\documentclass[reqno]{amsart} \usepackage{eepic} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 31, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2008 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2008/31\hfil Parabolic boundary-value problems] {Parabolic boundary-value problems with equivalued surface on the domain \\ with a thin layer} \author[M. Du, F. Li\hfil EJDE-2008/31\hfilneg] {Meihua Du, Fengquan Li} % in alphabetical order \address{Meihua Du \newline Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China} \email{meihua0815@126.com} \address{Fengquan Li \newline Department of Applied Mathematics, Dalian University of Technology, Dalian, 116024, China} \email{fqli@dlut.edu.cn} \thanks{Submitted August 29, 2007. Published March 6, 2008.} \thanks{Partially supported by grant 10401009 from NSFC, and by NCET of China} \subjclass[2000]{35A05, 35B40, 35K20} \keywords{Limit behavior of solutions; existence; uniqueness; equivalued surface; \hfill\break\indent equivalued interface} \begin{abstract} We study the existence, uniqueness and limit behavior of solutions to parabolic boundary-value problems with equivalued surface on a domain with a thin layer. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Motivated by the study of resistivity well-logging in petroleum exploitation, the boundary value problem with equivalued surface, a new kind of boundary value problem for partial differential equations was proposed in 1970's. It is a kind of non-local boundary value problem, which can also be used to give mathematical descriptions for other problems in physics and mechanics (see \cite{l1,l2,l4,s1}). In single-well system of heterogeneous synthetic reservoirs, for the cause of mud contamination in the process of well drilling and well completion, a polluted zone is formed. However, this zone is a thin layer compared with the whole heterogeneous reservoirs (see \cite{c2,j1,k1}). In practical calculation, the variation of solution near the thin layer should be quite large, and then in finite element procedure, it is necessary to have a refined partition of elements near the thin layer. This causes a complexity in computation. To get rid of this difficulty, when the thin layer is rather thin, the thin layer can be approximately regarded as an interface and corresponding the boundary value problem with equivalued surface on the thin layer can be approximately replaced by the boundary value problem with equivalued interface. To prove the above conclusion, we need to study existence, uniqueness and limit behavior of solutions for parabolic boundary value problems with equivalued surface on a domain with thin layer. Similar problem for elliptic equation has been studied in \cite{l3}. Here we consider the following parabolic boundary value problems with equivalued surface on the domain with thin layer: $$\label{eP} \begin{gathered} \frac{\partial u}{\partial t}-\sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i}}(\tilde{a}_{ij}(x,t)\frac{\partial u }{\partial x_{j}})=F(x,t)\quad \text{in } \tilde{Q},\\ u=0 \quad \text{on } \Sigma,\\ u =\tilde{C}(t)\quad \text{on } \tilde{\Sigma},\\ \int_{\tilde{\Gamma}_{1}}\frac{\partial u}{\partial n_{L}}\mathrm{d}s =\int_{\tilde{\Gamma}_{2}}\frac{\partial u}{\partial n_{L}}\mathrm{d}s+A(t) \quad \text{a.e. t } \in (0,T),\\ u(x,0)=\tilde{\varphi}_{0}(x) \quad \text{ in } \tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}, \end{gathered}$$ where $\tilde{Q}=(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})\times(0,T)$, $\Sigma=\Gamma\times(0,T)$, $\tilde{C}(t)$ is a function to be determined, $\tilde{\Sigma}=(\tilde{\Gamma}_{1}\cup\tilde{\Omega} \cup\tilde{\Gamma}_{2})\times(0,T)$, $T$ is a fixed positive constant, and the conormal derivative is $$\frac{\partial u}{\partial n_{L}}=\sum_{i,j=1}^{N}\tilde{a}_{ij}(x,t)\frac{\partial u}{\partial x_{j}}n_{i}\,.