\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:
\begin{equation}\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}
\end{equation}
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
\begin{equation}
\frac{\partial u}{\partial
n_{L}}=\sum_{i,j=1}^{N}\tilde{a}_{ij}(x,t)\frac{\partial u}{\partial
x_{j}}n_{i}\,.
\end{equation}

%{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
\begin{equation}
\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,
\end{equation}
 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 $\psi\in
U$,
\begin{equation}\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{equation}
\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:
\begin{equation}\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}
\end{equation}
Let
\begin{equation}\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} \}.
\end{equation}

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
\begin{equation}
 |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},
\end{equation}
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
$m\to \infty$,
\begin{equation}\label{e2.17}
c_{km}\to c_{k} \quad  \text{uniformly in } [0,T].
\end{equation}
For any positive integer $r\leq m$, \eqref{e2.14} yields
\begin{equation}\label{e2.18}
\sum_{k=1}^r c_{km}^2(t)\leq C, \quad \forall t\in (0, T),
\end{equation}
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
\begin{equation}
\sum_{k=1}^r c_{k}^2(t)\leq C.
\end{equation}
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
\begin{equation}\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].
\end{equation}
Noting $\{\omega_{k}\}_{k=1}^{\infty}$ is a complete orthonormal
basis in $L^{2}(\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2})$, we
deduce
\begin{equation}\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)).
\end{equation}
Thus \eqref{e2.13} and \eqref{e2.21} imply that
\begin{equation}\label{e2.22}
\tilde{u}_{m} \to\tilde{u}
 \quad \text{weakly in } L^{2}(0,T; V_{1}).
\end{equation}
Convergence \eqref{e2.21} yields
\begin{equation}\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}).
\end{equation}
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
\begin{equation}\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}
\end{equation}
According to \eqref{e2.22}-\eqref{e2.23}, let $m\to\infty$ in
\eqref{e2.24}, it is easy to prove that
\begin{equation}\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}
\end{equation}
For any positive integer $r$, let
\begin{equation}\label{e2.26}
\psi(x, t)=\sum_{k=1}^r\psi_k(t)\omega_k(x).
\end{equation}
Replacing $\psi_k(t)\omega_k(x)$ by the above $\psi(x, t)$ in
\eqref{e2.25}, we have
\begin{equation}\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}
\end{equation}
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
\begin{equation}u=\begin{cases}
\tilde{u} & \text{in }\tilde{Q}\, ,\\
\tilde{C}(t) & \text{in }\tilde{\Sigma}\, .
\end{cases}
\end{equation}
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
\begin{equation}\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.
\end{equation}
For a given $0<h<T$, let
\begin{gather} \label{e2.31}
u_{0h}=\frac{1}{h}\int_{t}^{t+h}u_{0}(x,\tau)\mathrm{d}\tau\,, \quad
\varphi= \begin{cases}
u_{0h} & 0\leq t<(T-h),\\
0 &( T-h)\leq t\leq T \,,
\end{cases} \\
\label{e2.32} \hat{\varphi}= \begin{cases}
 0 & t>(T-h)\,\\,
\varphi & 0<t\leq (T-h),\\
0 & t\leq0,
\end{cases} \quad
\hat{\varphi}_{h}=\frac{1}{h}\int_{t-h}^{t}
\hat{\varphi}(x,\tau)\mathrm{d}\tau.
\end{gather}
It is easy to prove that $\hat{\varphi}_{h}\in U$. Taking
$\psi=\hat{\varphi}_{h}$ in \eqref{e2.30}, we have
\begin{equation}\label{e2.33}
\begin{aligned}
&\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}u_{0}
\frac{\partial\hat{\varphi}_{h}}{\partial
t}\mathrm{d}x\mathrm{d}t \\
&=\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}u_{0}
\frac{\left[\hat{\varphi}(x,t)-\hat{\varphi}(x,t-h)\right]}{h}
