\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 25, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/25\hfil Parabolic equations]
{Parabolic equations with Robin type boundary conditions
in a non-rectangular domain}

\author[A. Kheloufi, B.-K. Sadallah\hfil EJDE-2010/25\hfilneg]
{Arezki Kheloufi, Boubaker-Khaled Sadallah} % in alphabetical order

\address{Arezki Kheloufi \newline
Department of Sciences and Techniques, Faculty of Technology \\
B\'{e}jaia University, 6000. B\'{e}jaia, Algeria}
\email{arezkinet2000@yahoo.fr}

\address{Boubaker-Khaled Sadallah \newline
Department of Mathematics, E.N.S. \\
16050 Kouba. Algiers, Algeria}
\email{sadallah@ens-kouba.dz}

\thanks{Submitted July 29, 2009. Published February 10, 2010.}
\thanks{Supported by grant 08MDU735 from EGIDE under the CMEP Program.}
\subjclass[2000]{35K05, 35K20}
\keywords{Parabolic equation; non-rectangular domains;
 Robin condition; \hfill\break\indent anisotropic Sobolev space}

\begin{abstract}
 In this article, we study the parabolic equation
 $\partial_{t}u-c^2(t)\partial_x^2u=f$
 in the  non-necessarily rectangular domain
 $$
 \Omega =\{ (t,x)\in\mathbb{R}^2:0<t<T,\,
 \varphi_1(t)<x<\varphi_2(t)\}.
 $$
 The boundary conditions are of Robin type, while the right-hand
 side  lies in the  Lebesgue space $L^2(\Omega )$.
 Our aim is to find  conditions on  $c$ and the functions
 $(\varphi_i)_{i=1,2}$ such that the solution
 belongs to the anisotropic Sobolev space
 $H^{1,2}(\Omega )=\{u\in L^2(\Omega ):\partial_{t}u,
 \partial_xu,\partial_x^2u\in L^2(\Omega )\} $.
 For goal we use the method of approximation of domains.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

Let $\Omega \subset\mathbb{R}^2$ be the triangular domain
\[
\Omega =\{ (t,x)\in \mathbb{R}^2:0<t<T,\,
\varphi_1(t)<x<\varphi_2(t)\},
\]
where $\varphi_1,\varphi_2$ are the functions of
parametrization with $ \varphi_1(0)=\varphi_2(0)$,
and $T$ is a finite positive number. In
$\Omega $, we consider the boundary-value problem
\begin{equation} \label{eP}
\begin{gathered}
\partial_{t}u-c^2(t)\partial_x^2u=f\quad \text{in }  L^2(\Omega )\\
b_i(t)\partial_xu+\alpha _i(t)u\big|_{x=\varphi_i(t)}=0,\quad i=1,2,
\end{gathered}
\end{equation}
where $(\alpha_i)$ and $(b_i)$ are given. We
look for conditions on the functions
$(b_i,\alpha_i,\varphi _i)_{i=1,2}$ and the coefficient $c$ such
that \eqref{eP} admits a unique solution $u$ belonging to the
anisotropic Sobolev space
\[
H^{1,2}(\Omega )=\{ u\in L^2(\Omega )
:\partial_{t}u,\partial_xu,\partial_x^2u\in L^2(\Omega)\} .
\]
We consider the case where
$\alpha_i(t)\neq 0$  and $b_i(t)\neq 0$
for all $t\in ] 0,T[ $. So, \eqref{eP}
may be written in the form
\begin{equation} \label{eP'}
\begin{gathered}
\partial_{t}u-c^2(t)\partial_x^2u=f\quad \text{in }L^2(\Omega )\\
\partial_xu+\beta_i(t) u\big|_{\Gamma_i}=0,\quad i=1,2,
\end{gathered}
\end{equation}
where $\beta_i(t)=\frac{\alpha_i(t)}{b_i(t)}$,
$\Gamma_i=\{ (t,\varphi_i(t)),t\in ] 0,T[ \} $, $i=1,2$.

In the sequel, the hypothesis
\begin{equation}
(-1)^{i}(c^2(t)\beta_i(t)-\frac{\varphi_i'(t)}{2})\geq 0
\quad\text{a.e. } t\in ] 0,T[ ,\;i=1,2, \label{1}
\end{equation}
is imposed in order to guarantee the uniqueness of the solution
of \eqref{eP'}. Indeed, if $u$ is the solution of the
\eqref{eP'} with a null right-hand side, the
calculations show that the inner product $\langle \partial
_{t}u-c^2(t)\partial_x^2u,u\rangle $ in
$L^2(\Omega )$ gives
\begin{align*}
0 & =  \sum_{i=1}^2\int_{\Gamma_i}(-1)^{i}\Big(
c^2(t)\beta_i(t)-\frac{\varphi
_i'(t)}{2}\Big)u^2(t,\varphi
_i(t))dt \\
&\quad +\frac{1}{2}\int_{\Gamma_3}u^2dx+\int_{\Omega
}c^2.(\partial_xu)^2\,dt\,dx
\end{align*}
where $\Gamma_3=\{ (T,x):\varphi_1(T)<x<\varphi_2(T)\} $
if $\varphi_1(T) \neq \varphi_2(T)$. The hypothesis \eqref{1}
implies that $\partial_xu=0$ and consequently $\partial_x^2u=0$.
Then,  \eqref{eP'} gives $\partial_{t}u=0$. Thus, $u$ is constant.
The boundary conditions and the fact that
$\beta_i(t)\neq 0$ for all $t\in ] 0,T[ $ imply
 $u=0$.

We also assume that the functions $(\beta_i)_{i=1,2}$ satisfy
the  assumption
\begin{equation}
\beta_1(t)<0, \quad \beta_2(t)>0\quad \text{for all }
t\in [0,T].    \label{2}
\end{equation}

The most interesting point of the parabolic problem studied here is
the fact that $\varphi_1(0)=\varphi_2(0)$ or
$\varphi_1(T)=\varphi_2(T)$. In this case the domain $\Omega $
is not rectangular and
cannot be transformed into a regular domain without the appearance
of some degenerate terms in the parabolic equation; see, for example
 Sadallah \cite{Sad1}.

The solvability of this kind of problems with Cauchy-Dirichlet
boundary conditions has been investigated in
 \cite{lab1,Lab1,Sad2, Sad3}. In Sadallah \cite{Sad3}, the same
equation is studied by another approach making use of the so-called
Schur's Lemma and gives the same result obtained in \cite{Sad2} by
the \textit{a priori} estimates technique. In \cite{lab1} and
\cite{Lab1}, the authors deal with the heat equation (i.e., the
case where $c(t)=1$) set in a non-rectangular domain
with a right-hand side taken in $L^{p}$, where
$p\in ] 1,\infty[ $, and have obtained optimal regularity
results by the operators sum method. These results are generalized
in \cite{Lab2} to a parabolic equation of the type
\[
\partial_{t}u(t,x)-\partial_x^2u(t,x)
+\lambda m(t,x)u(t,x)=f(t,x)
\]
where $\lambda $ is a positive spectral parameter and $m(.)$
some positive weight functions. Hofmann and Lewis \cite{Hof} have also
considered the classical heat equation with Neumann boundary condition in
noncylindrical domains satisfying some conditions of Lipschitz's type. The
authors showed that the optimal $L^{p}$ regularity holds for $p=2$ and the
situation gets progressively worse as $p$ approaches $1$. In Savar\'{e}
\cite{Sav}, parabolic problems in noncylindrical domains are considered in
the Hilbertian case. The author obtains some regularity results under
assumption on the geometrical behavior of the boundary which cannot include
our triangular domain.