$$ %{tt1.eps} \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(100,50)(0,0) \put(50,25){\ellipse{100}{50}} \put(50,25){\ellipse{70}{35}} \put(50,25){\ellipse{40}{20}} \put(7,24){$\widetilde{\Omega}_1$} \put(22.5,24){$\widetilde{\Omega}$} \put(48,24){$\widetilde{\Omega}_2$} \put(65,23){$\widetilde{\Gamma}_2$} \put(80.5,23){$\widetilde{\Gamma}_1$} \put(97,23){$\Gamma$} \end{picture} \end{center} \caption{The compostion of $\Omega$ with thin layer $\widetilde{\Omega}$} \end{figure} Let $\Omega\subset \mathbb{R}^{N}(N\geq2)$ be a bounded domain with smooth outside boundary $\Gamma$ (see Fig.1). Suppose that $\Omega$ is composed of three non-overlapping subdomains $\tilde{\Omega}_{1}$, $\tilde\Omega$ and $\tilde{\Omega}_{2}$, and $\tilde\Gamma_{1}$ and $\tilde\Gamma_{2}$ are the interfaces of $\tilde{\Omega}$ with $\tilde{\Omega}_{1}$ and $\tilde{\Omega}_{2}$ respectively. The unit normal $\vec{n}=(n_{1},n_{2},\dots,n_{N})$ takes the inward and outward directions (or vice versa) for the domain $\tilde{\Omega}$ on $\tilde{\Gamma}_{1}$ and $\tilde{\Gamma}_{2}$. This paper is organized as follows: In section 2, we will prove the existence and uniqueness of weak solution to problem \eqref{eP}. In section 3, we will discuss parabolic boundary value problem \eqref{eP'} with equivalued interface. In section 4, the limit behavior of solutions to problems \eqref{eP} will be studied. \section{Existence and uniqueness of weak solution to problem \eqref{eP}} In this section, we discuss the existence and uniqueness of weak solution to problem \eqref{eP}. We first state the following assumption: \begin{itemize} \item[(H0)] The functions $\tilde{a}_{ij}$ are piecewise smooth in $Q$ and $\tilde{a}_{ij}=\tilde{a}_{ji}$; there exist two constants $\alpha,\beta>0$ such that $$\alpha|\xi|^{2}\leq\sum_{i,j=1}^{N}\tilde {a}_{ij}(x,t)\xi_{i}\xi_{j}\leq\beta|\xi|^{2},\quad \forall\,\xi=(\xi_{1},\xi_{2},\dots, \xi_{N})\in \mathbb{R}^N,$$ a.e. $(x,t)\in Q$, where $Q=\Omega\times(0,T)$. We also assume $F\in L^{2}(Q)$, $A\in L^{2}(0,T)$, $\tilde{\varphi} _{0}\in L^{2}(\Omega)$. \end{itemize} Let \begin{gather*} V=\{v\in H_{0}^{1}(\Omega):v|_{\tilde{\Gamma}_{1}\cup\tilde{\Omega} \cup\tilde{\Gamma}_{2}}=\mathrm{constant}\}, \\ U=\{v\in{\mathring{W}_{2}^{1,1}(Q):v(x,T)=0,v|_{{ \tilde{\Sigma}}}=C(t)}\}, \end{gather*} where $C(t)$ is arbitrary function of $t$. \begin{definition} \label{def2.1} \rm A measurable function $u\in L^{2}(0,T; V)$ is called a weak solution to problem \eqref{eP}, if for any $\psi\in U$, \label{e2.4} \begin{aligned} &-\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}u\frac{\partial \psi}{\partial t} \mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\tilde{a}_{ij}\frac{\partial u}{\partial x_{j}}\frac{\partial \psi}{\partial x_{i}}\mathrm{d}x\mathrm{d}t\\ &=\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} F\psi\mathrm{d}x\mathrm{d}t +\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\tilde{\varphi}_{0}(x) \psi(x,0)\mathrm{d}x +\int_{0}^{T}A(t)\psi|_{{\tilde{\Sigma}}}\mathrm{d}t. \end{aligned} \end{definition} Now we can state the existence and uniqueness of weak solutions to \eqref{eP}. \begin{theorem} \label{thm2.2} Assume {\rm (H0)}, $\tilde\varphi_{0}\in L^{2}(\Omega)$, $F\in L^{2}(Q)$, $A\in L^{2}(0,T)$. Then \eqref{eP} admits a unique solution $u\in L^{2}(0,T; V)$ in the sense of Definition \eqref{def2.1}. \end{theorem} \begin{proof} (1) Existence: We will first consider the problem: $$\label{eP0} \begin{gathered} \frac{{\partial \tilde{u}}}{\partial t}-\sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i}}(\tilde{a}_{ij}(x,t)\frac{\partial \tilde{u}}{\partial x_{j}})=F(x,t)\quad \text{in } \tilde{Q},\\ \tilde{u}=0 \quad \text{ on }\Sigma,\\ \tilde{u} =\tilde{C}(t) \quad \text{on } (\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2})\times(0,T),\\ \int_{\tilde{\Gamma}_{1}}\frac{\partial \tilde{u}}{\partial n_{L}}\mathrm{d}s=\int_{\tilde{\Gamma}_{2}}\frac{\partial \tilde{u}}{\partial n_{L}}\mathrm{d}s+A(t)\quad \text{a.e. t }\in(0,T),\\ \tilde{u}(x,0)=\tilde{\varphi}_{0}(x)\quad \text{in }\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}. \end{gathered}$$ Let $$\label{e2.5} V_{1}=\{v\in H^{1}(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}): v|_{\Gamma}=0,\; v|_{\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2} } =\mathrm{constant} \}.