\mathrm{d}x\mathrm{d}t\\
& =\frac{1}{h}\Big[\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup
\tilde{\Omega}_{2}}u_{0}\hat{\varphi}(x,t)\mathrm{d}x
\mathrm{d}t-\int_{0}^{T}\int_{\tilde{\Omega}_{1}\cup
\tilde{\Omega}_{2}}u_{0}\hat{\varphi}(x,t-h)\mathrm{d}x\mathrm{d}t\Big]\\
&=\frac{1}{h}\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}\cup
\tilde{\Omega}_{2}}u_{0}\hat{\varphi}(x,t)\mathrm{d}x\mathrm{d}t
-\int_{-h}^{T-h}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}
u_{0}(x,t+h)\hat{\varphi}(x,t)
\mathrm{d}x\mathrm{d}t\\
& = \frac{1}{h}\Big[\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}
\cup\tilde{\Omega}_{2}}u_{0}\varphi(x,t)
 \mathrm{d}x\mathrm{d}t-\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}
\cup\tilde{\Omega}_{2}}u_{0}(x,t+h)
\varphi(x,t)\mathrm{d}x\mathrm{d}t\Big]\\
& =-\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}
\frac{[u_{0}(x,t+h)-u(x,t)]}{h}\varphi
(x,t)\mathrm{d}x\mathrm{d}t\\
& =-\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}
\frac{\partial u_{0h}}{\partial t}\varphi(x,t)\mathrm{d}x\mathrm{d}t\,.
\end{aligned}
\end{equation}
Similarly to the above equality, we can get
\begin{equation}\label{e2.34}
\begin{aligned}
&\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{j}}\frac{\partial\hat{\varphi}_{h}}{\partial
x_{i}}\mathrm{d}t\mathrm{d}x\\
&=\frac{1}{h}\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial
x_{j}}\int_{t-h}^{t}\frac{\partial\hat{\varphi}(x,\tau)}{\partial
x_{i}}\mathrm{d}\tau\mathrm{d}x\mathrm{d}t\\
&= \frac{1}{h}\int_{\Omega}\int_{0}^{T}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial
x_{j}}\int_{t-h}^{t}\frac{\partial\hat{\varphi}(x,\tau)}{\partial
x_{i}}\mathrm{d}\tau\mathrm{d}t\mathrm{d}x\\
&= \frac{1}{h}\int_{\Omega}\Big[\int_{-h}^{0}\frac{\partial\hat{\varphi}}{\partial
x_{i}}\int_{0}^{\tau+h}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{j}}\mathrm{d}t\mathrm{d}\tau
+\int_{0}^{T-h}\frac{\partial\hat{\varphi}}{\partial
x_{i}}\int_{\tau}^{\tau+h}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{j}}\mathrm{d}t\mathrm{d}\tau\\
&\quad +\int_{T-h}^{T}\frac{\partial\hat{\varphi}}{\partial
x_{i}}\int_{\tau}^{T}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial
x_{j}}\mathrm{d}t\mathrm{d}\tau\Big]\mathrm{d}x\\
&=\frac{1}{h}\int_{\Omega}\int_{0}^{T-h}\frac{\partial\varphi(x,\tau)}{\partial
x_{i}}\int_{\tau}^{\tau+h}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial
x_{j}}\mathrm{d}t\mathrm{d}\tau\mathrm{d}x\\
&= \int_{\Omega}\int_{0}^{T-h}\frac{\partial\varphi(x,t)}{\partial
x_{i}}\Big(\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{j}}\Big)_{h}\mathrm{d}t\mathrm{d}x.
\end{aligned}
\end{equation}
By \eqref{e2.30}, \eqref{e2.33} and \eqref{e2.34}, we  deduce
\[
\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}
\varphi\frac{\partial u_{0h}}{\partial
t}\mathrm{d}x\mathrm{d}t+\int_{0}^{T-h}
\int_{\Omega}\Big(\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{j}}\Big)_{h}
\frac{\partial\varphi(x,t)}{\partial x_{i}}\mathrm{d}x\mathrm{d}t=0\, .
\]
Taking $\varphi$ defined in \eqref{e2.31}, we get
\[
\int_{0}^{T-h}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}u_{0h}\frac{\partial
u_{0h}}{\partial t}
\mathrm{d}x\mathrm{d}t+\int_{0}^{T-h}\int_{\Omega}\Big(\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{j}}\Big)_{h}\frac{\partial u_{0h}}{\partial
x_{i}}\mathrm{d}x\mathrm{d}t=0 \, .
\]
Letting $h\to 0$ in in the above equation, we have
\[
\frac{1}{2}\int_{\tilde{\Omega}_{1}\cup\tilde{\Omega}_{2}}u_{0}^{2}
\mathrm{d}x\Big|_{0}^{T}+\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^N\tilde{a}_{ij}\frac{\partial
u_{0}}{\partial x_{i}}\frac{\partial u_{0}}{\partial
x_{j}}\mathrm{d}x\mathrm{d}t=0\,.
\]
Hence
\[
\int_{0}^{T}\int_{\Omega}|Du_{0}|^{2}\mathrm{d}x\mathrm{d}t=0.
\]
Thus we can prove $u_{0}=0$  a.e. in $\Omega$. Thus the proof of
uniqueness is completed.
\end{proof}