The plan of this paper is as follows. In Section 2, we derive some
technical lemmas which will allow us to prove an \textit{a priori}
estimate (in a sense to be defined later). In Section 3, there are
two main steps. First, we prove that \eqref{eP'} admits a (unique) solution in the case of a domain which
can be transformed into a rectangle. Secondly, for $T$ small enough,
we prove that the result holds true in the case of a triangular
domain under some assumptions on the coefficient $c$ and the
functions $(\beta_i,\varphi_i)_{i=1,2}$ to be
made more precise later on. The method used here is based on
the approximation of the triangular domain by a sequence of subdomains
$(\Omega_{n})_{n}$ which can be transformed into regular
domains (rectangles) and we establish an \textit{a priori} estimate
of the type
\[
\| u_{n}\|_{H^{1,2}(\Omega_{n})}\leq
K\| f\|_{L^2(\Omega_{n})},
\]
where $u_{n}$ is the solution of  \eqref{eP'}
in $\Omega_{n}$ and $K$ is a constant independent of $n$, which allows us
to pass to the limit.
Finally, in Section 4 we study \eqref{eP'} in the case where $T$
is not necessarily small.

\section{Preliminaries}

Let $(\beta_i)_{i=1,2}$ be continuous real-valued functions
on $] 0,T[ $. Assume that there exists a constant $l>0$ such that
\begin{gather}
\big| \frac{(1+\beta_2(t))}{A(t)}\big| \leq l  ,  \label{2.1} \\
\big| \frac{\beta_1(t)(1+\beta_2(t))}{A(t)}\big| \leq l,  \label{2.2}
\end{gather}
where
\begin{equation}
A(t)=\beta_1(t)\beta_2(t)+\beta
_1(t)-\beta_2(t)\neq 0,
\label{2.3}
\end{equation}
for every $t\in ] 0,T[ $.

\begin{lemma}\label{lem1}
Assume that $\beta_1$ and $\beta_2$ fulfil the conditions
\eqref{2.1}, \eqref{2.2} and \eqref{2.3}. Then, for a fixed
$t\in ] 0,1 [ $, there exists a positive constant $K_1$
independent of $t$, such that for each $u\in H_{\gamma }^2(0,1)$
\[
\| u^{^{(j)}}\|_{L^2(
0,1)}\leq K_1\| u^{^{(2)}}\|_{L^2(0,1)},j=0,1,
\]
where
\[
H_{\gamma }^2(0,1)
=\{ u\in H^2( 0,1):u'(0)+\beta_1(t)
u(0)=0 ,\; u'(1)+\beta_2(t)u(1)=0\} .
\]
\end{lemma}

\begin{proof}
Let $t\in ] 0,1[ $ and $f$ an arbitrary fixed element of
$ L^2(0,1)$. Then the solution of the problem
\begin{gather*}
u''=f \\
u'(0)+\beta_1(t)u(0)=0\\
u'(1)+\beta_2(t)u(1)=0,
\end{gather*}
can be written in the form
\[
u(y) =  \int_0^{y}\Big\{ \int_0^{x}f(s) ds\Big\} dx+yu'(0)+u(0),
\]
where
\begin{gather*}
u(0)=\frac{\int_0^{1}f(s)ds+\beta_2(
t)\int_0^{1}\{ \int_0^{x}f(s)ds\} dx}{
A(t)} \\
u'(0)=-\beta_1(t)u(0).
\end{gather*}
The uniqueness of the solution is easy to check, thanks to the boundary
conditions and the condition \eqref{2.3}.

Using the Cauchy-Schwarz inequality, we obtain the following two estimates
\begin{gather*}
| u(0)| \leq  C| \frac{(1+\beta_2(t))}{A(t)}|
\| f\|_{L^2(0,1)} \\
| u'(0)| \leq C| \frac{\beta_1(t)(1+\beta_2(t))}{
A(t)}| \| f\|_{L^2(0,1)},
\end{gather*}
which will allow us to obtain the desired estimates, thanks to the
conditions \eqref{2.1}, \eqref{2.2}.
\end{proof}

\begin{lemma}\label{lem2}
Under the assumptions \eqref{2.1}, \eqref{2.2} and \eqref{2.3} on
$(\beta_i)_{i=1,2}$ and for a fixed $t\in ]0,1[ $,
there exists a constant $C_1$ (independent of $a$ and $b$) such that
\[
\| v^{(j)}\|_{L^2(a,b)
}^2\leq C_1(b-a)^{2(2-j)}\|
v^{(2)}\|_{L^2(a,b) }^2,\quad j=0,1,
\]
for each $v\in H_{\gamma }^2(a,b)$, with
\[
H_{\gamma }^2(a,b)=\{ v\in H^2(a,b)
:v'(a)+\frac{\beta_1(t)}{b-a}
v(a)=0,\; v'(b)+\frac{\beta_2(t)}{b-a}v(b)=0\} .
\]
\end{lemma}

\begin{proof}
It is a direct consequence of Lemma \ref{lem1} by using the
 affine change of variable
$[0,1] \to  [a,b]$, $x  \to  (1-x)a+xb=y$.
\end{proof}

\section{Solution of the problem \eqref{eP'}}

\subsection{A domain that can be transformed into a
rectangle}
Let
\[
\Omega =\{ (t,x)\in \mathbb{R} ^2:0<t<T,\varphi_1(t)<x<\varphi_2(
t)\}
\]
where $T$ is a finite positive number, while $\varphi_1$ and
$\varphi_2 $ are Lipschitz continuous in $[0,T]$,
such that $\varphi_1(t)<\varphi_2(t)
$ for all $t\in [0,T]$. Consider $c$ a continuous function on
$[0,T]$, such that
\begin{equation}
0<d_1\leq c\leq d_2,    \label{5}
\end{equation}
where $d_1,d_2$ are two constants.