$$ Here we will use the Galerkin method (cf \cite{e1,z1}). Taking a basis $\{\omega_{k}\}_{k=1}^{\infty}$ of $V_{1}$ that is complete and orthonormal in $L^{2}(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})$. For any fixed $m$, let $S_{m}=\mathop{\rm span}\{\omega_{1}, \omega_{2}, \dots, \omega_{m}\}$. We set $\tilde{u}_{m}=\sum_{k=1}^{m}c_{km}\omega_{k}$, then Galerkin equations read as follows: \begin{gather} \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\frac{\partial \tilde{u}_{m}}{\partial t}\omega_{k}\mathrm{d}x+\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\sum_{i,j=1}^{N}\tilde{a}_{ij} \frac{\partial \omega_{k}}{\partial x_{i}}\frac{\partial \tilde{u}_{m}}{\partial x_{j}}\mathrm{d}x=\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}F\omega_{k}\mathrm{d}x+A(t)\omega_{k}| _{\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2}}, \label{e2.6} \\ \tilde{u}_{m}(x,0)=\sum_{k=1}^{m}(\tilde{\varphi}_{0},\omega_{k})\omega_{k}=\sum_{k=1}^{m}c_{k0}\omega_{k} =\tilde{\varphi}_{0m}(x).\label{e2.7} \end{gather} Namely, for almost all $t\in (0, T)$, \begin{gather} \frac{d}{dt}c_{km}(t)+\sum_{l=1}^{m}c_{lm}(t) \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial \omega_{l}}{\partial x_{j}}\frac{\partial \omega_{k}}{\partial x_{i}}\mathrm{d}x=\int_{\tilde{\Omega}_{1}\cup \tilde{\Omega}_{2}}F\omega_{k}\mathrm{d}x+A(t)\omega_{k}| _{\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2}},\label{e2.8} \\ c_{km}(0)=(\tilde{\varphi}_{0},\omega_{k})=c_{k0}.\label{e2.9} \end{gather} By the theory of system of ordinary differential equations, problem \eqref{e2.8}--\eqref{e2.9} has a unique solution vector $(c_{1m},c_{2m},\dots,c_{mm})$. Multiplying \eqref{e2.8} by $c_{km}(t)$ and summing over $k$, we obtain \begin{align*} &\frac{1}{2}\frac{d}{dt}\|\tilde{u}_{m}(\cdot,t) \|_{L^{2}(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})}^{2}+\int_ {\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial \tilde{u}_{m}}{\partial x_{j}}\frac{\partial \tilde{u}_{m}}{\partial x_{i}}\,\mathrm{d}x\\ &=\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}F\tilde{u}_{m} \mathrm{d}x+A(t)\tilde{u}_{m}|_{{\tilde{\Gamma}_{1} \cup\tilde{\Gamma}_{2}}} . \end{align*} Integrating over $(0,\tau)$ with respect to $t$, we have \begin{align*} &\frac{1}{2}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{u}_{m}^{2}(x,\tau)\,\mathrm{d}x+\int_{0}^{\tau} \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial \tilde{u}_{m}}{\partial x_{j}}\frac{\partial \tilde{u}_{m}}{\partial x_{i}}\,\mathrm{d}x\mathrm{d}t\\ &=\int_{0}^{\tau}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} F\tilde{u}_{m}\,\mathrm{d}x\mathrm{d}t+\int_{0}^{\tau}A(t)\tilde{u}_{m}| _{{\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2}}}\,\mathrm{d}t +\frac{1}{2}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{\varphi}_{{0m}}^{2}\,\mathrm{d}x. \end{align*} Let $\tilde{Q}_{\tau}=(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})\times(0,\tau)$, by (H0), H\"{o}lder's inequality and the trace theorem, we have \begin{align*} &\frac{1}{2} \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{u}_{m}^{2}(x,\tau)\,\mathrm{d}x +\alpha\|D\tilde{u}{_{m}}\|_{{L^{2}(\tilde{Q}_{\tau})}}^{2}\\ &\leq\|F\|_{{L^{2}(\tilde{Q}_{\tau})}}\|\tilde{u}_{m} \|_{{L^{2}(\tilde{Q}_{\tau})}}+\frac{1}{2}\|\tilde{\varphi}_{0}\| ^{2}_{L^{2}(\Omega)}+\frac{1}{|\tilde{\Gamma}_{2}|^{1/2}} \int_{0}^{\tau}|A(t)|\Big(\int_{\tilde {\Gamma}_{2}}\tilde{u}_{m}^{2}\,\mathrm{d}s\Big)^{1/2}\mathrm{d}t\\ &\leq\|F\|_{{L^{2}(\tilde{Q}_{T})}}\|\tilde{u}_{m} \|_{{L^{2}(\tilde{Q}_{\tau})}}+\frac{1}{2}\|\tilde{\varphi}_{0}\| ^{2}_{{L^{2}(\Omega)}}+\frac{C}{|\tilde{\Gamma}_{2}|^{1/2}} \|A(t)\|_{{L^{2}(0,T)}}(\|\tilde{u}_{m}\|_{{L^{2}(\tilde{Q}_{\tau})}} \\ &\quad +\|D\tilde{u}_{m} \|_{{L^{2}(\tilde{Q}_{\tau})}})\\ &\leq\Big(\|F\|_{{L^{2}(\tilde{Q}_{T})}}+\frac{C}{|\tilde{\Gamma}_{2}|^{1/2}} \|A(t)\|_{{L^{2}(0,T)}}\Big)\|\tilde{u}_{m}\|_{{L^{2}(\tilde{Q}_{\tau})}}+\frac{1}{2}\|\tilde{\varphi}_{0}\| ^{2}_{{L^{2}(\Omega)}} \\ &\quad +\frac{C}{|\tilde{\Gamma}_{2}|^{1/2}} \|A(t)\|_{{L^{2}(0,T)}}\|D\tilde{u}_{m}\|_{{L^{2}(\tilde{Q}_{\tau})}} . \end{align*} By Young's inequality and Gronwall's inequality, we obtain \begin{gather} \|\tilde{u}_{m}\|_{{L^{2}(0,T; V_{1})}}\leq C, \label{e2.13}\\ \|\tilde{u}_{m}\|_{{L^{\infty}(0,T; L^{2}(\tilde{\Omega}_{1} \cup\tilde{\Omega}_{2}))}}\leq C,\label{e2.14} \end{gather} where $C$ is a positive constant independent of $m$. Integrating \eqref{e2.6} over $(t, t+\Delta t)$, we get \begin{align*} &c_{km}(t+\Delta t)-c_{km}(t)+\int_{t}^{t+\Delta