\section{Parabolic boundary value problem with equivalued interface}

To study the limit behavior of solutions to problem \eqref{eP}, we
need to study the following equivalued interface problem
\eqref{eP'}. Here we give another division of $\Omega$ as shown in
Fig.2. $\Omega$ is composed of two non-overlapping subdomains
$\Omega_{1}$ and $\Omega_{2}$, and $\tilde{\Gamma}$ is the interface
of $\Omega_{1}$ and $\Omega_{2}$. Denote
$Q_{0}=(\Omega_{1}\cup\Omega_{2})\times(0,T),
\tilde{\Sigma}_{0}=\tilde{\Gamma}\times(0,T)$.

%{tt2.eps}
\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(70,35)(0,0)
\put(35,17.5){\ellipse{70}{35}} \put(35,17.5){\ellipse{40}{20}}
\put(7.5,16){$\Omega_1$}
\put(33.5,16){$\Omega_2$}
\put(51,16){$\widetilde{\Gamma}$} \put(67,16){$\Gamma$}
\end{picture}
\end{center}
\caption{The compostion of $\Omega$ with thin interface
$\widetilde{\Gamma}$}
\end{figure}

In this section we  consider the  boundary value
problem with equivalued interface:
\begin{equation}\label{eP'}
\begin{gathered}
\frac{\partial u}{\partial
t}-\sum_{i,j=1}^{N}\frac{\partial}{\partial
 x_{i}}(a_{ij}(x,t)\frac{\partial u}{\partial
 x_{j}})=F(x,t) \quad \text{in } Q_{0},\\
u=0 \quad \text{on } \Sigma, \\
u_{+}=u_{-} =C(t)\quad \text{on } \tilde{\Sigma}_{0}, \\
\int_{\tilde{\Gamma}}\big(\frac{\partial u}{\partial
n_{L}}\big)_{+}\mathrm{d}s=\int_{\tilde{\Gamma}}\big(\frac{\partial
u}{\partial n_{L}}\big)_{-}\mathrm{d}s
+A(t) \quad \text{a.e. t }\in (0, T),\\
u(x,0)=\varphi_{0}(x)\quad \text{in }\Omega,
\end{gathered}
\end{equation}
where the subscripts $+$ and $-$ denote the values on both sides of
$\tilde{\Gamma}$, and the unit normal vector
$\vec{n}=(n_{1},\dots,n_{N})$ takes the same direction on both
sides of $\tilde{\Gamma}$.

