\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \markboth{\hfil Solutions to parabolic-hyperbolic equations \hfil EJDE--2003/09} {EJDE--2003/09\hfil L. Bougoffa \& M. S. Moulay \hfil} \begin{document} \title{\vspace{-1in} \parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 09, pp. 1--6. \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} \\ % Generalized solutions to parabolic-hyperbolic equations % \thanks{\emph{Mathematics Subject Classifications:} 35M20. \hfil\break \indent {\em Key words:} Composite type, parabolic-hyperbolic equations, generalized solutions, \hfil\break \indent energy inequality. \hfil\break \indent \copyright 2003 Southwest Texas State University. \hfil\break \indent Submitted December 2, 2002. Published February 4, 2003.} } \date{} \author{Lazhar Bougoffa \& Mohamed Said Moulay} \maketitle \begin{abstract} We study boundary-value problems for composite type equations: pa\-rabolic-hyperbolic equations. We prove the existence and uniqueness of generalized solutions, using energy inequality and the density of the range of the operator generated by the problem. \end{abstract} \newtheorem{theorem}{Theorem}[section] \numberwithin{equation}{section} \section{Introduction} The equations of compsite type, as independent mathematical objects, arose first in the works of Hadamard \cite{h1}. Then they were continued by Sjostrand \cite{s1}, and other \cite{b1,d1,e1}. In all these works the equations in question are investigated mainly in the plane and with the model operators in the principal part. In recent years, special equations of composite type have received attention in several papers. Most of the papers were directed to parabolic-elliptic equations, and to hyperbolic-elliptic equations, see for instance \cite {b2,b3,b4}. Motivated by this, we study a boundary-value problem for a class of composite equations of parabolic-hyperbolic type. Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with sufficiently smooth boundary $\partial \Omega$. Points in this space are denoted by $% x=(x_{1},x_{2},\dots ,x_{n})$. In the cylinder $Q=\Omega \times (0,T)$, we consider the boundary-value problem $$\begin{gathered} lu:=( \frac{\partial }{\partial t}-\Delta ) ( \frac{\partial ^{2}u}{\partial t^{2}}-\Delta u) =f(x,t),\quad\text{on }Q, \\ u(x,0)=\frac{\partial u}{\partial t}(x,0) =\frac{\partial ^{2}u}{\partial t^{2}}(x,0)=0,\quad \text{on }\Omega , \\ \frac{\partial u}{\partial \upsilon }=\frac{\partial ^{3}u}{\partial \upsilon ^{3}} =0,\quad\text{on }S \end{gathered} \label{1.1}$$ where $S=\partial \Omega \times (0,T)$, $\upsilon$ is the unit exterior vector, and $\Delta =\sum_{i=1}^{n}\frac{\partial ^{2}}{\partial x_{i}^{2}}$. The aim is to prove existence and uniqueness of a generalized solution to the above equation. The proof is based on an energy inequality and the density of the range of the operator generated by this problem. Analogous to problem \eqref{1.1}, we consider its dual problem. We denote by $l^{\ast }$ the formal dual of the operator $l$, which is defined with respect to the inner product in the space $L_{2}(Q)$ using $$(lu,v)=(u,lv)\quad \mbox{for all }u,v\in