t} \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial \tilde{u}_{m}}{\partial x_{j}}\frac{\partial {\omega}_{{k}}}{\partial x_{i}}\,\mathrm{d}x\mathrm{d}\tau\\ &=\int_{t}^{t+\Delta t}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} F{\omega}_{{k}}\,\mathrm{d}x\mathrm{d}\tau+\int_{0}^{\tau}A(\tau){\omega}_{{k}}| _{{\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2}}}\,\mathrm{d}\tau. \end{align*} Condition (H0), \eqref{e2.13} and trace theorem imply $$|c_{km}(t+\Delta t)-c_{km}(t)|\leq C(1+||F||_{L^2(Q)}+||A||_{L^2(0, T)})||\omega_k||_{V_1}|\Delta t|^\frac{1}{2},$$ where $C$ is a positive constant independent of $m, k$. From the above inequality, we can deduce that for any fixed positive integer $k$, $c_{km}$ is equicontinuous with respect to $m$ in $[0,T]$. Thus by Ascoli-Arzela theorem, we can extract a subsequence of $\{c_{km}\}$ (still denoted by $\{c_{km}\}$) such that as $m\to \infty$, $$\label{e2.17} c_{km}\to c_{k} \quad \text{uniformly in } [0,T].$$ For any positive integer $r\leq m$, \eqref{e2.14} yields $$\label{e2.18} \sum_{k=1}^r c_{km}^2(t)\leq C, \quad \forall t\in (0, T),$$ where $C$ is a positive constant independent of $m, k, t$. Let $m\to \infty$ in \eqref{e2.18}, then for any positive integer $r$ we have $$\sum_{k=1}^r c_{k}^2(t)\leq C.$$ Let $\tilde{u}(x, t)=\sum_{k=1}^{\infty}c_k(t)\omega_{k}(x)$, (2.14) imply that $\tilde{u}(x, t)\in L^2(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})$ for all $t\in [0, T]$. For any fixed positive integer $k$, from \eqref{e2.17} it follows that $$\label{e2.20} (\tilde{u}_{m}(\cdot, t)-\tilde{u}(\cdot, t), \omega_{k})=c_{km}-c_{k}\to 0, \quad \text{uniformly in } [0,T].$$ Noting $\{\omega_{k}\}_{k=1}^{\infty}$ is a complete orthonormal basis in $L^{2}(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})$, we deduce $$\label{e2.21} \tilde{u}_{m} \to \tilde{u} \quad \text{weakly in } C([0,T]; L^2(\tilde{\Omega}_1\cup\tilde{\Omega}_2)).$$ Thus \eqref{e2.13} and \eqref{e2.21} imply that $$\label{e2.22} \tilde{u}_{m} \to\tilde{u} \quad \text{weakly in } L^{2}(0,T; V_{1}).$$ Convergence \eqref{e2.21} yields $$\label{e2.23} u_{_m}(\cdot,0)\to \tilde{u}(\cdot,0) \quad \text{weakly in } L^{2}(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}).$$ Consequently $\tilde{u}(0)=\tilde{\varphi}_{0}$. For any given a sequence of smooth function $\{\psi_k(t)\}_{k=1}^\infty$ defined in [0, T] with $\psi_k(T)=0$, multiplying the Galerkin equation \eqref{e2.6} by $\psi_k(t)$ and using integration by parts, we obtain \label{e2.24} \begin{aligned} &-\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{u}_m\frac{\partial \psi_k}{\partial t} \omega_{k}\mathrm{d}x\mathrm{d}t+\int_{0}^{T} \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial\omega_{k}}{\partial x_{i}}\frac{\partial\tilde{u}_m}{\partial x_{j}}\psi_k\mathrm{d}x\mathrm{d}t \\ &=\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} F\psi_k\omega_{k}\mathrm{d}x\mathrm{d}t+ \int_{0}^{T}A\psi\omega_{k}|_{{\tilde{\Gamma}_{1} \cup\tilde{\Gamma}_{2}}}\mathrm{d}t +\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{\varphi}_{0m}(x)\psi(0)\omega_{k}\mathrm{d}x. \end{aligned} According to \eqref{e2.22}-\eqref{e2.23}, let $m\to\infty$ in \eqref{e2.24}, it is easy to prove that \label{e2.25} \begin{aligned} &-\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{u}\frac{\partial \psi_k}{\partial t}\omega_{k}\mathrm{d}x \mathrm{d}t+\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup \tilde{\Omega}_{2}}\sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial\omega_{k}}{\partial x_{i}}\frac{\partial\tilde{u}}{\partial x_{j}}\psi_k\mathrm{d}x\mathrm{d}t \\ &= \int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} F\psi_k\omega_{k}\mathrm{d}x\mathrm{d}t+ \int_{0}^{T}A\psi\omega_{k}|_{{\tilde{\Gamma}_{1}\cup \tilde{\Gamma}_{2}}}\mathrm{d}t +\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\tilde{\varphi}_{0}(x) \psi(0)\omega_{k}\mathrm{d}x. \end{aligned} For any positive integer $r$, let $$\label{e2.26} \psi(x, t)=\sum_{k=1}^r\psi_k(t)\omega_k(x).