\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 18, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2011/18\hfil Existence of solutions] {Existence of solutions for quasilinear parabolic equations with nonlocal boundary conditions} \author[B. Chen\hfil EJDE-2011/18\hfilneg] {Baili Chen} \address{Baili Chen \newline Department of Mathematics and Computer Science\\ Gustavus Adolphus College\\ Saint Peter, MN 56082, USA} \email{bchen@gustavus.edu} \thanks{Submitted April 22, 2010. Published February 3, 2011.} \subjclass[2000]{35K20, 35K59, 35B45, 35D30} \keywords{Faedo-Galerkin method; nonlocal boundary conditions; a priori \hfill\break\indent estimates; quasilinear parabolic equations; generalized solution} \begin{abstract} We prove the existence of a generalized solution a quasilinear parabolic equation with nonlocal boundary conditions, using the Faedo-Galerkin approximation. \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} In this paper, we are concerned with the existence of a generalized solution of the following quasilinear parabolic equation with nonlocal boundary conditions: \begin{gather} \frac{\partial{u}}{\partial{t}} - \sum_{i=1}^{n}\frac{\partial}{\partial x_i}(|u|^{p-2} \frac{\partial{u}}{\partial{x_i}}) + |u|^{p-2}u = f(x,t),\quad x\in\Omega,\; t\in [0,T] \label{e1}\\ u(x,t) = \int_\Omega k(x,y)u(y,t)dy, \quad x\in \Gamma \label{e2}\\ u(x,0) = u_0(x). \label{e3} \end{gather} As a physical motivation, problem \eqref{e1}--\eqref{e3} arises from the study of quasi-static thermoelasticity. The main difficulty of this problem is related to the presence of both quasilinear term in \eqref{e1} and nonlocal boundary condition \eqref{e2}. Literatures to this type of problem are very limited. We only found [4] in which the authors study a quasilinear parabolic equation with nonlocal boundary conditions different from \eqref{e2}. The quasilinear term in \eqref{e1} makes it difficult to apply classical methods like semi-group method or method of upper and lower solutions. However, we found that Faedo-Galerkin method serves as a convenient tool for this type of problem. We proved the existence of a generalized solution of problem \eqref{e1}--\eqref{e3} by constructing approximate solution using Faedo-Galerkin method and applying weak convergence and compactness arguments. It is well known that Faedo-Galerkin method is used to prove the existence of solutions for linear parabolic equations in \cite{Lady}. In \cite{Gerbi}, Faedo-Galerkin method is coupled with contraction mapping theorems to prove the existence of weak solutions of semilinear wave equations with dynamic boundary conditions. Bouziani et al. use Faedo-Galerkin method to show the existence of a unique weak solution for a linear parabolic equation with nonlocal boundary conditions. Lion's book \cite[Chapter 1]{lions}, collects the work of Dubinskii and Raviart, in which they use Faedo-Galerkin method to prove the existence and uniqueness of weak solution for a quasilinear parabolic equation with homogeneous boundary condition. Problem \eqref{e1}--\eqref{e3} is the extension of the problem in \cite[p. 140]{lions} in which the boundary conditions are homegeneous. This article is organized as follows: in section 2, we give the