\begin{theorem}\label{theo1}
Under  assumptions \eqref{1}, \eqref{2.1}, \eqref{2.2} and
\eqref{2.3} on
$(\beta_i)_{i=1,2}$, the problem
\begin{equation} \label{eP1}
\begin{gathered}
\partial_{t}u-c^2(t)\partial_x^2u=f \quad \text{in }L^2(\Omega ), \\
u\big|_{t=0}=0, \\
\partial_xu+\beta_i(t)u\big|_{x=\varphi_i(t)}=0,\quad i=1,2,
\end{gathered}
\end{equation}
admits a (unique) solution $u\in H^{1,2}(\Omega )$.
\end{theorem}

\begin{proof}
 The uniqueness of the solution is easy to check, thanks to
\eqref{1}. Let us prove the existence. The change of variables
\[
(t,x) \mapsto  (t,y)=\Big(t,\frac{x-\varphi
_1(t)}{\varphi_2(t)-\varphi_1(t)}\Big)
\]
transforms $\Omega $ into the rectangle
$R=] 0,T[ \times ]0,1[ $. Putting $u(t,x)=v(t,y)$
and $ f(t,x)=g(t,y)$, then Problem \eqref{eP1}
becomes
\begin{equation} \label{eP2}
\begin{gathered}
\partial_{t}v(t,y)+a(t,y)\partial_{y}v(
t,y)-\frac{1}{b^2(t)}\partial_{y}^2v(t,y)=g(t,y)\\
v\big|_{t=0}=0 \\
\frac{1}{\varphi (t)}\partial_{y}v+\beta_1(t)
v\big|_{y=0}=0, \\
\frac{1}{\varphi (t)}\partial_{y}v+\beta_2(t)
v\big|_{y=1}=0,
\end{gathered}
\end{equation}
where
\begin{gather*}
\varphi (t)=\varphi_2(t)-\varphi_1(t)\\
b(t)=\frac{\varphi (t)}{c(t)} \\
a(t,y)=-\frac{y\varphi '(t)+\varphi_1'(t)}{\varphi (t)}.
\end{gather*}
This change of variables conserves the spaces $H^{1,2}$ and $L^2$.
In other words
\begin{gather*}
f \in L^2(\Omega )\Leftrightarrow g\in L^2(R)\\
u \in H^{1,2}(\Omega )\Leftrightarrow v\in H^{1,2}(R).
\end{gather*}
\end{proof}

\begin{lemma}\label{lem3}
The operator
\[
\begin{array}{llll}
B: & H_{\gamma }^{1,2}(R)& \to & L^2(R)
\\
& v & \mapsto & Bv=a(t,y)\partial_{y}v
\end{array}
\]
is compact, where for a fixed $t\in ] 0,T[ $,
\[
H_{\gamma }^{1,2}(R)=\{ v\in H^{1,2}(R)
:v\big|_{\Gamma_0}=0,\partial_{y}v+\varphi (t)\beta_i(
t)v\big|_{\Gamma_{i,R}}=0,\;i=1,2\} ,
\]
with $\Gamma_0=\{ 0\} \times ] 0,1[ $,
$\Gamma _{1,R}=] 0,T[ \times \{ 0\} $ and
$\Gamma_{2,R}=] 0,T[ \times \{ 1\} $.
\end{lemma}

\begin{proof}
$R$ has the ``horn property" of Besov \cite{Bes}, so
\[
\begin{array}{llll}
\partial_{y}: & H_{\gamma }^{1,2}(R)& \to & H^{\frac{1
}{2},1}(R)\\
& v & \mapsto & \partial_{y}v
\end{array}
\]
is continuous. Since $R$ is bounded, the canonical injection is
compact from
$H^{\frac{1}{2},1}(R)$ into $L^2(R)$, see for instance \cite{Bes}.
Here
\[
H^{\frac{1}{2},1}(R)=L^2(0,T;H^{1}] 0,1[ )\cap H^{\frac{1}{2}
}(0,T;L^2] 0,1[ ).
\]
See \cite{Lions} for the complete definitions of the $H^{r,s}$
Hilbertian Sobolev spaces.

Then $\partial _{y}$ is a compact operator from $H_{\gamma }^{1,2}(R)$
to $L^2(R)$. Furthermore, since $a(.,.)$ is a bounded function,
the operator $B=a\partial_{y}$ is then
compact from $H_{\gamma }^{1,2}(R)$ into $L^2(R)$.
\end{proof}

So, it is sufficient to show that the operator
\[
\partial_{t}-\frac{c^2}{\varphi ^2}\partial_{y}^2:  H_{\gamma
}^{1,2}(R) \to  L^2(R)
\]
is an isomorphism.
A simple change of variable $t=h(s)$ with
$h'(s)=\frac{\varphi ^2}{c^2}(t)$, transforms the
problem
\begin{gather*}
\partial_{t}v(t,y)-\frac{c^2}{\varphi ^2}(t)
\partial_{y}^2v(t,y)=g(t,y)\in L^2(R), \\
v\big|_{t=0}=0, \\
\frac{1}{\varphi (t)}\partial_{y}v+\beta_1(t)v\big|_{y=0}=0, \\
\frac{1}{\varphi (t)}\partial_{y}v+\beta_2(t)v\big|_{y=1}=0,
\end{gather*}
into
\begin{equation} \label{eP3}
\begin{gathered}
\partial_{s}w(s,y)-\partial_{y}^2w(s,y)=\zeta
(s,y)\\
w\big|_{s=h^{-1}(0)}=0 \\
\frac{1}{\varphi (h(s))}\partial_{y}w+\beta
_1(h(s))w\big|_{y=0}=0, \\
\frac{1}{\varphi (h(s))}\partial_{y}w+\beta
_2(h(s))w\big|_{y=1}=0,
\end{gathered}
\end{equation}
with $\zeta (s,y)=\frac{g(t,y)}{h'(s)}$ and $w(s,y)=v(t,y)$.
Note that this change of variables preserves the
spaces $L^2$ and $H^{1,2}$. It follows from \eqref{2} that there
exists a unique $w\in H^{1,2}$ solution of the problem \eqref{eP3}.
 This implies that Problem \eqref{eP1} admits a unique solution
$u\in H^{1,2}(\Omega )$. We obtain the function $u$ by setting
$u(t,x)=v(t,y)=w(h^{-1}(t),y)$. This completes the proof of
Theorem \ref{theo1}.

We shall need the following result in order to justify the calculus
of the next section.

\begin{lemma}\label{lem4}
The space
\[
W=\{ u\in D([0,T];H^2(0,1))
:\partial_xu+\beta_i(t)u\big|_{\Gamma_i}=0,\;i=1,2\}
\]
is dense in
\[
H_{\gamma }^{1,2}(] 0,T[ \times ] 0,1[ )
=\{ u\in H^{1,2}(] 0,T[ \times ] 0,1[
):\partial_xu+\beta_i(t)u\big|_{\Gamma_i}=0,\;i=1,2\}
\]
where $\Gamma_1=] 0,T[ \times \{ 0\} $ and
$\Gamma _2=] 0,T[ \times \{ 1\} $.
\end{lemma}

The above lemma is a particular case of \cite[Theorem 2.1]{Lions}.

\begin{remark}\label{remq1} \rm
We can replace in Lemma \ref{lem4}
$R=] 0,T[ \times ] 0,1[$ by $\Omega $ with the help of the change of
variables defined above.
\end{remark}