$$ Replacing $\psi_k(t)\omega_k(x)$ by the above $\psi(x, t)$ in \eqref{e2.25}, we have \label{e2.27} \begin{aligned} &-\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{u}\frac{\partial \psi}{\partial t}\mathrm{d}x\mathrm{d}t+\int_{0}^{T} \int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}\sum_{i,j=1}^N\tilde{a}_{ij} \frac{\partial\psi}{\partial x_{i}}\frac{\partial\tilde{u}}{\partial x_{j}}\mathrm{d}x\mathrm{d}t\\ &=\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} F\psi\mathrm{d}x\mathrm{d}t+ \int_{0}^{T}A\psi|_{{\tilde{\Gamma}_{1}\cup\tilde{\Gamma}_{2}}}\mathrm{d}t +\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{\varphi}_{0}(x)\psi(x,0)\mathrm{d}x. \end{aligned} Since the set composed of functions such as \eqref{e2.26} is dense in the space $U$, then for any $\psi\in U$ \eqref{e2.27} holds. Thus we obtain $\tilde{u}$ is the weak solution to problem \eqref{eP0}. Let $$u=\begin{cases} \tilde{u} & \text{in }\tilde{Q}\, ,\\ \tilde{C}(t) & \text{in }\tilde{\Sigma}\, . \end{cases}$$ It is easy to verify that $u\in L^{2}(0,T;\ V)$ and satisfy \eqref{e2.4}. Thus we obtain $u$ is the weak solution to problem \eqref{eP}. \smallskip \noindent(2) Proof of uniqueness: If $u_{1}$ and $u_{2}$ are two weak solutions to problem \eqref{eP}, by Definition 2.1 we get \begin{align*} &-\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}u_{i} \frac{\partial\psi}{\partial t} dx dt+\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial\psi}{\partial x_i} \,dx\,dt\\ &=\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}F\psi dx\,dt +\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} \tilde{\varphi}_{0}(x)\psi(x,0) dx+ \int_{0}^{T}A(t)\psi|_{{\tilde{\Sigma}}} dt, \quad i=1,2. \end{align*} Let $u_{0}=u_{1}-u_{2}$, then the above equality yields $$\label{e2.30} -\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}} u_{0}\frac{\partial \psi}{\partial t}\mathrm{d}x\mathrm{d}t +\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial u_{0}}{\partial x_{j}}\frac{\partial\psi}{\partial x_i}\mathrm{d}x\mathrm{d}t=0.$$ For a given $0(T-h)\,\\, \varphi & 00$ such that $\alpha|\xi|^{2}\leq\sum_{i,j=1}^{N}a_{ij}(x,t)\xi_{i}\xi_{j} \leq\beta|\xi|^{2}\,,\quad \forall\xi=(\xi_{1},\xi_{2},\dots, \xi_{N}) \in \mathbb{R}^N, \; \text{a.e. } (x,t)\in Q.$ \end{itemize} Let \begin{gather}\label{e3.2} V_{0}=\big\{v\in H_{0}^{1}(\Omega):v|_{{\tilde{\Gamma}}} =\mathrm{constant}\big\}, \\ U_{0}=\big\{v\in\mathring{W}_{2}^{1,1}(Q):v|_{{\tilde{\Sigma}_{0}}} =C(t),\; v(x,T)=0\big\}.\label{e3.3} \end{gather} \begin{definition} \label{def3.1} \rm A measurable function $u\in L^{2}(0,T;V_{0})$ is called a weak solution to problem \eqref{eP'}, if for any $\psi\in U_{0}$, \label{e3.4} \begin{aligned} &-\int_{0}^{T}\int_{\Omega}u\frac{\partial \psi}{\partial t}\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij}\frac{\partial \psi}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}\mathrm{d}x\mathrm{d}t \\ &=\int_{0}^{T}\int_{\Omega}Fu\mathrm{d}x\mathrm{d}t+\int_{\Omega}\varphi_{0}(x)\psi(x,0)\mathrm{d}x +\int_{0}^{T}A(t)\psi|_{{\tilde{\Sigma}_{0}}}\mathrm{d}t. \end{aligned} \end{definition} Now we state the main result of this section as follows: \begin{theorem}\label{thm3.2} Suppose that $\varphi_{0}\in L^{2}(\Omega), F\in L^{2}(Q)$, $A\in L^{2}(0,T)$ and {\rm (H1)} hold. Then there exists a unique weak solution $u\in L^{2}(0,T; V_{0})$ to \eqref{eP'}. \end{theorem} The proof of this theorem is similar to Theorem \ref{thm2.2}, so we omit it. \section{Limiting behavior of solutions to problem \eqref{eP}} Let $\varepsilon>0$ be a small parameter and replace $\tilde{\Omega}_{1},\tilde{\Omega}_{2},\tilde{\Omega}$ by $\Omega_{1}^{\varepsilon},\Omega_{2}^{\varepsilon}, \tilde{\Omega}^{\varepsilon}$, also interface $\tilde{\Gamma}_{1}$ and $\tilde{\Gamma}_{2}$ by the interface $\tilde{\Gamma}_{1}^{\varepsilon}$ and $\tilde{\Gamma}_{2}^{\varepsilon}$, respectively as shown in Figure 3. Let $\tilde{Q}^{\varepsilon}=(\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon})\times(0,T), \tilde{\Sigma}_{\varepsilon}=(\tilde{\Gamma}_{1}^{\varepsilon}\cup\tilde{\Omega}^{\varepsilon}\cup \tilde{\Gamma}_{2}^{\varepsilon})\times(0,T)$. %{tt3.eps} \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(100,50)(0,0) \put(50,25){\ellipse{100}{50}} \put(50,25){\ellipse{70}{35}} \put(50,25){\ellipse{40}{20}} \put(7,24){$\Omega_1^\epsilon$} \put(22.5,24){$\widetilde{\Omega}^\epsilon$} \put(48,24){$\Omega_2^\epsilon$} \put(65,23){$\widetilde{\Gamma}_2^\epsilon$} \put(80.5,23){$\widetilde{\Gamma}_1^\epsilon$} \put(97,23){$\Gamma$} \end{picture} \end{center} \caption{The