definition of the generalized solution of problem \eqref{e1}--\eqref{e3} and introduce the function spaces related to the generalized solution. In section 3, we demonstrate the construction of an approximation solution by Faedo-Galerkin method and derive a priori estimates for the approximation solution. Section 4 is devoted to the proof of existence of the generalized solution by compactness arguments. \section{Preliminaries} In this article, we use the following notation:\\ $\Omega$: regular and bounded domain of $R^n$; $\Gamma$: boundary of $\Omega$;\\ $(\cdot,\cdot )$: usual inner product in $L^2(\Omega)$;\\ $W^{k,p}(\Omega)$: Sobelev space on $\Omega$; $H^{r}(\Omega)$: Sobelev space $W^{r,2}(\Omega)$;\\ $L^p(\Omega)$: $L^p$ space defined on $\Omega$; $|\cdot |_p$: norm in $L^p(\Omega)$; $|\cdot |_{p,\Gamma}$: norm in $L^p(\Gamma)$;\\ $H^{-r}(\Omega)$: dual space of $H^{r}(\Omega)$; $|\cdot |_{H^{-r}(\Omega)}$: norm in $H^{-r}(\Omega)$;\\ $c$: nonzero constant which may take different values on each occurrence;\\ $C$: nonnegative constant which may take different values on each occurrence;\\ $\hookrightarrow$: continuous embedding;\\ $K(x)$: norm of $k(x,y)$ in $L^q(\Omega)$ with respect to $y$,\\ i.e., $K(x)=(\int_{\Omega}|k(x,y)|^q dy)^{1/q}$;\\ $K_i (x)$: norm of $D_i k(x,y)$ in $L^q(\Omega)$ with respect to $y$,\\ i.e., $K_i(x)=(\int_{\Omega}|\frac{\partial k(x,y)}{\partial x_i}|^q dy)^{1/q}$. In this article, we make the following assumptions: \begin{itemize} \item[(A1)] $n\ge 2$, $p>n$, $r>\frac{n}{2} + 2$; \item[(A2)] $\frac{1}{p} + \frac{1}{q} = 1$; \item[(A3)] $f\in L^q(0,T; L^{q}(\Omega))$ and $u_0 \in L^\infty(\Omega)$; \item[(A4)] For any $x\in \Gamma$, $K(x)<\infty$, $K_i(x)<\infty$; \item[(A5)] $\sum_{i=1}^n \int_\Gamma K(x)^{p-1} K_i(x) d\Gamma < 1-\frac{1}{p}$. \end{itemize} Here we give an example of a function $k(x,y)$ which satisfies assumptions (A4) and (A5): When $n=2$, $p=3$ and $\Omega$ is an unit square, let $k(x,y)=x_1x_2(y_1y_2)^{2/3}$. It is easy to verify that $K(x)=(\int_{\Omega}|k(x,y)|^q dy)^{1/q}$ and $K_i(x)=(\int_{\Omega}|\frac{\partial k(x,y)}{\partial x_i}|^q dy)^{1/q}$ satisfy assumptions (A4) and (A5). With assumption (A1), using Sobelev embedding theorems, see \cite{Adam}, we have $$H^r(\Omega) \hookrightarrow W^{2,p}(\Omega) \hookrightarrow W^{1,p}(\Omega) \hookrightarrow L^p(\Omega) \hookrightarrow L^2(\Omega).$$ Define a space $V$: $$\label{e4} V=\{v\in H^r(\Omega):\ v(x)=\int_\Omega k(x,y)v(y)dy, \ for\ x\in \Gamma\}$$ It is easy to see that $V$ is a subspace of $H^r(\Omega)$. \begin{definition}\label{def2.1} \rm Define a generalized solution of problem \eqref{e1}--\eqref{e3} as a function $u$, such that \begin{itemize} \item[(i)] $u\in L^\infty (0,T;L^2(\Omega))\cap C([0,T],H^{-r}(\Omega))$; \item[(ii)] $\frac{du}{dt}\in L^q(0,T;H^{-r}(\Omega))$; \item[(iii)] $u(x,0)=u_0(x)$; \item[(iv)] for all $v\in V$ and a.e. $t\in [0,T]$, $$\label{e5} (\frac{du}{dt}, v) - (\sum_{i=1}^{n}\frac{\partial}{\partial x_i} (|u|^{p-2}\frac{\partial u}{\partial x_i}) , v) + (|u|^{p-2}u, v) = (f,v)\,.