We state the following assumption to $a_{ij}$:
\begin{itemize}
\item[(H1)]  The functions $a_{ij}$ are piecewise smooth in $Q$,
$a_{ij}=a_{ji}$, and there exist two constants $\alpha,\beta>0$
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 $\psi\in U_{0}$,
\begin{equation}\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{equation}
\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
\begin{equation}\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}
\end{equation}
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 $\psi\in U_{\varepsilon}$,
\begin{equation}\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{equation}
\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 $\varepsilon\to 0$,
\begin{equation}\label{e4.6}
\psi_{\varepsilon}\to\psi\quad  \text{strongly in }U_{0},
\end{equation}
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 $\varepsilon\to 0$,
\begin{equation}\label{e4.7}
\varphi_{0\varepsilon}\to\varphi_{0}\quad \text{weakly in }
L^{2}(\Omega),
\end{equation}
then for every weak solution $u_{\varepsilon}$ to \eqref{ePe'} we
have
\begin{equation}\label{e4.8}
u_{\varepsilon}\to u\quad \text{weakly in }L^{2}(0,T; V_{0}),
\end{equation}
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 $0<h<T$, let $u_{0}$ and $u_{0h}$ be replaced by
$u_{\varepsilon}$ and $u_{\varepsilon h}$ in \eqref{e2.33}
respectively. Taking $\psi=\hat{\varphi}_{h}$ (see \eqref{e2.32}) in
\eqref{e4.5}, we have
\begin{align*}
&-\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}
u_{\varepsilon}\frac{\partial\hat{\varphi}_{h}}
{\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\hat{\varphi}_{h}}{\partial x_{i}}
 \mathrm{d}x\mathrm{d}t\\
&=\int_{0}^{T}
 \int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}
 F\hat{\varphi}_{h} \mathrm{d}x\mathrm{d}t
 +\int_{0}^{T}A(t)\hat{\varphi}_{h}(x,t)|_{{\tilde{\Sigma}_{\varepsilon}}}
\mathrm{d}t.
\end{align*}
Similar to \eqref{e2.33} and \eqref{e2.34}, it is easy to prove that
\begin{gather*}
-\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}
\cup\Omega_{2}^{\varepsilon}}u_{\varepsilon}
\frac{\partial\hat{\varphi}_{h}}
{\partial t}\mathrm{d}x\mathrm{d}t
=\int_{0}^{T-h}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}\frac{\partial
u _{\varepsilon h}}{\partial t}u_{\varepsilon
h}\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\hat{\varphi}_{h}}{\partial
x_{i}}\mathrm{d}x\mathrm{d}t
=\int_{0}^{T-h}\int_{\Omega}\sum_{i,j=1}^{N}\frac{\partial
u_{\varepsilon h}}{\partial
x_{i}}\big(a_{ij}^{\varepsilon}\frac{\partial
u_{\varepsilon}}{\partial x_{j}}\big)_{h}\mathrm{d}x\mathrm{d}t,
\\
\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup
\Omega_{2}^{\varepsilon}}F\hat{\varphi}_{h}\mathrm{d}x\mathrm{d}t
=\int_{0}^{T-h}\int_{\Omega_{1}^{\varepsilon}\cup