\subsection{Case of a triangular domain}
In this case, we define $\Omega $ by
\[
\Omega =\{ (t,x)\in\mathbb{R}^2:0<t<T,\varphi_1(t)<x<\varphi_2(t)\}
\]
with
\begin{equation}
\begin{gathered}
\varphi_1(0)=\varphi_2(0)\\
\varphi_1(T)<\varphi_2(T).
\end{gathered}  \label{3.2}
\end{equation}
We assume that the functions $(\varphi_i)_{i=1,2}$ satisfy
\begin{equation}
\varphi_i'(t)(\varphi_2(t)
-\varphi_1(t)) \to  0 \quad \text{as } t\to 0, \; i=1,2.  \label{3.3}
\end{equation}
For each $n\in \mathbb{N}$, we define $\Omega_{n}$ by
\[
\Omega_{n}=\{ (t,x)\in\mathbb{R}^2:a_{n}<t<T,\varphi_1(t)<x<\varphi_2(
t)\}
\]
where $(a_{n})_{n}$ is a decreasing sequence to zero. Thus, we
have
\begin{gather*}
\varphi_1(a_{n})<\varphi_2(a_{n}), \\
\varphi_1(T)<\varphi_2(T).
\end{gather*}
Setting $f_{n}=f\big|_{\Omega_{n}}$, where $f\in L^2(\Omega
)$, we denote $u_{n}\in H^{1,2}(\Omega_{n})$
the solution of \eqref{eP1} in $\Omega_{n}$
\begin{equation} \label{eP1n}
\begin{gathered}
\partial_{t}u_{n}-c^2(t)\partial_x^2u_{n}=f_{n}\quad \text{in }
L^2(\Omega_{n})\\
u_{n/t=a_{n}}=0 \\
\partial_xu_{n}+\beta_i(t)u_{n/\Gamma_{n,i}}=0,\quad
i=1,2,
\end{gathered}
\end{equation}
here $\Gamma_{n,i}=\{ (t,\varphi_i(t)),
a_{n}<t<T\} $, $i=1,2$, and $c$ is a bounded differentiable
coefficient depending on time such that
\begin{equation}
0<\alpha \leq c(t)c'(t)\leq \beta  \label{8}
\end{equation}
for every $t\in ]0,T[ $, where $\alpha $ and $\beta $ are two
constants. We also assume that
\begin{gather}
(\beta_1c^2)\text{ is an increasing function on }] 0,T[  \label{3.5a}
\\
(\beta_2c^2)\text{ is a decreasing function on }] 0,T[ .  \label{3.5b}
\end{gather}
Such a solution $u_{n}$ exists by Theorem \ref{theo1}.

\begin{theorem}\label{theo2}
There exists a constant $K>0$ independent of $n$ such that
\[
\| u_{n}\|_{H^{1,2}(\Omega_{n})}^2\leq K\| f_{n}\|_{L^2(\Omega
_{n})}^2\leq K\| f\|_{L^2(\Omega)}^2.
\]
\end{theorem}

To prove Theorem \ref{theo2}, we need some preliminary
results.

\begin{lemma}\label{lem6}
For every $\epsilon >0$ satisfying
$(\varphi_2(t)-\varphi_1(t))\leq \epsilon $, there
exists a constant $C>0$ independent of $n$, such that
\[
\| \partial_x^{j}u_{n}\|_{L^2(\Omega
_{n})}^2\leq C\epsilon ^{2(2-j)}\| \partial
_x^2u_{n}\|_{L^2(\Omega_{n})}^2, \quad
j=0,1.
\]
\end{lemma}

\begin{proof}
Replacing in Lemma \ref{lem2} $v$ by $u_{n}$ and
$] a,b[$ by $] \varphi_1(t),\varphi_2(t)[ $, for a
fixed $t$, we obtain
\begin{align*}
\int_{\varphi_1(t)}^{\varphi_2(t)
}(\partial_x^{j}u_{n})^2dx
& \leq C(\varphi_2( t)-\varphi_1(t))^{2(2-j)
}\int_{\varphi_1(t)}^{\varphi_2(t)}(
\partial_x^2u_{n})^2dx \\
& \leq  C\epsilon ^{2(2-j)}\int_{\varphi_1(t)
}^{\varphi_2(t)}(\partial_x^2u_{n})^2dx
\end{align*}
where $C$ is the constant of Lemma \ref{lem2}. Integrating with
respect to $t$, we obtain the desired estimates.
\end{proof}

\begin{proposition}\label{prop1}
There exists a constant $C>0$ independent of $n$ such that
\[
\| \partial_{t}u_{n}\|_{L^2(\Omega_{n})
}^2+\| \partial_x^2u_{n}\|_{L^2(\Omega
_{n})}^2\leq C\| f\|_{L^2(\Omega )}^2.
\]
\end{proposition}

Then Theorem \ref{theo2} is a direct consequence of Lemma
\ref{lem6} and Proposition \ref{prop1}, since $\epsilon $ is
independent of $n$.

\begin{proof}[Proposition \ref{prop1}]
  Thanks to the density results, Lemma
\ref{lem2} and Remark \ref{remq1}, it is sufficient to prove the
first part of the proposition (Relationship \eqref{9} below) in the
case when $u_{n}\in\{v\in
H^2(\Omega_{n}),\partial_xv+\beta_i(t)v\big|_{\Gamma_{n,i}}=0,i=1,2\}$
without assuming the Cauchy condition $u_{n/t=a_{n}}=0$.