$$ \end{itemize} \end{definition} \begin{remark} \label{rmk2.2} \rm From the proof of existence theorem in section 4, we will see that each inner product in the identity \eqref{e5} is a function of $t$ in $L^q(0,T)$, hence the identity holds for a.e. $t\in [0,T]$. On the other hand, since $u(t)\in V$, the boundary condition \eqref{e2} is satisfied. \end{remark} \section{Construction of an approximate solution and a priori estimates} Since $V$ is a subspace of $H^{r}(\Omega)$, which is separable. We can choose a countable set of distinct basis elements ${w_j},\ j=1,2,\cdots,$ which generate $V$ and are orthonormal in $L^2 (\Omega)$. Let $V_m$ be the subspace of $V$ generated by the first $m$ elements: ${w_1, w_2,\cdots,w_m}$. We construct the approximate solution of the form: $$\label{e6} u_m(x,t) = \sum_{j=1}^{m} g_{jm}(t) w_j(x),\quad (x,t)\in\Omega\times[0,T].$$ where $(g_{jm}(t))_{j=1}^m$ remains to be determined. Denote the orthogonal projection of $u_0$ on $V_m$ as $u_m^0 = P_{V_m} u_0$, then $u_m^0 \to u_0$ in $V$, as $m\to\infty$. Let $(g_{jm}^0)_{j=1}^m$ be the coordinate of $u_m^0$ in the basis $(w_j)_{j=1}^m$ of $V_m$; i.e., $u_m^0 = \sum_{j=1}^m g_{jm}^0 w_j$, let $g_{jm}(0) = g_{jm}^0$. We need to determine $(g_{jm}(t))_{j=1}^m$ to satisfy \begin{equation*} ({u_m}', w_j) - (\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(|u_m|^{p-2}\frac{\partial u_m}{\partial x_i}) , w_j) + (|u_m|^{p-2}u_m, w_j) = (f,w_j),\quad 1\le j\le m. \end{equation*} Integrating by parts on the second term of left-hand side, we have \label{e7} \begin{aligned} &({u_m}', w_j) + \sum_{i=1}^{n}\int_{\Omega} (|u_m|^{p-2} D_i u_m)(D_i w_j) \,dx \\ & - \sum_{i=1}^{n}\int_{\Gamma} (|u_m|^{p-2} D_i u_m) w_j d\Gamma + (|u_m|^{p-2}u_m, w_j) = (f,w_j),\quad 1\le j\le m. \end{aligned} The above system is a system of ordinary differential equations in $(g_{jm}(t))_{j=1}^m$. By Caratheodory theorem \cite{Codd}, there exists solution $(g_{jm}(t))_{j=1}^m,\ t\in [0,t_m)$. We need a priori estimates that permit us to extend the solution to the whole domain $[0,T]$. We derive a priori estimates for the approximate solution as follows: Multiply \eqref{e7} by $g_{jm}(t)$, then sum over $j$ from $1$ to $m$, we have \begin{align*} &({u_m}', u_m) + \sum_{i=1}^{n}\int_{\Omega} (|u_m|^{p-2} D_i u_m)(D_i u_m) dx\\ &- \sum_{i=1}^{n}\int_{\Gamma} (|u_m|^{p-2} D_i u_m) u_m d\Gamma + (|u_m|^{p-2}u_m, u_m) = (f,u_m),\quad 1\le j\le m. \end{align*} which gives \label{e8} \begin{aligned} &\frac{1}{2}\frac{d}{dt} |u_m(t)|_2^2 + \frac{4}{p^2} \sum_{i=1}^n \int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx + |u_m(t)|_p^p \\ &= (f, u_m) + \sum_{i=1}^n\int_\Gamma (|u_m|^{p-2} D_i u_m) u_m d\Gamma. \end{aligned} Integrating with respect to $t$ from $0$ to $T$ on both sides, we obtain \label{e9} \begin{aligned} &\frac{1}{2} |u_m(T)|_2^2 + \int_0^T \frac{4}{p^2} \sum_{i=1}^n \int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx dt + \int_0^T |u_m(t)|_p^p dt \\ &= \int_0^T (f, u_m) dt + \int_0^T \sum_{i=1}^n\int_\Gamma (|u_m|^{p-2} D_i u_m) u_m d\Gamma dt + \frac{1}{2} |u_m(0)|_2^2. \end{aligned} This gives \label{e10} \begin{aligned} &\frac{1}{2} |u_m(T)|_2^2 + \int_0^T \frac{4}{p^2} \sum_{i=1}^n \int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx dt + \int_0^T |u_m(t)|_p^p dt\\ &\le \int_0^T |(f, u_m)| dt + \int_0^T \sum_{i=1}^n\int_\Gamma |(|u_m|^{p-2} D_i u_m) u_m | d\Gamma dt + \frac{1}{2} |u_m(0)|_2^2. \end{aligned} The