\Omega_{2}^{\varepsilon}}u_{\varepsilon
h}F_{h}\mathrm{d}x\mathrm{d}t,
\\
\int_{0}^{T}A(t)\hat{\varphi}_{h}|_{{\tilde{\Sigma_\varepsilon}}}
\mathrm{d}t
=\frac{1}{|\tilde{\Gamma}|}\int_{0}^{T}\int_{\tilde{\Gamma}}A(t)
\hat{\varphi}_{h}\mathrm{d}s\mathrm{d}t
=\frac{1}{|\tilde{\Gamma}|}\int_{0}^{T-h}
\int_{\tilde{\Gamma}}(A(t))_{h}u_{\varepsilon h}\mathrm{d}s\mathrm{d}t.
\end{gather*}
Thus we can write the above expression  as
\begin{align*}
&\int_{0}^{T-h}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}\frac{\partial
u _{\varepsilon h}}{\partial t}u_{\varepsilon
h}\mathrm{d}x\mathrm{d}t
+\int_{0}^{T-h}\int_{\Omega}\frac{\partial u_{\varepsilon
h}}{\partial x_{i}}\big(a_{ij}^{\varepsilon}\frac{\partial
u_{\varepsilon}}{\partial x_{j}}\big)_{h}\mathrm{d}x\mathrm{d}t\\
&=\int_{0}^{T-h}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}u_{\varepsilon
h}F_{h}\mathrm{d}x\mathrm{d}t+\frac{1}{|\tilde{\Gamma}|}\int_{0}^{T-h}\int_{\tilde{\Gamma}}(A(t))_{h}u_{\varepsilon
h}\mathrm{d}s\mathrm{d}t.
\end{align*}
Let $h\to0$ in the above expression, we have
\begin{equation}\label{e4.15}
\begin{aligned}
&\frac{1}{2}\int_{\Omega_{1}^{\varepsilon}\cup\Omega_{2}^{\varepsilon}}
u_{\varepsilon}^{2}\mathrm{d}x\Big|_{0}^{T}+
\int_{0}^{T}\int_{\Omega}\sum_{i,j=1}^{N}a_{ij}^{\varepsilon}\frac{\partial
u_{\varepsilon}}{\partial x_{i}}\frac{\partial
u_{\varepsilon}}{\partial
x_{j}}\mathrm{d}x\mathrm{d}t\\
&=\int_{0}^{T}\int_{\Omega_{1}^{\varepsilon}\cup
\Omega_{2}^{\varepsilon}}u_{\varepsilon}F
\mathrm{d}x\mathrm{d}t+\frac{1}{|\tilde{\Gamma}|}
\int_{0}^{T}\int_{\tilde{\Gamma}}Au_{\varepsilon}\mathrm{d}s\mathrm{d}t.
\end{aligned}
\end{equation}
Condition (H4), H\"{o}lder's inequality, the trace theorem, and
Poinc\'{a}re's inequality yield
\begin{align*}
&\alpha\| Du_{\varepsilon}\|_{{L^{2}(Q)}}^{2}\\
&\leq\| u_{\varepsilon}\|_{{L^{2}(Q)}}\|
F\|_{{L^{2}(Q)}}+\frac{1}{|\tilde{\Gamma}|^{1/2}}
\int_{0}^{T}|A|\Big(\int_{\tilde{\Gamma}}
u_{\varepsilon}^{2}\mathrm{d}s\Big)^{1/2}\mathrm{d}t\\
&\leq\| Du_{\varepsilon}\|_{{L^{2}(Q)}}\|
F\|_{{L^{2}(Q)}}+\frac{C}{|\tilde{\Gamma}|^{1/2}}
 \int_{0}^{T}|A|\Big(\int_{\Omega_{2}}|u_{\varepsilon}|^{2}
+|Du_{\varepsilon}|^{2}\mathrm{d}x\Big)^{1/2}\mathrm{d}t\\
&\leq\| Du_{\varepsilon}\|_{{L^{2}(Q)}}\|
F\|_{{L^{2}(Q)}}+C\|A\|_{{L^{2}(0,T)}}\|
Du_{\varepsilon}\|_{{L^{2}(Q)}}\\
&\leq(C\| A\|_{{L^{2}(0,T)}}+\|F\|_{{L^{2}(Q)}})\|
Du_{\varepsilon}\|_{{L^{2}(Q)}}\,,
\end{align*}
where $C$ is a positive constant independent of $\varepsilon$.
By Young's inequality we obtain
\begin{equation}\label{e4.17}
\|Du_{\varepsilon}\|_{{L^{2}(Q)}}\leq C.
\end{equation}
Hence we get
\begin{equation}\label{e4.18}
\| u_{\varepsilon}\|_{{L^{2}(0,T; V_{0})}}\leq C,
\end{equation}
where $C$ a positive constant independent of $\varepsilon$.