For this end, we develop the inner product in $L^2(\Omega_{n})$
\begin{align*}
\| f_{n}\|_{L^2(\Omega_{n})}^2
& = \langle \partial_{t}u_{n}-c^2\partial_x^2u_{n},\partial
_{t}u_{n}-c^2\partial_x^2u_{n}\rangle \\
& =  \| \partial_{t}u_{n}\|_{L^2(\Omega
_{n})}^2+\| c^2.\partial_x^2u_{n}\|
_{L^2(\Omega_{n})}^2-2\langle \partial
_{t}u_{n},c^2\partial_x^2u_{n}\rangle .
\end{align*}
Calculating the last term of the previous relation, we obtain
\begin{align*}
\langle \partial_{t}u_{n},c^2\partial_x^2u_{n}\rangle
& = \int_{\Omega_{n}}\partial_{t}u_{n}.c^2\partial_x^2u_{n}
\,dt\,dx \\
& =  -\int_{\Omega_{n}}c^2\partial_x\partial
_{t}u_{n}.\partial_xu_{n}\,dt\,dx+\int_{\partial \Omega
_{n}}c^2\partial
_{t}u_{n}.\partial_xu_{n}\nu_xd\sigma .
\end{align*}
So,
\begin{align*}
&-2\langle \partial_{t}u_{n},c^2\partial
_x^2u_{n}\rangle\\
& =  \int_{\Omega_{n}}c^2\partial
_{t}(\partial_xu_{n})^2\,dt\,dx-2\int_{\partial \Omega
_{n}}c^2\partial_{t}u_{n}.\partial
_xu_{n}\nu_xd\sigma \\
& =  -\int_{\Omega_{n}}2cc'(\partial
_xu_{n})^2\,dt\,dx+\int_{\partial \Omega_{n}}c^2(
\partial_xu_{n})^2\nu_{t}d\sigma
 -2\int_{\partial \Omega_{n}}c^2\partial_{t}u_{n}.\partial
_xu_{n}\nu_xd\sigma \\
& =  \int_{\partial \Omega_{n}}c^2[(\partial
_xu_{n})^2\nu_{t}-2\partial_{t}u_{n}.\partial_xu_{n}\nu_x
]d\sigma
 -\int_{\Omega_{n}}2cc'(\partial_xu_{n})^2\,dt\,dx,
\end{align*}
where $\nu_{t},\nu_x$ are the components of the outward normal vector
at the boundary of $\Omega_{n}$. We shall rewrite the boundary
integral making use of the boundary conditions. On the part of the
boundary of $\Omega_{n}$ where $t=a_{n}$, we have $\nu_x=0$ and
$\nu_{t}=-1$. The corresponding boundary integral
\[
A_1=-\int_{\varphi_2(a_{n})}^{\varphi_1(
a_{n})}c^2(\partial_xu_{n})
^2dx=\int_{\varphi_1(a_{n})}^{\varphi_2(
a_{n})}c^2(\partial_xu_{n})^2dx \ge0.
\]
On the part of the boundary of $\Omega_{n}$ where $t=T$, we have
$\nu_x=0$ and $\nu_{t}=1$.
Accordingly the corresponding boundary integral
\[
A_2=\int_{\varphi_1(T)}^{\varphi_2(T)
}c^2(\partial_xu_{n})^2dx
\]
is nonnegative. On the parts of the boundary where
$x=\varphi_i(t)$, $i=1,2$, we have
\[
\nu_x=\frac{(-1)^{i}}{\sqrt{1+(\varphi_i')^2(t)}}, \quad
\nu_{t}=\frac{(-1)^{i+1}\varphi_i'(t)}{
\sqrt{1+(\varphi_i')^2(t)}}
\]
 and
$\partial_xu_{n}(t,\varphi_i(t))+\beta
_i(t)u_{n}(t,\varphi_i(t))=0$, $i=1,2$.
Consequently, the corresponding integral is
\begin{align*}
&\int_{a_{n}}^{T}c^2\varphi_1'(t)
[\partial_xu_{n}(t,\varphi_1(t))]
^2dt-2\int_{a_{n}}^{T}(\beta_1c^2)(t)
\partial_{t}u_{n}(t,\varphi_1(t)).u_{n}(
t,\varphi_1(t))dt\\
&-\int_{a_{n}}^{T}c^2\varphi_2'(t)
[\partial_xu_{n}(t,\varphi_2(t))]
^2dt+2\int_{a_{n}}^{T}(\beta_2c^2)(t)
\partial_{t}u_{n}(t,\varphi_2(t)).u_{n}(
t,\varphi_2(t))dt.
\end{align*}
By setting
\begin{gather*}
I_{n,k} =(-1)^{k+1}\int_{a_{n}}^{T}c^2\varphi
_{k}'(t)[\partial_xu_{n}(
t,\varphi _{k}(t))]^2dt,k=1,2, \\
J_{n,k} =(-1)^{k}2\int_{a_{n}}^{T}(\beta
_{k}c^2)(t)\partial_{t}u_{n}(t,\varphi
_{k}(t)).u_{n}(t,\varphi_{k}(t))dt,k=1,2,
\end{gather*}
we have
\begin{equation}
-2\langle \partial_{t}u_{n},c^2\partial
_x^2u_{n}\rangle \geq -| I_{n,1}|
-| I_{n,2}| -| J_{n,1}|
-| J_{n,2}| -\int_{\Omega_{n}}2cc'(\partial_xu_{n})^2\,dt\,dx.
 \label{9}
\end{equation}
\end{proof}

\subsection*{1. Estimation of $I_{n,k},k=1,2$}

\begin{lemma}\label{lem7}
There exists a constant $K>0$ independent of $n$ such that
\[
| I_{n,k}| \leq  K\epsilon \| \partial
_x^2u_{n}\|_{L^2(\Omega_{n})}^2, \quad k=1,2.
\]
\end{lemma}