first term in the right-hand side of \eqref{e10} can be estimated as follows: \label{e11} \begin{aligned} \int_0^T |(f,u_m)| dt &= \int_0^T \int_\Omega |fu_m| dx dt\\ &\le \int_0^T |f|_q |u_m|_p dt \quad \text{(h\"older's\ inequality)} \\ &\le \int_0^T (\frac{1}{p} |u_m|_p^p + \frac{p-1}{p}|f|_q^\frac{p}{p-1}) dt. \quad \text{(Young's inequality)} \end{aligned} Next, we estimate second term in the right-hand side of \eqref{e10}: For $x\in\Gamma$, we have |u_m(x,t)| = \int_\Omega |k(x,y) u_m(y,t)|dy \le |k(x,y)|_q |u_m|_p. \end{equation*} Then we have |u_m(x,t)| \le K(x) |u_m|_p\ for\ x\in \Gamma. Similarly, we have |D_i u_m(x,t)| \le K_i(x) |u_m|_p for x\in \Gamma. Then using h\"older's inequality and assumptions (A4) and (A5), we have \label{e12} \begin{aligned} &\int_0^T \big|\sum_{i=1}^n \int_\Gamma (|u_m|^{p-2} D_i u_m ) u_m d\Gamma \big| dt\\ &\le \int_0^T \sum_{i=1}^n \int_\Gamma K(x)^{p-1} |u_m|_p^{p-1} K_i(x) |u_m|_p d\Gamma dt\\ &\le \int_0^T \Big(\sum_{i=1}^n\int_\Gamma K(x)^{p-1} K_i(x) d\Gamma\Big)|u_m|_p^p dt\\ &=C\int_0^T |u_m|_p^p dt \end{aligned} where \[ C=\sum_{i=1}^n\int_\Gamma K(x)^{p-1} K_i(x) d\Gamma < 1-\frac{1}{p}\,. With the above estimates and \eqref{e10}, we have \begin{align*} &\frac{1}{2} |u_m(T)|_2^2 + \int_0^T \frac{4}{p^2} \sum_{i=1}^n \int_\Omega (D_i(|u_m|^\frac{p-2}{2} u_m))^2 dx dt + \int_0^T (1-\frac{1}{p} - C) |u_m(t)|_p^p dt \\ &\le \int_0^T (\frac{p-1}{p}|f|_q^\frac{p}{p-1}) dt + \frac{1}{2}|u_m(0)|_2^2. \end{align*} which holds for any finite $T>0$. Under assumption (A1)-(A5), we have the following a priori estimates: \begin{itemize} \item[(B)] $u_m$ is bounded in $L^\infty(0, T;\ L^2(\Omega))$; \item[(C)] $|u_m|^\frac{p-2}{2}|u_m|$ is bounded in $L^2(0,T; H^1(\Omega))$; \item[(D)] $u_m$ is bounded in $L^p (0,T; L^p(\Omega))$. \end{itemize} Since $T$ is an arbitrary positive number, we have $\ |u_m|_p^p < \infty \quad \text{a.e. } t$ \section{Existence of a generalized solution} To prove the existence of a generalized solution, we first prove the following lemma: \begin{lemma} \label{lem4.1} Let $u_m$, constructed in \eqref{e6}, be the approximate solution of \eqref{e1}--\eqref{e3} in the sense of Definition \ref{def2.1}. Then $u_m'$ is bounded in $L^q(0,T;\ H^{-r}(\Omega))$. \end{lemma} \begin{proof} For $v\in V\subset H^r$, from \eqref{e7}, we have \label{e13} \begin{aligned} &({u_m}', v) + (\sum_{i=1}^{n}\int_\Omega (|u_m|^{p-2}D_i u_m) (D_i v) dx \\ &- \sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma + (|u_m|^{p-2}u_m,v) = (f,v). \end{aligned} The last term in the left-hand side can be estimated as in \cite{lions}: \begin{align*} |(|u_m|^{p-2}u_m,v) | &\le |\ |u_m|^{p-1}|_q |v|_p\\ &\le (|u_m|_p^p)^{1/q} |v|_p \\ &\le (|u_m|_p^p)^{1/q} C|v|_{H^r}, \end{align*} since $H^r\hookrightarrow L^p$. Hence $|\,|u_m|^{p-2} u_m|_{H^{-r}(\Omega)} \le C(|u_m|_p^p)^{1/q} < \infty$. The norm of $|u_m|^{p-2} u_m$ in $L^q(0,T;H^{-r}(\Omega))$ is bounded by $\Big(\int_0^T (C(|u_m|_p^p)^{1/q})^q dt\Big)^{1/q} = \Big(\int_0^T C^q |u_m|_p^p dt\Big)^{1/q} < \infty.$ Therefore, $|u_m|^{p-2} u_m$ is bounded in $L^q(0,T;H^{-r}(\Omega))$. Next, we consider the term $\sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma$ in the left-hand side of \eqref{e13}: $v \to \sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma = (a(u_m), v).