C_{0}^{3,4}(Q), \label{1.2}$$ where $(,)$ is the inner product in $L_{2}(Q)$. We consider the dual problem % \eqref{1.3}: $$\begin{gathered} l^{\ast }v:=( -\frac{\partial }{\partial t}-\Delta ) ( \frac{\partial ^{2}v}{\partial t^{2}}-\Delta v) =g(x,t),\quad \text{on }Q,\\ v(x,T)=\frac{\partial v}{\partial t}(x,T)=\frac{\partial ^{2}v}{\partial t^{2}}(x,T)=0,\quad \text{on }\Omega , \\ \frac{\partial v}{\partial \upsilon }=\frac{\partial ^{3}v}{\partial \upsilon ^{3}}=0, \quad\mbox{on } S \end{gathered} \label{1.3}$$ \section{Functional Spaces} The domain $D(l)$ of the operator $l$ is $D(l)=H_{+}^{3,4}(Q)$, the subspace of the Sobolev space $H^{3,4}(Q)$, which consists of all the functions $u\in H^{3,4}(Q)$ satisfying the conditions of \eqref{1.1}. The domain of $l^{\ast }$ is $D(l^{\ast })=H_{-}^{3,4}(Q)$, which consists of functions $v\in H^{3,4}(Q)$ satisfying the conditions of \eqref{1.3}. Let $H_{\sigma }^{2,3}(Q)$ be the Sobolev space \begin{align*} H_{\sigma }^{2,3}(Q)=\Big\{ &u\in H_{0}^{1}(Q): \sigma (t)^{1/2}u_{tt}\in L_{2}(Q),\; \sigma (t)^{1/2}\nabla u_{t}\in L_{2}(Q), \\ &\nabla u_{t}\in L_{2}(Q),\; \sigma (t)\nabla u_{tt}\in L_{2}(Q),\; \sigma(t)\Delta u_{t}\in L_{2}(Q), \\ &\Delta u\in L_{2}(Q),\;\sigma (t)^{1/2}\Delta u_{t}\in L_{2}(Q),\; \sigma (t)^{1/2}\nabla \Delta u\in L_{2}(Q) \Big\}, \end{align*} where $\sigma (t)=(T-t)$. We introduce the function space $H_{0,\sigma }^{2,3}(Q)=\big\{ u\in H_{\sigma }^{2,3}(Q)\text{ satisfying the conditions of }\eqref{1.1}\big\}$. Note that $H_{0,\sigma }^{2,3}(Q)$ is Hilbert space with the inner product: \begin{align*} ( u,v) _{\sigma }=&( u,v) _{1}+(u_{tt},v_{tt}) _{0,\sigma } +( \nabla u_{t},\nabla v_{t})_{0,\sigma } +( \nabla u_{t},\nabla v_{t}) _{0} \\ &+( \Delta u,\Delta v) _{0}+( \Delta u_{t},\Delta v_{t}) _{0,\sigma } +( \nabla \Delta u,\nabla \Delta v)_{0,\sigma } \end{align*} where the symbols $(,) _{0},(,) _{1}$, and $(,)_{0,\sigma }$ denote the inner product in $L_{2}(Q)$, $H^{1}(Q)$, and $L_{2,\sigma }(Q)$ respectively. This space is equipped with the norm \begin{align*} \| u\| _{2,3,\sigma }^{2} =&\int_{Q}[ u^{2}+u_{t}^{2}+|\nabla u| ^{2}] dx\,dt +\int_{Q} [ |\nabla u_{t}|^{2}+( \Delta u) ^{2}] dx\,dt \\ &+\int_{Q}( T-t) [u_{tt}^{2}+| \nabla u_{t}| ^{2}+( \Delta u_{t}) ^{2} +(\nabla \Delta u) ^{2}] dx\,dt. \end{align*} The dual of this space is denoted by $H_{\sigma }^{-2,-3}(Q)$ with respect to the canonical bilinear form $\langle u,v\rangle$ for $u\in H_{0,\sigma }^{2,3}(Q)$ and $v\in H_{\sigma }^{-2,-3}(Q)$, which is the extension by continuity of the bilinear form $(u,v)$, where $u\in L_{2}(Q)$ and $v\in H_{0,\sigma }^{2,3}(Q)$. \paragraph{Definition} The solution of \eqref{1.1} will be seen as a solution of the operational equation $$lu=f,\quad u\in D(l). \label{2.1}$$ The solution of \eqref{1.3} will be seen as a solution of the operational equation $$l^{\ast }v=g,\quad v\in D(l). \label{2.2}$$ To solve the equation (2.1) for every $f\in H_{\sigma }^{-2,-3}(Q)$, we construct, through the bilinear form $v\rightarrow a_{u}(v)=\langle l^{\ast }v,u\rangle$ for all $v\in D(l)$, the extension $L$ of the operator $l$, whose range $R(L)$ coincides with $H_{\sigma }^{-2,-3}(Q)$, meaning that $L$ is invertible. Then we have the fundamental relation $\langle l^{\ast }v,u\rangle = \langle v,Lu\rangle$ for all $u\in D(l)$ and all $\in H_{0,\sigma }^{2,3}(Q)$, which is obtained by analytic form of Hann-Banach's theorem. In the same manner, we construct, through the bilinear form: $u\to a_{v}(u)=\langle v,lu\rangle$ for all $u\in D(l)$, the extension $L^{\ast}$ of the operator $l^{\ast }$. We obtain, \begin{equation*} \langle v,lu\rangle =\langle L^{\ast }v,u\rangle ,\quad \forall u\in H_{0, \sigma}^{2,3}(Q),\forall v\in D(L^{\ast }). \end{equation*} We denote the norm of $Lu$ in $H_{\sigma }^{-2,-3}(Q)$ by $\|Lu\| _{-2,-3,\sigma }$. \paragraph{Definition} The solution of the operational equation \begin{equation*} Lu=f, \quad u\in D(L), \end{equation*} is called generalized solution of \eqref{1.1}, and the solution of the operational equation \begin{equation*} L^{\ast }v=g, \quad v\in D(L^{\ast }), \end{equation*} is called generalized solution of \eqref{1.3}. \section{A priori estimates} \begin{theorem} \label{thm3} For Problem \eqref{1.1}, we have the following a priori estimates: \begin{gather} \| u\| _{2,3,\sigma }^{{}}\leq c\| Lu\| _{-2,-3,\sigma }^{{}},\quad \forall u\in D(L), \label{3.1}\\ \| v\| _{2,3,\sigma }^{{}}\leq c^{\ast}\| L^{\ast }v\| _{-2,-3,\sigma }^{{}}, \quad \forall v\in D(L^{\ast }), \label{3.2} \end{gather} where the positive constants $c$ and $c^{\ast }$ are independent of $u$ and $v$. \end{theorem} \paragraph{Proof.} We first prove the inequality (3.1) for the functions $u\in D(l)$. For $u\in D(l)$ define the operator \begin{equation*} Mu=\Phi (t)u_{tt}-\Phi (t)\Delta u_{t}, \end{equation*} where $\Phi (t)=(t-T)^{2}$. Consider the scalar product $(lu,Mu)_{0}$. Employing integration by parts and taking into account of conditions of % \eqref{1.1}, we see that \begin{aligned} ( lu,(t-T)^{2}u_{tt}) _{0}=&\int_{Q}(T-t)( u_{tt}) ^{2}dxdt+\int_{Q}(T-t)| \nabla u_{t}| ^{2}dx\,dt\\ &+\int_{Q}(T-t)^{2}| \nabla u_{tt}|^{2}dx\,dt +\int_{Q}( \Delta u) ^{2}dx\,dt\\ &-\int_{Q}(T-t)^{2}( \Delta u_{t}) ^{2}dx\,dt \end{aligned} \label{3.3} and \begin{aligned} &( lu,-(t-T)^{2}\Delta u_{t}) _{0}\\ &=-\int_{Q}(T-t)^{2}| \nabla u_{tt}| ^{2}dx\,dt +\int_{Q}| \nabla u_{t}|^{2}dx\,dt+\int_{Q}(T-t)^{2}( \Delta u_{t})^{2}dx\,dt\\ &\quad +\int_{Q}(T-t)( \Delta u_{t}) ^{2}dxdt+\int_{Q}(T-t)^{{}}( \nabla \Delta u) ^{2}dx\,dt\,. \end{aligned} \label{3.4} Hence \begin{aligned} &( lu,(t-T)^{2}u_{tt}-(t-T)^{2}\Delta u_{t})_{0}\\ &=\int_{Q}(T-t)( u_{tt}) ^{2}dx\,dt +\int_{Q}(T-t)| \nabla u_{t}| ^{2}dx\,dt +\int_{Q}( \Delta u) ^{2}dx\,dt\\ &\quad+\int_{Q}| \nabla u_{t}| ^{2}dx\,dt +\int_{Q}(T-t)( \Delta u_{t}) ^{2}dx\,dt +\int_{Q}(T-t)^{{}}( \nabla \Delta u) ^{2}dx\,dt \end{aligned} \label{3.5} For the function $u\in D(l)$, we have the following Poincar\'{e} estimates $$\begin{gathered} \int_{Q}u^{2}dx\,dt\leq4T^{2}\int_{Q}u_{t}^{2}dx\,dt,\quad \forall u\in D(l),\\ \int_{Q}u_{t}^{2}dx\,dt\leq 4T\int_{Q}(T-t)u_{tt}^{2}dx\,dt, \quad \forall u\in D(l) \\ \int_{Q}| \nabla u| ^{2}dx\,dt\leq 4T\int_{Q}(T-t)| \nabla u_{t}|^{2}dx\,dt,\quad \forall u\in D(l). \end{gathered} \label{3.6}$$ We now apply the $\varepsilon$-inequality to the left hand side of (3.5). Using inequalities (3.6), we obtain (3.1). For $u\in D(L)$, we use the regularization operators of Freidrich \cite {a1,g1} to conclude (3.1). This completes the proof. \hfill$\square$ \section{Solvability Problem} \begin{theorem} \label{thm4} For each function $f\in H_{\sigma }^{-2,-3}(Q)$ (resp. $g\in H_{\sigma }^{-2,-3}(Q)$) there exists a unique solution of \eqref{1.1} (resp.