\begin{proof}
We convert the boundary integral $I_{n,1}$ into a surface integral
by setting
\begin{align*}
[\partial_xu_{n}(t,\varphi_1(t))]^2
& =  -\frac{\varphi_2(t)
-x}{\varphi_2(t)-\varphi_1(t)}[
\partial_xu_{n}(t,x)]^2\big|_{x=\varphi_1(t)
}^{x=\varphi_2(t)} \\
&= -\int_{\varphi_1(t)}^{\varphi_2(
t)} \frac{\partial }{\partial x}\{ \frac{\varphi
_2(t)-x}{ \varphi_2(t)-\varphi_1(t)}[\partial
_xu_{n}(t,x)]^2\} dx\\
&= -2\int_{\varphi_1(t)}^{\varphi_2(
t)} \frac{\varphi_2(t)-x}{\varphi_2(
t)-\varphi_1(t)}\partial_xu_{n}(
t,x)\partial_x^2u_{n}(t,x)dx \\
&\quad +\int_{\varphi_1(t)}^{\varphi_2(t)
} \frac{1}{\varphi_2(t)-\varphi_1(t)}[
\partial_xu_{n}(t,x)]^2dx.
\end{align*}
Then
\begin{align*}
I_{n,1} & =  \int_{a_{n}}^{T}c^2(t)\varphi
_1'(t)[\partial_xu_{n}(t,\varphi_1(t))]^2dt \\
& =  \int_{\Omega_{n}}c^2(t)\frac{\varphi
_1'(t)}{\varphi_2(t)-\varphi_1(t)
}(\partial_xu_{n})^2\,dt\,dx \\
&\quad -2\int_{\Omega_{n}}c^2(t)\frac{\varphi_2(
t)-x}{\varphi_2(t)-\varphi_1(t)}
\varphi_1'(t)(\partial_xu_{n})
(\partial_x^2u_{n})\,dt\,dx.
\end{align*}
Thanks to Lemma \ref{lem6}, we can write
\[
\int_{\varphi_1(t)}^{\varphi_2(t)
}[
\partial_xu_{n}(t,x)]^2dx \leq  C[\varphi
_2(t)-\varphi_1(t)]
^2\int_{\varphi_1(t)}^{\varphi_2(t)}[\partial
_x^2u_{n}(t,x)]^2dx.
\]
Therefore,
\[
\int_{\varphi_1(t)}^{\varphi_2(t)}[
\partial_xu_{n}(t,x)]^2\frac{| \varphi_1'| }{\varphi_2-\varphi_1}dx
\leq C| \varphi_1'| [\varphi
_2-\varphi_1]\int_{\varphi_1(t)
}^{\varphi_2(t)}[\partial_x^2u_{n}(
t,x)]^2dx,
\]
consequently,
\begin{align*}
| I_{n,1}|
& \leq  C\int_{\Omega _{n}}c^2(t)| \varphi_1'|
[\varphi _2-\varphi_1](\partial_x^2u_{n})^2\,dt\,dx
 +2\int_{\Omega_{n}}c^2(t)| \varphi
_1'| | \partial _xu_{n}|
| \partial_x^2u_{n}| \,dt\,dx,
\end{align*}
since $| \frac{\varphi_2(t)-x}{\varphi_2(
t)-\varphi_1(t)}| \leq 1$. So, for all $
\epsilon >0$, we have
\begin{align*}
| I_{n,1}| & \leq  C\int_{\Omega
_{n}}c^2(t)| \varphi_1'| [\varphi
_2-\varphi_1](\partial_x^2u_{n})^2\,dt\,dx \\
& \quad +\int_{\Omega_{n}}\epsilon c^2(t)(
\partial_x^2u_{n})^2\,dt\,dx+\frac{1}{\epsilon
}\int_{\Omega_{n}}c^2(t)(\varphi
_1')^2(
\partial_xu_{n})^2\,dt\,dx.
\end{align*}
Lemma \ref{lem6} yields
\[
\frac{1}{\epsilon }\int_{\Omega_{n}}c^2(t)(
\varphi_1')^2(\partial_xu_{n})^2\,dt\,dx
 \leq  C\frac{1}{\epsilon }\int_{\Omega
_{n}}c^2(t)(\varphi_1')
^2[\varphi_2-\varphi_1]
^2(\partial_x^2u_{n})^2\,dt\,dx.
\]
Thus, there exists a constant $M>0$ independent of $n$ such that
\begin{align*}
| I_{n,1}|
& \leq  C\int_{\Omega
_{n}}c^2(t)[| \varphi_1'| | \varphi_2-\varphi_1|
+\frac{1}{\epsilon }(\varphi_1')
^2| \varphi_2-\varphi_1|
^2](\partial_x^2u_{n})^2\,dt\,dx
\\
&\quad +\int_{\Omega_{n}}c^2(t)\epsilon (
\partial_x^2u_{n})^2\,dt\,dx
\\
& \leq  M\epsilon \int_{\Omega_{n}}(\partial
_x^2u_{n})^2\,dt\,dx,
\end{align*}
because $| \varphi_1'(\varphi _2-\varphi_1)| \leq \epsilon $ and
$c^2(t)$ is bounded. The inequality
\[
| I_{n,2}|  \leq  K\epsilon \| \partial
_x^2u_{n}\|_{L^2(\Omega_{n})}^2
\]
can be proved by a similar argument.

\subsection*{Estimation of $J_{n,k}$, $k=1,2$}

We have
\begin{align*}
J_{n,1} & =  -2\int_{a_{n}}^{T}(\beta_1c^2)(
t)\partial_{t}u_{n}(t,\varphi_1(t)
).u_{n}(t,\varphi_1(t))dt
\\
& =  -\int_{a_{n}}^{T}(\beta_1c^2)(t)
[\partial_{t}u_{n}^2(t,\varphi_1(t))] dt.
\end{align*}
By setting
$ h(t)=u_{n}^2(t,\varphi_1(t))$,
we obtain
\begin{align*}
J_{n,1}
& =  -\int_{a_{n}}^{T}\beta_1c^2.
\big[h'(t)-\varphi_1'(t)\partial
_xu_{n}^2(t,\varphi_1(t))\big]dt \\
& =   -\beta_1c^2.h(t)\big]_{a_{n}}^{T}+
\int_{a_{n}}^{T}(\beta_1c^2)'.h(
t)dt+\int_{a_{n}}^{T}\beta_1c^2.\varphi_1'(t)
\partial_xu_{n}^2(t,\varphi_1(t))dt.
\end{align*}
Thanks to \eqref{2}, \eqref{3.5a} and the fact that
$u_{n}^2(a_{n},\varphi_1(a_{n}))=0$, we have
\[
 -\beta_1c^2.h(t)\big]_{a_{n}}^{T}+
\int_{a_{n}}^{T}(\beta_1c^2)'.h(
t)dt\geq 0.
\]
The last boundary integral in the expression of $J_{n,1}$ can be
treated by a similar argument used in Lemma \ref{lem7}. So, we
obtain the existence of a
positive constant $K$ independent of $n$, such that
\begin{equation}
\big| \int_{a_{n}}^{T}\beta_1c^2.\varphi_1'(t)\partial_xu_{n}^2(t,\varphi_1(
t))dt\big| \leq K\epsilon \| \partial
_x^2u_{n}\|_{L^2(\Omega_{n})}^2.
  \label{3.7}
\end{equation}
By a similar method, we obtain
\begin{align*}
J_{n,2} & =   \beta_2(t)c^2(t)
u_{n}^2(t,\varphi_2(t))\big]
_{a_{n}}^{T}-\int_{a_{n}}^{T}(\beta_2c^2)
'.u_{n}^2(t,\varphi_2(t))dt \\
& \quad -\int_{a_{n}}^{T}\beta_2c^2.\varphi_2'(t)
\partial_xu_{n}^2(t,\varphi_2(t))dt.
\end{align*}
Thanks to \eqref{2}, \eqref{3.5b} and the fact that
$u_{n}^2(a_{n},\varphi_2(a_{n}))=0$, we have
\[
 \beta_2(t)c^2(t)u_{n}^2(
t,\varphi_2(t))\big]_{a_{n}}^{T}-\int_{a_{n}}^{T}
(\beta_2c^2)'.u_{n}^2(t,\varphi
_2(t))dt\geq 0.
\]
Then
\begin{equation}
| -\int_{a_{n}}^{T}\beta_2c^2.\varphi_2'(t)\partial_xu_{n}^2(t,\varphi_2(
t))dt| \leq K\epsilon \| \partial
_x^2u_{n}\|_{L^2(\Omega_{n})}^2  \label{3.8}
\end{equation}
where $K$ is a positive constant independent of $n$.