$ We have \begin{align*} &\sum_{i=1}^n \int_\Gamma (|u_m|^{p-2}D_i u_m) v d\Gamma\\ &\le \sum_{i=1}^n |\ (|u_m|^{p-2}D_i u_m)|_{q,\Gamma} |v|_{p,\Gamma} \\ &= \sum_{i=1}^n \Big|\Big(|\int_\Omega k(x,y) u_m(y,t) dy|^{p-2} \int_\Omega D_i k(x,y) u_m(y,t)dy\Big)\Big|_{q,\Gamma} \\ &\quad\times \Big|\int_\Omega k(x,y)v(y,t)dy\Big|_{p,\Gamma}\\ &\le \sum_{i=1}^n |(K(x)^{p-2} K_i (x)|u_m|_p^{p-1})|_{q,\Gamma} |(K(x)|v|_p)|_{p,\Gamma}\\ &\le \sum_{i=1}^n |K(x)^{p-2} K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma} |u_m|_p^{p-1} |v|_p\\ &\le \sum_{i=1}^n |K(x)^{p-2} K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma} |u_m|_p^{p-1} C|v|_{H^r}. \end{align*} Therefore, $|a(u_m)|_{H^{-r}(\Omega)} \le \sum_{i=1}^n |K(x)^{p-2} K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma} |u_m|_p^{p-1} C < \infty.$ Then the norm of $a(u_m)$ in $L^q(0,T; H^{-r}(\Omega))$ is bounded by $\Big(\int_0^T \sum_{i=1}^n (|K(x)^{p-2} K_i (x)|_{q,\Gamma} |K(x)|_{p,\Gamma}C)^q |u_m|_p^p dt\Big)^{1/q} < \infty.$ Hence, $a(u_m)$ is bounded in $L^q(0,T; H^{-r}(\Omega))$. Next, we consider the second term in the left-hand side of \eqref{e13}. Integrating by parts gives \label{e14} \begin{aligned} &\sum_{i=1}^n \int_\Omega (|u_m|^{p-2} D_i u_m) (D_i v) dx \\ &= \frac{1}{c} (\sum_{i=1}^n \int_\Gamma |u_m|^{p-2}u_m D_i v d\Gamma - \int_\Omega |u_m|^{p-2}u_m \Delta v dx). \end{aligned} Consider $v \to \sum_{i=1}^n \int_\Gamma |u|^{p-2} u D_i v d\Gamma= (I_1(u),v)$, we have: \begin{align*} |(I_1(u), v)| &\le \sum_{i=1}^n |\ |u|^{p-2}u|_{q,\Gamma} |D_i v|_{p,\Gamma}\\ &= \sum_{i=1}^n \Big| \Big(\int_\Omega k(x,y) u(y,t) dy \Big)^{p-1} \Big|_{q,\Gamma} \Big|\int_\Omega D_i k(x,y) v(y,t) dy \Big|_{p,\Gamma}\\ &\le \sum_{i=1}^n |(K(x)^{p-1} |u|_p^{p-1})|_{q,\Gamma} |(K_i(x)|v|_p)|_{p,\Gamma}\\ &= \sum_{i=1}^n |K(x)^{p-1}|_{q,\Gamma} |K_i(x)|_{p,\Gamma} |u|_p^{p-1} |v|_p\\ &\le \sum_{i=1}^n |K(x)^{p-1}|_{q,\Gamma} |K_i(x)|_{p,\Gamma} |u|_p^{p-1}C |v|_{H^r} \end{align*} So we have $|I_1(u_m)|_{H^{-r}(\Omega)} \le \sum_{i=1}^n |K(x)^{p-1}|_{q,\Gamma} |K_i(x)|_{p,\Gamma} |u_m|_p^{p-1}C <\infty.$ With this, it is easy to see that norm of $I_1(u_m)$ in $L^q(0,T; H^{-r}(\Omega))$ is bounded. Next, consider $v \to \int_\Omega |u|^{p-2}u \Delta v dx = (I_2(u), v)$. From the proof of \cite[Theorem 12.2]{lions}, we know $I_2(u_m)$ is bounded in $L^q(0,T; H^{-r}(\Omega))$. Since $f\in L^q(0,T; L^{q}(\Omega))\subset L^q(0,T; H^{-r}(\Omega))$, from \eqref{e13} and the above discussion, we have $u_m'$ is bounded in $L^q(0,T; H^{-r}(\Omega))$. \end{proof} With Lemma \ref{lem4.1}, we can use \cite[Theorem 12.1]{lions}. We quote the theorem here. \begin{theorem} \label{thm4.2} Let $B,B_1$ be Banach spaces, and $S$ be a set. Define $M(v) = (\sum_{i=1}^n\int_\Omega |v|^{p-2}(\frac{\partial v} {\partial x_i})^2 dx)^{1/p}$ on $S$ with: \begin{itemize} \item[(a)] $S\subset B\subset B_1$, and $M(v)\ge 0$ on $S$, $M(\lambda v)=|\lambda| M(v)$; \item[(b)] the set $\{v|v\in S, M(v)\le 1\}$ is relatively compact in $B$. \end{itemize} Define the set $F=\{v: v$ is locally summable on $[0,T]$ with value in $B_1$,\\ $\int_0^T (M(v(t)))^{p_0} dt \le C$, $v'$ bounded in $L^{p_1}(0,T; B_1)\}$. Where \$1