega )\right)& \\ &\theta \in L^2\left([0, T); H_0^1(\Omega )\right) \cap C^1\left([0, T);L^2(\Omega )\right)& \nonumber \end{eqnarray} and satisfying the system in the following sense [7]: For any $(v,\varphi )\in [H_0^1(\Omega )]^2$, \begin{eqnarray} \frac{\partial }{\partial t}\int_{\Omega }u_tv\,dx &=&\int_{\Omega }A(x)\nabla u\cdot\nabla v\,dx+\int_{\Omega }vb(x)\cdot \nabla \theta \,dx \\ &&+\int_{\Omega }vD(x)\cdot \nabla u\,dx-\int_{\Omega }vm(x)u_t\,dx+\int_{\Omega }ve^{\beta t}u|u|^{p-2}\,dx \nonumber \\ \frac{\partial }{\partial t}\int_{\Omega }c(x)\theta \varphi \,dx &=&\int_{\Omega}K(x)\nabla \theta \cdot\nabla \varphi \,dx +\frac{\partial }{\partial t} \int_{\Omega }ub(x)\cdot \nabla \varphi \,dx \int_{\Omega}\varphi R(x) \cdot \nabla u\,dx \nonumber \end{eqnarray} for almost every $t\in [0,T)$. \paragraph{Remark.} The condition on $p$ in (H3) is imposed so that $\int_{\Omega }ve^{\beta t}u|u|^{p-2}\,dx$ makes sense. \section{Main Result} In this section we prove our main result. For this purpose we set \begin{equation} E(t):=\frac{1}{2}\int_{\Omega }[u_t^2+A(x)\nabla u\cdot \nabla u+c(x)\theta ^2]\,dx-\frac{1}{p}\int_{\Omega }e^{\beta t}|u|^{p}\,dx\,. \end{equation} \begin{lemma} If $E(0)<0$ and \begin{equation} \beta \geq 2\sqrt{\frac{n(cd^2+r^2)}{4c}} \end{equation} Then \begin{equation} E^{\prime }(t)\leq -\int_{\Omega }K(x)\nabla \theta \cdot \nabla \theta \,dx\leq -k\int_{\Omega }|\nabla \theta |^2\,dx\leq 0. \end{equation} \end{lemma} \paragraph{Proof.} By taking a derivative of (2.1) and using the equations of (1.1) we get \begin{eqnarray} E^{\prime}(t)&=&\int_{\Omega }[u_tu_{tt}+A(x)\nabla u\cdot \nabla u_t+c(x)\theta \theta_t]\,dx \nonumber \\ &&-\frac{\beta }{p}\int_{\Omega }e^{\beta t}|u|^{p}\,dx-\int_{\Omega }e^{\beta t}|u|^{p-2}uu_t\,dx \\ &=&\int_{\Omega }u_tD(x)\cdot \nabla u-\int_{\Omega }m(x)u_t^2\,dx+\int_{\Omega }\theta R(x)\cdot \nabla u(x,t) \nonumber \\ &&-\int_{\Omega }K(x)\nabla \theta \cdot\nabla \theta \,dx-\frac{\beta }{p} \int_{\Omega }e^{\beta t}|u|^{p}\,dx \nonumber \end{eqnarray} We then use Young's inequality and (H1) to obtain \begin{eqnarray} E^{\prime}(t)&\leq& -k\int_{\Omega }|\nabla \theta |^2\,dx-m_0 \int_{\Omega }u_t^2\,dx+\varepsilon_1\int_{\Omega }u_t^2\,dx +\frac{nd^2}{4\varepsilon_1} \int_{\Omega }|\nabla u|^2\,dx \nonumber \\ &&+\varepsilon_2\int_{\Omega }\theta ^2\,dx+\frac{nr^2}{4\varepsilon_2} \int_{\Omega }|\nabla u|^2\,dx-\frac{\beta }{p}\int_{\Omega }e^{\beta t} |u|^{p}\,dx \end{eqnarray} where $d:=\|D\|_\infty $ and $r:=\|R\|_\infty $. By using (2.1) we obtain \begin{eqnarray} E^{\prime}(t)&\leq& -k\int_{\Omega }|\nabla \theta |^2\,dx+2(\varepsilon _1-m_0)E(t)+[\varepsilon_2-c(\varepsilon_1-m_0)]\int_{\Omega }\theta ^2\,dx \nonumber \\ &&-[a(\varepsilon_1-m_0)-\frac{n}{4}(\frac{d^2}{\varepsilon_1}+\frac{r^2}{% \varepsilon_2})]\int_{\Omega }|\nabla u|^2\,dx \\ &&-\frac{1}{p}[\beta-2(\varepsilon_1-m_0)]\int_{\Omega } e^{\beta t}|u|^{p}\,dx \nonumber \end{eqnarray} At this point we take $\varepsilon_2=c(\varepsilon_1-m_0)$ and $% \varepsilon_1 $ large enough so that \[ a(\varepsilon_1-m_0)-\frac{n}{4}(\frac{d^2}{\varepsilon_1}+\frac{r^2}{% c(\varepsilon_1-m_0)})\geq 0.