compostion of $\Omega$ described by parameter $\varepsilon$} \end{figure} Here we will discuss the problem $$\label{ePe'} \begin{gathered} \frac{\partial u_{\varepsilon}}{\partial t}-\sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i}}(a_{ij}^\varepsilon(x,t)\frac{\partial u_{\varepsilon}}{\partial x_{j}})=F(x,t)\quad \text{ in } \tilde{Q}^{\varepsilon}\\ u_{\varepsilon}=0 \quad \text{on } \Sigma,\\ u_{\varepsilon} =\tilde{C}_{\varepsilon}(t)\quad \text{on } \tilde{\Sigma}_{\varepsilon},\\ \int_{\tilde{\Gamma}_{1}^{\varepsilon}}\frac{\partial u_{\varepsilon}}{\partial n_{L^{\varepsilon}}}\mathrm{d}s=\int_{\tilde{\Gamma}_{2}^{\varepsilon}}\frac{\partial u_{\varepsilon}}{\partial n_{L^{\varepsilon}}}\mathrm{d}s+A(t) \quad \text{ a.e. t } \in (0,T),\\ u_{\varepsilon}(x.0)=\varphi_{0\varepsilon}(x) \quad \text{in } \Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}. \end{gathered}$$ We state the following assumptions: \begin{itemize} \item[(H2)] $\tilde{\Gamma}\subset\tilde{\Omega}^{\varepsilon}$, for all $\varepsilon>0$; $\tilde{\Omega}^{\varepsilon}$ shrinks to $\tilde{\Gamma}$, as $\varepsilon\to 0$. \item[(H3)] For any given domain $\tilde{\Omega}$ such that $\tilde{\Gamma}\subset\tilde{\Omega}\subset\Omega$, then for any $\varepsilon>0$ small enough, we have $\tilde{\Omega}_{\varepsilon}\subset\tilde{\Omega}$. \item[(H4)] $a_{ij}^{\varepsilon}$ are piecewise smooth functions in $Q$, $a_{ij}^{\varepsilon}=a_{ji}^{\varepsilon}$, and there exist two constants $\alpha,\beta>0$ independent of $\varepsilon$ such that $\alpha|\xi|^{2}\leq\sum _{i,j=1}^{N}a_{ij}^{\varepsilon}(x,t)\xi_{i}\xi_{i}\leq\beta|\xi|^{2}, \quad \forall\ \xi=(\xi_{1},\xi_{2},\dots,\xi_{N})\in \mathbb{R}^{N},$ a.e. $(x,t)\in Q$. \item[(H5)] For any given domain $\tilde{\Omega}$ such that $\tilde{\Gamma}\subset\tilde{\Omega}\subset\Omega$, then $%4.2 a_{ij}^{\varepsilon}(x,t) \to a_{ij}(x,t)\quad\text{strongly in } L^{\infty} (0,T; L^{\infty}(\Omega\backslash\tilde{\Omega})).$ Set \begin{gather*} V_{\varepsilon}=\{v\in H_{0}^{1}(\Omega): v|_{\tilde\Gamma_{1}^{\varepsilon}\cup\tilde\Omega^{\varepsilon}\cup \tilde\Gamma_{2}^{\varepsilon}}=\mathrm{constant}\}, \\ U_{\varepsilon}=\{v\in{\mathring{W}_{2}^{1,1}(Q):v(x,T)=0,\; v|_{{\tilde{\Sigma}_{\varepsilon}}}=C(t)}\}, \end{gather*} where $C(t)$ is arbitrary function of $t$. \end{itemize} \begin{definition} \label{def4.1} \rm A measurable function $u_{\varepsilon}\in L^{2}(0,T; V_{\varepsilon})$ is a weak solution to problem \eqref{ePe'}, if for any $\psi\in U_{\varepsilon}$, \label{e4.5} \begin{aligned} &-\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}} u_{\varepsilon}\frac{\partial \psi}{\partial t} \mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij} ^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\frac{\partial \psi}{\partial x_{i}}\mathrm{d}x\mathrm{d}t \\ &=\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}} F\psi\mathrm{d}x\mathrm{d}t +\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}} \varphi_{0\varepsilon}(x)\psi(x,0)\mathrm{d}x +\int_{0}^{T}A(t)\psi|_{{\tilde{\Sigma}_{\varepsilon}}}\mathrm{d}t. \end{aligned} \end{definition} \begin{remark}\label{rm4.2} \rm For every fixed $\varepsilon >0$, if (H4) and $\varphi_{0\varepsilon}\in L^{2}(\Omega)$, $F\in L^{2}(Q)$, $A\in L^{2}(0,T)$ hold, we can similarly prove that \eqref{ePe'} admits a unique weak solution $u_\varepsilon\in L^{2}(0,T; V_{\varepsilon})$ in the sense of Definition \ref{def4.1}. \end{remark} To prove the main result in this section, we need the following Lemma. \begin{lemma}\label{lem4.3} Under hypothesis {\rm (H2)--(H3)}, for any given $\psi\in U_{0}$, there exist $\psi_{\varepsilon}\in U_{\varepsilon}$ such that as $\varepsilon\to 0$, $$\label{e4.6} \psi_{\varepsilon}\to\psi\quad \text{strongly in }U_{0},$$ where $U_{0}$ can be seen in \eqref{e3.3}. \end{lemma} \begin{proof} For convenience, we assume the origin is the interior point of $\Omega_{2}$ (see Figure 