Now, we can complete the proof of Proposition \ref{prop1}. Summing
up the estimates \eqref{8}, \eqref{9}, \eqref{3.7} and \eqref{3.8},
and making use of Lemma \ref{lem6}, we then
obtain
\begin{align*}
&\| f_{n}\|_{L^2(\Omega_{n})}^2\\
&\geq  \| \partial_{t}u_{n}\|_{L^2(\Omega
_{n})}^2+\| c^2\partial_x^2u_{n}\|
_{L^2(\Omega_{n})}^2-K\epsilon \|
\partial_x^2u_{n}\|_{L^2(\Omega_{n})}^2  -K_2\epsilon\|
\partial_x^2u_{n}\|
_{L^2(\Omega_{n})}^2
\\
& \geq  \| \partial_{t}u_{n}\|_{L^2(
\Omega_{n})}^2+(d_1^2-K\epsilon
-K_2\epsilon)\|
\partial_x^2u_{n}\|_{L^2(\Omega_{n})}^2
\end{align*}
where $K_2$ is a positive number. Then, it is sufficient to choose
$\epsilon $ such
that $(d_1^2-K\epsilon -K_2\epsilon)>0$, to get a constant
$K_0>0$ independent of $n$ such that
\[
\| f_{n}\|_{L^2(\Omega_{n})}^2
\geq K_0(\| \partial_{t}u_{n}\|_{L^2(\Omega
_{n})}^2+\| \partial_x^2u_{n}\|_{L^2(\Omega_{n})}^2).
\]
However,
\[
\| f_{n}\|_{L^2(\Omega_{n})}  \leq
\| f\|_{L^2(\Omega )},
\]
then, there exists a constant $C>0$, independent of $n$ satisfying
\[
\| \partial_{t}u_{n}\|_{L^2(\Omega_{n})
}^2+\| \partial_x^2u_{n}\|_{L^2(\Omega
_{n})}^2\leq C\| f_{n}\|_{L^2(\Omega
_{n})}^2\leq C\| f\|_{L^2(\Omega )
}^2.
\]
This completes  the proof of Proposition \ref{prop1}.
\end{proof}

\subsection*{Passage to the limit}

We are now in position to prove the first main result of this work.

\begin{theorem}\label{theo3}
Assume that the following conditions are satisfied
\begin{itemize}
\item[(1)] $(\varphi_i)_{i=1,2}$ fulfil the assumptions \eqref{3.2}
 and \eqref{3.3}.

\item[(2)] the coefficient $c$ verifies the conditions \eqref{5} and
\eqref{8}.

\item[(3)] $(\beta_i)_{i=1,2}$ fulfil the conditions \eqref{2},
\eqref{2.1}, \eqref{2.2} and \eqref{2.3}.

\item[(4)] $(\varphi_i,\beta_i,c)_{i=1,2}$ fulfil the
conditions \eqref{1}, \eqref{3.5a} and \eqref{3.5b}.
\end{itemize}
Then, for $T$ small enough, \eqref{eP'} admits a (unique) solution
$u$ belonging to
\[
H_{\gamma }^{1,2}(\Omega )=\{ u\in H^{1,2}(
\Omega );(\partial_xu+\beta_i(t)
u)\big|_{\Gamma_i}=0,i=1,2\} ,
\]
where $\Gamma_i$, $i=1,2$ are the parts of the boundary of $\Omega $
where $x=\varphi_i(t)$.
\end{theorem}

\begin{proof}
Choose a sequence $(\Omega_{n})_{n\in \mathbb{N}}$ of
the domains defined above, such that $\Omega_{n}\subseteq \Omega $
with $(a_{n})$ a decreasing sequence to $0$, as
$n\to \infty $. Then, we have $\Omega_{n}\to \Omega$, as
$n\to \infty $.

Consider the solution $u_{n}\in H^{1,2}(\Omega_{n})$
of the Robin boundary value problem
\begin{gather*}
\partial_{t}u_{n}-c^2(t)\partial_x^2u_{n}=f_{n} \quad
\text{in }\Omega_{n} \\
u_{n/t=a_{n}}=0 \\
\partial_xu_{n}+\beta_i(t)u_n\big|_{\Gamma_{n,i}}=0,\quad i=1,2,
\end{gather*}
where $\Gamma_{n,i}$ are the parts of the boundary of $\Omega_{n}$ where
$ x=\varphi_i(t)$, $i=1,2.$\ Such a solution $u_{n}$
exists by Theorem \ref{theo1}. Let $\widetilde{u_{n}}$ the
$0-$extension of $u_{n}$ to $\Omega $. In virtue of Theorem
\ref{theo2}, we know that there exists a constant $K>0$ such that
\[
\| \widetilde{u_{n}}\|_{L^2(\Omega )
}^2+\| \widetilde{\partial_{t}u_{n}}\|_{L^2(
\Omega )}^2+\| \widetilde{\partial_xu_{n}}\|
_{L^2(\Omega )}^2+\| \widetilde{\partial
_x^2u_{n}}\|_{L^2(\Omega )}^2\leq K\|
f\|_{L^2(\Omega )}^2.
\]
This means that $\widetilde{u_{n}}$, $\widetilde{\partial
_{t}u_{n}}$, $ \widetilde{\partial_x^{j}u_{n}}$, for $j=1,2$ are
bounded functions in $ L^2(\Omega )$. So, for a
suitable increasing sequence of integers $n_{k}$, $k=1,2,\dots$, there
exist functions
$u,v$  and $v_{j}$, $j=1,2$ in $L^2(\Omega )$ such that
\begin{gather*}
\widetilde{u_{n_{k}}}  \rightharpoonup  u \quad \text{weakly in }L^2(
\Omega ),\; k\to \infty \\
\widetilde{\partial_{t}u_{n_{k}}}  \rightharpoonup  v \quad
\text{weakly in } L^2(\Omega ), \; k\to \infty \\
\widetilde{\partial_x^{j}u_{n_{k}}}  \rightharpoonup  v_{j} \quad
\text{weakly in }L^2(\Omega ), \; k\to \infty ,\; j=1,2.
\end{gather*}
Clearly
$v=\partial_{t}u$, $v_1=\partial_xu$ and
$v_2=\partial_x^2u$
in the sense of distributions in $\Omega $. So,
$u\in H^{1,2}(\Omega )$ and
\[
\partial_{t}u-c^2(t)\partial_x^2u=f\quad \text{in } \Omega .
\]
On the other hand, the solution $u$ satisfies the boundary
conditions
\[
\partial_xu+\beta_i(t)u\big|_{\Gamma_i}=0,\quad i=1,2,
\]
since
for all $n\in \mathbb{N}$, $u\big|_{\Omega_{n}}=u_{n}$.
This proves the existence of solution to \eqref{eP'}.

The uniqueness of the solution is easy to check, thanks to the
hypothesis \eqref{1}.
\end{proof}

\section{The case of an arbitrary $T$}

Assume that $\Omega$ satisfies \eqref{3.2}. In the case where $T$ is
not in the neighborhood of zero, we set $\Omega =D_1\cup D_2\cup
\Gamma_{T_1}$ where
\begin{gather*}
D_1=\{ (t,x)\in \mathbb{R}^2:0<t<T_1,\varphi
_1(t)<x<\varphi_2(t)\}
\\
D_2=\{ (t,x)\in \mathbb{R}^2:T_1<t<T,\varphi
_1(t)<x<\varphi_2(t)\}
\\
\Gamma_{T_1}=\{ (T_1,x)
\in\mathbb{R}^2:\varphi_1(T_1)<x<\varphi
_2(T_1)\}
\end{gather*}
with $T_1$ small enough.

In the sequel, $f$ stands for an arbitrary fixed element of
$L^2(\Omega )$ and $f_i=f\big|_{D_i}$, $i=1,2$.