\newline \] It suffices in this case to have \begin{equation} a(\varepsilon_1-m_0)-\frac{n}{4}\frac{cd^2+r^2}{c(\varepsilon _1-m_{9})}\geq 0, \end{equation} which is equivalent to \begin{equation} \varepsilon_1\geq m_0+\sqrt{\frac{n(cd^2+r^2)}{4c}}\newline \end{equation} By combining all above and using (2.2) we arrive at \[ E^{\prime}(t)\leq -k\int_{\Omega }|\nabla \theta |^2\,dx+2(\varepsilon _1-m_0)E(t). \] Therefore (2.3) is established provided that $E(t)\leq 0$. This is of course true since $E(0)\leq 0$. \begin{lemma} Suppose that (H3) holds. Then there exists a positive constant $C>1$ depending on $n,p$ only such that \begin{equation} \Vert u\Vert _p^{s}\leq C\left( \Vert \nabla u\Vert _2^2+\Vert u\Vert _p^{p}\right) \end{equation} for any $u\in H_0^1(\Omega )$ and $2\leq s\leq p$. \end{lemma} \paragraph{Proof.} If $\|u\|_p\leq 1$ then $\|u\|_p^{s}\leq \|u\|_p^2\leq C\|\nabla u\|_2^2$ by Sobolev embedding theorems. If $\|u\|_p>1$ then $% \|u\|_p^{s}\leq C\|u\|_p^{p}$. Therefore (2.9) follows. As a result of (2.1), (2.9), and the lemma, we have \begin{corollary} Assume that (H3) holds. Then we have \begin{equation} \Vert u\Vert _p^{s}\leq C\left( E(t)+\Vert u_t\Vert _2^2+e^{\beta t}\Vert u\Vert _p^{p}+\Vert \theta \Vert _2^2\right) \end{equation} for any $u\in H_0^1(\Omega )$ and $2\leq s\leq p$. \end{corollary} \begin{theorem} Let (H1) and (H3) be fulfilled. Then given $T>0$ there exists $\lambda >0$ such that, for any initial data satisfying (H2) and \begin{equation} E(0)<-\lambda , \end{equation} the solution (1.2) blows up in a time $T^{*}\leq T$. \end{theorem} \paragraph{Proof.} Set $H(t)=-E(t)$. Then, by virtue of (2.3), $H^{\prime}(t)\geq k\int_{\Omega }|\nabla \theta |^2\,dx\geq 0$; hence \begin{equation} \lambda <-E(0)=H(0)\leq H(t)\leq \frac{b}{p}e^{\beta t}\|u\|_p^{p}. \end{equation} We then define \begin{equation} L(t):=H^{1-\alpha }(t)+\varepsilon \int_{\Omega }uu_t(x,t)\,dx+\frac{ \varepsilon }{2}\int_{\Omega }m(x)u^2(x,t)\,dx \end{equation} for $\varepsilon $ small to be chosen later and $\alpha =(p-2)/2p$. By taking a derivative of (2.13) and using equation (1.1) we obtain \begin{eqnarray} L^{\prime}(t)&=&(1-\alpha )H^{-\alpha }(t)H^{\prime}(t)-\varepsilon \int_{\Omega }A(x)\nabla u\cdot\nabla u\,dx \nonumber \\ &&+\varepsilon \int_{\Omega }u_t^2+\varepsilon e^{\beta t}\int_{\Omega }|u|^{p}\,dx+\varepsilon \int_{\Omega }ub(x)\cdot\nabla \theta \,dx \\ &=&(1-\alpha )H^{-\alpha }(t)H^{\prime}(t)+\varepsilon \int_{\Omega }u_t^2-\varepsilon \int_{\Omega }A(x)\nabla u\cdot\nabla u\,dx \nonumber \\ && +\varepsilon \int_{\Omega }ub\cdot\nabla \theta \,dx +\varepsilon p\left[ H(t)+\frac{1}{2}\int_{\Omega }[u_t^2+A(x)\nabla u \cdot \nabla u+c(x)\theta ^2]\,dx\right] \nonumber \end{eqnarray} Then use Young's inequality to estimate $\int_{\Omega }ub\cdot\nabla \theta \,dx$ in (2.14). For all $\delta >0$, \begin{eqnarray} L^{\prime}(t)&\geq& k(1-\alpha )H^{-\alpha }(t)\|\nabla \theta \|_2^2+\varepsilon (\frac{p}{2}-1)\int_{\Omega }A(x)\nabla u\cdot\nabla u\,dx \nonumber \\ &&+\varepsilon (\frac{p}{2}+1)\int_{\Omega }u_t^2\,dx+\frac{p\varepsilon }{2} \int_{\Omega }c(x)\theta ^2\,dx+p\varepsilon H(t) \nonumber \\ &&-B\varepsilon \left[ \frac{1}{4\delta }\|\nabla \theta \|_2^2+\delta \int_{\Omega }u^2\,dx\right] \\ &\geq& \left[ k(1-\alpha )H^{-\alpha }(t)-\frac{B\varepsilon }{4\delta } \right] \|\nabla \theta \|_2^2+\varepsilon (\frac{p}{2}-1)\int_{\Omega }A(x)\nabla u\cdot\nabla u\,dx \nonumber \\ &&+\varepsilon (\frac{p}{2}+1)\int_{\Omega }u_t^2+p\varepsilon H(t)+\frac{% p\varepsilon }{2}\int_{\Omega }c(x)\theta ^2\,dx-B\varepsilon \delta \|u\|_2^2, \nonumber \end{eqnarray} where $B=$ $\|b\|_\infty $. We then take $\delta $ $=H^{\alpha }(t)/M$, for large $M$ to be specified later. Substitute in (2.15) to arrive at \begin{eqnarray} \lefteqn{L'(t)} \\ &\geq& \left[ k(1-\alpha )-\frac{M}{4}\varepsilon B\right] H^{-\alpha }(t)\|\nabla \theta \|_2^2+\varepsilon (\frac{p}{2}-1)\int_{\Omega }A(x)\nabla u\cdot\nabla u\,dx \nonumber \\ && +\varepsilon (\frac{p}{2}+1)\int_{\Omega }u_t^2+\frac{p\varepsilon }{2} \int_{\Omega }c(x)\theta ^2\,dx+\varepsilon \left[ pH(t)-\frac{B}{M}% H^{\alpha }(t)\|u\|_2^2\right] . \nonumber \end{eqnarray} By (2.10) and the inequality $\|u\|_2^2\leq C\|u\|_p^2$, we obtain \[ H^{\alpha }(t)\|u\|_2^2\leq C\left(\frac{b}{p}\right) ^{\alpha }e^{\alpha \beta t}\|u\|_p^{2+\alpha p}\leq C_{T}\|u\|_p^{2+\alpha p}; \] where $C_{T}=C\left(\frac{b}{p}\right) ^{\alpha }e^{\alpha \beta T}$; consequently (2.16) yields \begin{eqnarray} \lefteqn{L'(t)} \\ &\geq &\left[ k(1-\alpha )-\frac{M}{4}\varepsilon B\right] H^{-\alpha }(t)\|\nabla \theta \|_2^2+\varepsilon (\frac{p}{2} -1)\int_{\Omega }A(x)\nabla u\cdot\nabla u\,dx \nonumber \\ && +\varepsilon (\frac{p}{2}+1)\int_{\Omega }\rho (x)u_t^2+\frac{ p\varepsilon }{2}\int_{\Omega }c(x)\theta ^2\,dx+\varepsilon \left[ pH(t)- \frac{B}{M}C_{T}\|u\|_p^{2+\alpha p}\right] . \nonumber \end{eqnarray} We then use corollary 2.3, for $s=2+\alpha p0$ is a constant depending on $C_{T}$ (hence on $T$)$.$ Once $M$ is chosen we then pick $\varepsilon $ small enough so that $k(1-\alpha )-\varepsilon BM/4\geq 0$ and \begin{equation} L(0)=H^{1-\alpha }(0)+\varepsilon \int_{\Omega }u_0u_1(x)\,dx+\frac{ \varepsilon }{2}\int_{\Omega }m(x)u_0^2(x)\,dx>\frac{\lambda }{2}; \end{equation} therefore (2.19) takes the form \begin{equation} L^{\prime}(t)\geq \Gamma \left[ H(t)+\|u_t\|_2^2+\|u\|_p^{p}+\|\theta \|_2^2\right] , \end{equation} where $\Gamma =\Gamma_1\varepsilon >0$; hence $L(t)\geq L(0)>\frac{\lambda }{% 2}$ for all $t\geq 0$. Now the estimate \begin{equation} |\int_{\Omega }uu_t\,dx|\leq \|u\|_2\|u_t\|_2\leq C\|u\|_p\|u_t\|_2; \end{equation} implies \[ \Big| \int_{\Omega }uu_t\,dx\Big|^{1/(1-\alpha )}\leq C\|u\|_p^{1/(1-\alpha )}\|u_t\|_2^{1/(1-\alpha )}. \] Again Young's inequality, by virtue of corollary 2.3, yields \begin{eqnarray} \Big|\int_{\Omega }uu_t\,dx\Big|^{1/(1-\alpha )} &\leq &C\left[\|u\|_p^{p}+\|u_t\|_2^2\right] \\ &\leq& C_{T}\left[H(t)+\|u\|_p^{p}+\|u_t\|_2^2+\|\theta \|_2^2\right] \,. \nonumber \end{eqnarray} Now we have the estimate \begin{equation} |\int_{\Omega }m(x)u^2\,dx|^{1/(1-\alpha )}\leq C\|u\|_p^{2/(1-\alpha )}\leq C_{T}\left[ H(t)+\|u\|_p^{p}+\|u_t\|_2^2+\|\theta \|_2^2\right] , \end{equation} since $2/(1-\alpha )