From the above inequality, we can extract a subsequence of
$\{u_{\varepsilon}\}$ (still denoted by $\{u_{\varepsilon}\}$) such
that
\begin{equation}\label{e4.19}
u_{\varepsilon}\to u\quad\text{weakly in } L^{2}(0,T; V_{0}).
\end{equation}
By Lemma \ref{lem4.3}, for any given $\psi\in U_{0}$, there exists
$\psi_{\varepsilon}\in U_{\varepsilon}$ such that
\begin{equation}\label{e4.20}
\psi_{\varepsilon}\to\psi \quad\text{strongly in }U_{0}, \text{ as }
\varepsilon\to 0.
\end{equation}
Fixed $\varepsilon _{0}>0$ 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
\begin{equation}\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}
\end{equation}
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
\begin{equation}\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.
\end{equation}
 For any given $\tilde{\Omega}$ such that
$\tilde{\Gamma}\subset\tilde{\Omega}\subset\Omega$, we have
\begin{equation}\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}
\end{equation}
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
\begin{equation}\label{e4.27}
|\mathrm{III}|+|\mathrm{IV}|<\frac{\delta}{2}.
\end{equation}

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 $0<\varepsilon<\varepsilon_{1}$,
\begin{equation}\label{e4.28}
|\mathrm{I}|+|\mathrm{II}|<\frac{\delta}{2}.
\end{equation}
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
\begin{equation}\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}
\end{equation}
Using \eqref{e4.20}, we get
\begin{equation}\label{e4.30}
\psi_{\varepsilon_{0}}\to\psi\quad \text{ strongly in } C([0,T];
L^{2}(\Omega)),   \text{ as } \varepsilon_{0}\to 0.
\end{equation}
Hence as $\varepsilon_{0}\to 0$, we also have
\begin{equation}\label{e4.31}
\psi_{\varepsilon_{0}}(x,0)\to\psi(x,0)\quad\text{strongly in
}L^{2}(\Omega).
\end{equation}
By \eqref{e4.20} and trace theorem, we can deduce
\begin{equation}\label{e4.32}
\psi_{\varepsilon_{0}}\to\psi\quad\text{strongly in
}L^{2}(\tilde{\Sigma}_{0}),  \text{ as } \varepsilon_{0}\to 0.
 \end{equation}
Hence
\begin{equation}\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).
\end{equation}
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}. By the uniqueness of weak solution to problem
\eqref{eP'}, \eqref{e4.19} holds for the whole sequence
$\{u_{\varepsilon}\}$. This completes the proof.
\end{proof}

\begin{thebibliography}{00}

\bibitem{c1} Yazhe Chen;
 \emph{Second Order Parabolic Differential Equations}(in Chinese),
Peking University Press, Beijing, 2003.

\bibitem{c2} Zhong-xiang Chen and  Li-shang Jiang;
\emph{Exact solution for the system of flow equations through a
medium with double-porosity }, Sci. Sinica, 7(1980), 880-896.

\bibitem{d1}  E. Dibenedetto;
 \emph{Degenerate Parabolic Equations},
Springer-Verlag, Berlin, 1993.

\bibitem{e1}  L. C. Evans;
 \emph{Partial Differential Equations },
Graduate Studies in Math, Vol 19, AMS, Providence, RI, 1998.

\bibitem{j1} Li-shang Jiang, Zhong-xiang Chen;
 \emph{Theory of Well Test Analysis} (in Chinese),
Oil Industry Press, Beijing, 1985.

\bibitem{k1} Xiangyan Kong;
 \emph{Advanced Mechanics of Fluids in Porous Media
}(in Chinese), University Of Science And Technology of China Press,
Hefei, 1991.

\bibitem{l1} Ta-tsien Li;
 \emph{A class of non-local boundary value problems for partial
differential equations and its applications in numerical analysis},
J. Comput. Appl. Math., 28(1989), 49-62.

\bibitem{l2} Ta-tsien Li, et al;
 \emph{Boundary value problems with equivalued surface boundary
conditons for self- adjoint ellipic differential equations I,II}(in
Chinese), Fudan J. (Nat. Sci), 1, 3-4 (1976), 61-71, 136-145.

\bibitem{l3} Ta-tsien Li, Jinhai Yan;
 \emph{Limit behaviour of solutions to certain kinds of boundary
value problems with equivalued surface}, Asymptotic Analysis
21(1999), 23-35.

\bibitem{l4} Ta-tsien Li,  Song-mu Zheng, Yong-ji Tan, Wei-xi Shen;
 \emph{Boundary Value Problem with
Equivalued Surface and Resistivity Well-Logging}, Pitman Research
Notes in Math Series 382, Longman, 1998.

\bibitem{l5}  J. L. Lions;
 \emph{Quelques Methodes des de Resolutions des Problemes aux Limites
Nonlineaires}, Dunod Gauthier-Villars Paris, 1969

\bibitem{s1} Wei-xi Shen;
 \emph{On mixed initial-boundary value problems of
second order parabolc equations with equivalued surface boundary
conditions} (in Chinese), Fudan J.(Nat.Sci), 4(1978), 15-24.

\bibitem{z1} E. Zeidler;
 \emph{Nonlinear Functional Analysis and its Applications,
II/A: Linear Monotone Operators}, Springer-Verlag, New York, 1990.

\end{thebibliography}

\end{document}