2). %{tt4.eps} \begin{figure}[ht] \begin{center} \setlength{\unitlength}{1mm} \begin{picture}(100,50)(0,0) \put(50,25){\ellipse{100}{50}} \put(50,25){\ellipse{80}{40}} \put(50,25){\ellipse{60}{30}} \put(50,25){\ellipse{40}{20}} \put(50,25){\ellipse{20}{10}} \put(13.5,24){${\Omega}_1^\epsilon$} \put(24.5,24){$\widetilde{\Omega}^\epsilon$} \put(47,24){${\Omega}_2^\epsilon$} \put(55,24){$\widetilde{\Gamma}_2^\epsilon$} \put(66.5,24){$\widetilde{\Gamma}$} \put(75.5,24){$\widetilde{\Gamma}_1^\epsilon$} \put(86,24){$\Gamma^\epsilon$} \put(96.5,24){$\Gamma$} \end{picture} \end{center} \caption{Scaling down or up the compostion of $\Omega$ in Figure 2} \end{figure} For fixed $\varepsilon >0$ small enough, let $\Omega_{2}^{\varepsilon}=\{x(1-\varepsilon)|x\in \Omega_{2}\}$, $\Omega_{1}'=\{\frac{x}{1-\varepsilon}|x\in\Omega_{2}\}$, $\Omega_{1}^{\varepsilon}=\Omega \backslash\Omega_{1}'$, $\tilde{\Omega}^{\varepsilon}=\Omega_{1}'\backslash\Omega_{2}^{\varepsilon}$. Defining $\Gamma^{\varepsilon}=\{x(1-\varepsilon)|x\in\Gamma\}$ and assuming $\tilde{\Gamma}_{1}^{\varepsilon}$, $\tilde{\Gamma}_{2}^{\varepsilon}$ are the interfaces of $\tilde{\Omega}^{\varepsilon}$ with $\Omega_{1}^{\varepsilon}$ and $\Omega_{2}^{\varepsilon}$, we can write $\Gamma^{\varepsilon}\times(0,T)=\Sigma_{\varepsilon}$ (see Figure 4). Let $$\psi_{\varepsilon}=\psi_{\varepsilon}^{+}-\psi_{\varepsilon}^{-},$$ where $\psi_{\varepsilon}^{+}=\begin{cases} \big(\psi((1-\varepsilon)x,t)-\underset{\sup_{\varepsilon}} \psi^{+}(x,t)\big)^{+}, & (x,t)\in \Omega_{1}^{\varepsilon}\times(0,T),\\ \big(\psi(x,t)|_{{\tilde{\Gamma}\times(0,T)}} -\sup_{\Sigma_{\varepsilon}} \psi^{+}(x,t)\big)^{+}, & (x,t)\in(\tilde{\Gamma}_{1}^{\varepsilon}\cup \tilde{\Omega} ^{\varepsilon}\cup \tilde{\Gamma}_{2}^{\varepsilon})\times(0,T),\\ \big(\psi(\frac{x}{1-\varepsilon},t)-\sup_{\Sigma_{\varepsilon}} \psi^{+}(x,t)\big)^{+}, & (x,t)\in \Omega_{2}^{\varepsilon}\times(0,T)\,, \end{cases}$ and $\psi_{\varepsilon}^{-}=\begin{cases} \big(\psi((1-\varepsilon)x,t)-\inf_{\Sigma_{\varepsilon}} \psi^{-}(x,t)\big)^{-}, & (x,t)\in \Omega_{1}^{\varepsilon}\times(0,T),\\ \big(\psi(x,t)|_{{\tilde{\Gamma}\times(0,T)}} -\inf_{\Sigma_{\varepsilon}}\psi^{-}(x,t)\big)^{-}, & (x,t)\in(\tilde{\Gamma}_{1}^{\varepsilon}\cup \tilde{\Omega}^{\varepsilon}\cup \tilde{\Gamma}_{2}^{\varepsilon})\times(0,T),\\ \big(\psi(\frac{x}{1-\varepsilon},t)-\inf_{\Sigma_{\varepsilon}} \psi^{-}(x,t)\big)^{-}, & (x,t)\in \Omega_{2}^{\varepsilon}\times(0,T)\,. \end{cases}$ Obviously, $\psi_{\varepsilon}^{+}\in U_{\varepsilon}$, $\psi_{\varepsilon}^{-}\in U_{\varepsilon}$, so we have $\psi_{\varepsilon}\in U_{\varepsilon}$. It is easy to prove that $\psi_{\varepsilon}^{+}$ and $\psi_{\varepsilon}^{-}$ converge strongly to $\psi^{+}$ and $\psi^{-}$ in $U_{0}$ respectively. We omit the details. \end{proof} Now we give the limit behavior of solutions to \eqref{ePe'} as follows. \begin{theorem}\label{thm4.4} Suppose that {\rm (H1)-(H5)} and $F\in L^{2}(Q)$, $A\in L^{2}(0,T)$ hold, if as $\varepsilon\to 0$, $$\label{e4.7} \varphi_{0\varepsilon}\to\varphi_{0}\quad \text{weakly in } L^{2}(\Omega),$$ then for every weak solution $u_{\varepsilon}$ to \eqref{ePe'} we have $$\label{e4.8} u_{\varepsilon}\to u\quad \text{weakly in }L^{2}(0,T; V_{0}),$$ where $u$ is the weak solution to problem \eqref{eP} and definition of $V_{0}$ can be seen in \eqref{e3.2}. \end{theorem} \begin{proof} Let $u_{\varepsilon}$ be the solution to problem \eqref{ePe'}. For a given $00$ and for any $0<\varepsilon<\varepsilon_{0}$, we have $\tilde{\Omega}^{\varepsilon}\subset\tilde{\Omega}^{\varepsilon _{0}}$ and $\psi_{\varepsilon_{0}}\in U_{\varepsilon}$, taking $\psi=\psi_{\varepsilon_{0}}$ in \eqref{e4.5}, we have \label{e4.21} \begin{aligned} &-\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon} \cup\Omega_{2}^{\varepsilon}}u_{\varepsilon}\frac{\partial \psi_{\varepsilon_{0}}}{\partial t}\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega} \sum_{i,j=1}^{N}a_{ij}^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t\\ &=\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}} F\psi_{\varepsilon_{0}} \mathrm{d}x\mathrm{d}t+\int_{\Omega_{1}^{\varepsilon}\cup \Omega_{2}^{\varepsilon}}\varphi_{0\varepsilon} \psi_{\varepsilon_{0}}(x,0)\mathrm{d}x\mathrm{d}t +\int_{0}^{T}A(t)\psi_{\varepsilon_{0}}| _{{\tilde{\Sigma}_{\varepsilon_{0}}}} \mathrm{d}t. \end{aligned} By \eqref{e4.7}, \eqref{e4.19} and the absolute continuity of Lebegue