\eqref{1.3} ). \end{theorem} \paragraph{Proof.} The uniqueness of the solution follows immediately from inequality (3.1). This inequality also ensures the closure of the range $R(L)$ of the operator $L$. To prove that $\overline{R(L)}$ equals the space $H_{\sigma }^{-2,-3}(Q)$, we obtain the inclusion $\overline{R(L)}\subseteq R(L)$, and $R(L)=H_{\sigma }^{-2,-3}(Q)$. Indeed, let $\{f_{k}\}_{k\in N}$ be a Cauchy sequence in the space $H_{\sigma}^{-2,-3}(Q)$ , which consists of elements of set$R(L)$. Then it corresponds to a sequence $\{u_{k}\}_{k\in N}\subseteq D(L)$ such that: $Lu_{k}=f_{k}$, $k\in \mathbb{N}$. From the inequality (3.1), we conclude that the sequence $\{u_{k}\}$ is also a Cauchy sequence in the space $H_{\sigma }^{-2,-3}(Q$ and converges to an element $u$ in $H_{0,\sigma }^{2,3}(Q)$. It remains to obtain the density of the set$R(L)$ in the space $% H_{\sigma}^{-2,-3}(Q)$ when $u$ belongs to $D(L)$. Therefore, we establish an equivalent result which amounts to proving that $R(L)^{\bot }=\{0\}$. Indeed, let $v\in$ $H_{\sigma }^{-2,-3}(Q)$ be such that $\langle Lu,v\rangle = 0$ for all $u\in D(L)$, that is $\langle l^{\ast }v,u\rangle = 0$ for all $u\in D(L)$. By virtue of the equality $\langle l^{\ast }v,u\rangle = ( v,Lu)$ for all $u\in D(L)$, we have $\langle v,Lu \rangle =0$ for all $u\in D(L)$ and $v\in H_{\sigma }^{-2,-3}(Q)$. From the last equality, by virtue of the estimate (3.2), we conclude that $v=0$ in the space $H_{\sigma }^{-2,-3}(Q)$ when $u$ belongs to $D(L)$. The second part of the theorem can be proved in a similar way by using the operator $M^{\ast }v=t^{2}v_{tt}-t^{2}\Delta v_{t}$. \hfill$\square$ \begin{thebibliography}{99} \bibitem{} \frenchspacing \bibitem{a1} R. A. Adams, Sobolev Spaces, Academic press, (1975). \bibitem{b2} N. Benouar, Probl\{e}mes aux limites pour une classe d' \'{e}quations composites, C. R. Acad. Sci. Paris, t, 319, S\'{e}rie I. pp. 953-958, 1994. \bibitem{b1} P. E. Berkhin, Sur un probl\{e}me aux limites pour une \'{e}quation de type composite, Soviet Math. Dokl., 214, no. 3, 1974, p. 496-498. \bibitem{b3} L. Bougoffa, Existence and uniqueness of solutions for certain non-classical equation, Far East J. Appl. Math. 3(1)(1999), 49-58. \bibitem{b4} A. Bouziani, Probl\{e}mes aux limites pour certaines \'{e}quations de type non classique du troisi\{e}me ordre, Bull. Cla. 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Och Fisik 26A(1)(1937), 1-11. \end{thebibliography} \noindent \textsc{Lazhar Bougoffa} (email: abogafah@kku.edu.sa)\newline \textsc{Mohamed Said Moulay} (email: msmolai@kku.edu.sa)\\[2pt] King Khalid University \newline Department of Mathematics \newline P.O. Box 9004, Abha, Saudi Arabia. \end{document}