integral, it is easy to prove that \begin{gather} \int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}} u_{\varepsilon}\frac{\partial \psi_{\varepsilon_{0}}}{\partial t}\mathrm{d}x\mathrm{d}t\to\int_{0}^{T}\int_{\Omega}u\frac{\partial \psi_{\varepsilon_{0}}}{\partial t}\mathrm{d}x\mathrm{d}t, \label{e4.22} \\ \int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}F\psi_{\varepsilon_{0}}\mathrm{d}x \mathrm{d}t\to\int_{0}^{T}\int_{\Omega}F\psi_{\varepsilon_{0}} \mathrm{d}x\mathrm{d}t,\label{e4.23} \\ \int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2} ^{\varepsilon}}\varphi_{0\varepsilon}(x)\psi_{\varepsilon_{0}}(x,0)\mathrm{d}x\to\int_{\Omega} \varphi_{0}(x)\psi_{\varepsilon_{0}}(x,0)\mathrm{d}x.\label{e4.24} \end{gather} Next we prove that $$\label{e4.25} \int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij}^{\varepsilon} \frac{\partial u_{\varepsilon}}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t\to\int_{0}^{T}\int_{\Omega} \sum_{i,j=1}^{N}a_{ij}\frac{\partial u}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_0}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t.$$ For any given $\tilde{\Omega}$ such that $\tilde{\Gamma}\subset\tilde{\Omega}\subset\Omega$, we have \label{e4.26} \begin{aligned} &\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij}^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t-\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij}\frac{\partial u}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t \\ =&\int_{0}^{T}\int_{\Omega\backslash\tilde{\Omega}}\sum_{i,j=1}^{N}(a_{ij}^{\varepsilon}-a_{ij})\frac{\partial u_{\varepsilon}}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega\backslash\tilde{\Omega}}\sum_{i,j=1}^{N}a_{ij}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\frac{\partial(u_{\varepsilon}-u)}{\partial x_{j}}\mathrm{d}x\mathrm{d}t\\ &\quad +\int_{0}^{T}\int_{\tilde{\Omega}}\sum_{i,j=1}^{N}a_{ij}^{\varepsilon}\frac{\partial u_{\varepsilon}}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t-\int_{0}^{T}\int_{\tilde{\Omega}}\sum_{i,j=1}^{N}a_{ij}\frac{\partial u}{\partial x_{j}}\frac{\partial \psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t\\ = &\mathrm{I}+\mathrm{II}+\mathrm{III}+\mathrm{IV}. \end{aligned} For any given $\delta>0$, by (H1), (H4), \eqref{e4.19} and the absolute continuity of Lebegue integral, we can take $\tilde{\Omega}$ so small that $$\label{e4.27} |\mathrm{III}|+|\mathrm{IV}|<\frac{\delta}{2}.$$ Following the above $\tilde{\Omega}$ is chosen, by (H5) and noting \eqref{e4.19} and \eqref{e4.20}, then there exists $0<\varepsilon_{1}<\varepsilon_{0}$ such that for any $\varepsilon$ with $0<\varepsilon<\varepsilon_{1}$, $$\label{e4.28} |\mathrm{I}|+|\mathrm{II}|<\frac{\delta}{2}.$$ From \eqref{e4.26}--\eqref{e4.28} we get \eqref{e4.25}. Let $\varepsilon\to 0$ in \eqref{e4.21}, \eqref{e4.22}--\eqref{e4.25} yield \label{e4.29} \begin{aligned} &-\int_{0}^{T}\int_{\Omega}u\frac{\partial\psi_{\varepsilon_{0}}} {\partial t}\mathrm{d}x\mathrm{d}t+\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij}\frac{\partial u}{\partial x_{j}}\frac{\partial\psi_{\varepsilon_{0}}}{\partial x_{i}}\mathrm{d}x\mathrm{d}t \\ &= \int_{0}^{T}\int_{\Omega}F\psi_{\varepsilon_{0}}\mathrm{d}x \mathrm{d}t+\int_{\Omega}\varphi_0\psi_{\varepsilon_{0}}(x,0)\mathrm{d}x +\int_{0}^{T}A(t)\psi_{\varepsilon_0}|_{{\tilde{\Sigma}_{\varepsilon_0}}}\mathrm{d}t \end{aligned} Using \eqref{e4.20}, we get $$\label{e4.30} \psi_{\varepsilon_{0}}\to\psi\quad \text{ strongly in } C([0,T]; L^{2}(\Omega)), \text{ as } \varepsilon_{0}\to 0.$$ Hence as $\varepsilon_{0}\to 0$, we also have $$\label{e4.31} \psi_{\varepsilon_{0}}(x,0)\to\psi(x,0)\quad\text{strongly in }L^{2}(\Omega).$$ By \eqref{e4.20} and trace theorem, we can deduce $$\label{e4.32} \psi_{\varepsilon_{0}}\to\psi\quad\text{strongly in }L^{2}(\tilde{\Sigma}_{0}), \text{ as } \varepsilon_{0}\to 0.$$ Hence $$\label{e4.33} \psi_{\varepsilon_{0}}|_{{\tilde{\Sigma}_{\varepsilon_0}}} =\psi_{\varepsilon_0}|_{{\tilde{\Sigma}_0}} \to\psi|_{{\tilde{\Sigma}_0}}\quad\text{strongly in }L^{2}(0,T).$$ Let $\varepsilon_{0}\to 0$ in \eqref{e4.29}, by \eqref{e4.20}, \eqref{e4.31} and \eqref{e4.33} we can deduce $u$ satisfies \eqref{e3.4}. 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