Theorem \ref{theo3} applied to the triangular domain $D_1$, shows
that there exists a unique solution $u_1\in H^{1,2}(D_1)$ of the problem
\begin{equation} \label{ePD1}
\begin{gathered}
\partial_{t}u_1-c^2(t)\partial_x^2u_1=f_1\quad\text{in }L^2(D_1)\\
\partial_xu_1+\beta_i(t)u_{1/\Gamma_{i,1}}=0, \quad
i=1,2,
\end{gathered}
\end{equation}
where $\Gamma_{i,1}$ are the parts of the boundary of $D_1$, and
$x=\varphi_i(t)$, $i=1,2$.

\begin{lemma}\label{lem8}
If $u\in H^{1,2}(] 0,T[ \times ] 0,1[)$, then
$u\big|_{t=0}\in H^{1}(\gamma_0)$,
$u\big|_{x=0}\in H^{\frac{3}{ 4}}(\gamma_1)$ and
$u\big|_{x=1}\in H^{\frac{3}{4}}(\gamma_2)$, where
$\gamma_0=\{ 0\} \times ] 0,1[ $,
$\gamma _1=] 0,T[ \times \{ 0\} $ and
$\gamma _2=] 0,T[ \times \{ 1\} $.
\end{lemma}

The above lemma is a particular case of \cite[Theorem 2.1, Vol.2]{Lions}.
The transformation
\[
(t,x)\longmapsto (t',x')
=(t,(\varphi_2(t)-\varphi_1(t))x+\varphi_1(t))
\]
leads to the following lemma.

\begin{lemma}\label{lem9}
If $u\in H^{1,2}(D_2)$, then $u\big|_{\Gamma_{T_1}}\in
H^{1}(\Gamma_{T_1})$, $u\big|_{x=\varphi_1(
t)}\in H^{\frac{3}{4}}(\Gamma_{1,2})$ and
$u\big|_{x=\varphi_2(t)}\in H^{\frac{3}{4}}(\Gamma
_{2,2})$, where $\Gamma_{i,2}$ are the parts of the boundary
of $D_2$ where $x=\varphi_i(t)$, $i=1,2$.
\end{lemma}

Hereafter, we denote the trace $u_1\big|_{\Gamma_{T_1}}$ by
$\psi $ which is in the Sobolev space $H^{1}(\Gamma_{T_1})$
because $u_1\in H^{1,2}(D_1)$ (see Lemma \ref{lem9}).

Now, consider the following problem in $D_2$
\begin{equation} \label{ePD2}
\begin{gathered}
\partial_{t}u_2-c^2(t)\partial_x^2u_2=f_2
\quad\text{in }L^2(D_2)\\
u_2\big|_{\Gamma_{T_1}}=\psi \\
\partial_xu_2+\beta_i(t)u_{2/\Gamma_{i,2}}=0, \quad
i=1,2,
\end{gathered}
\end{equation}
where $\Gamma_{i,2}$ are the parts of the boundary of $D_2$, and
$x=\varphi _i(t)$, $i=1,2$.

We use the following result, which is a consequence of
\cite[Theorem 4.3, Vol.2]{Lions} to solve \eqref{ePD2}.

\begin{proposition}\label{prop2}
Let $Q$ be the rectangle $] 0,T[ \times ] 0,1[$,
$ f\in L^2(Q)$ and $\psi \in H^{1}(\gamma_0)$.
Then the problem
\begin{gather*}
\partial_{t}u-c^2(t)\partial_x^2u=f\quad \text{in }Q \\
u\big|_{\gamma_0}=\psi \\
\partial_xu+\beta_i(t)
u\big|_{\gamma_i}=0,\quad i=1,2,
\end{gather*}
where $\gamma_0=\{ 0\} \times ] 0,1[ $,
$\gamma_1=] 0,T[ \times \{ 0\} $ and
$\gamma_2=] 0,T [ \times \{ 1\} $, admits a
(unique) solution $u\in H^{1,2}(Q)$.
\end{proposition}

\begin{remark}\label{remq2} \rm
In the application of \cite[Theorem 4.3, Vol 2]{Lions}, we can
observe that there are no compatibility conditions to satisfy
because $\partial_x\psi$ is only in $L^2(\gamma_0)$.
\end{remark}

Thanks to the transformation
\[
(t,x)\longmapsto (t,y)=(t,(\varphi
_2(t)-\varphi_1(t))x+\varphi_1(t)),
\]
we deduce the following result.

\begin{proposition}\label{prop3}
Problem \eqref{ePD2} admits a (unique) solution
$u_2\in H^{1,2}(D_2)$.
\end{proposition}

So, the function
\[
u=\begin{cases}
u_1& \text{in }D_1 \\
u_2& \text{in }D_2
\end{cases}
\]
is the (unique) solution of \eqref{eP'} for an
arbitrary $T$.
Our second main result is as follows.

\begin{theorem}\label{theo4}
Assume that the following conditions are satisfied
\begin{itemize}
\item[(1)]  $(\varphi_i)_{i=1,2}$ satisfies  assumptions \eqref{3.2}
 and \eqref{3.3}.

\item[(2)] the coefficient $c$ satisfies conditions \eqref{5} and
\eqref{8}.

\item[(3)] $(\beta_i)_{i=1,2}$ fulfil the conditions \eqref{2},
\eqref{2.1}, \eqref{2.2} and \eqref{2.3}.

\item[(4)]  $(\varphi_i,\beta_i,c)_{i=1,2}$ fulfil the
conditions \eqref{1}, \eqref{3.5a} and \eqref{3.5b}.
\end{itemize}
Then, \eqref{eP'} admits a (unique) solution $u$ belonging to
\[
H_{\gamma }^{1,2}(\Omega )=\{ u\in H^{1,2}(\Omega );
(\partial_xu+\beta_i(t)u)\big|_{\Gamma_i}=0,i=1,2\} ,
\]
where $\Gamma_i$, $i=1,2$ are the parts of the boundary of
$\Omega $ where $x=\varphi_i(t)$.
\end{theorem}

\begin{remark}\label{remq3} \rm
Using the same method in the case where $\varphi_1(T)
=\varphi_2(T)$ we can obtain a result similar to
Theorem \ref{theo4}.
\end{remark}

\subsection*{Acknowledgment}
The authors are thankful to the anonymous referee for
his/her careful reading of a previous version of the manuscript,
which led to a substantial improvement of this manuscript.

\begin{thebibliography}{00}

\bibitem{Bes} V. Besov;
\emph{The continuation of function in $L_{p}^{1}$ and
$W_{p}^{1}$}, Proc. Steklov Inst. Math. 89 (1967),
5--17.

\bibitem{Hof} S. Hofmann, J. L. Lewis;
\emph{The $L^{p}$ regularity problems for
the heat equation in non-cylindrical domains}, J. of Functional
Analysis 220 (2005), 1--54.

\bibitem{lab1} R. Labbas, A. Medeghri, B.-K. Sadallah;
\emph{Sur une \'{e}quation parabolique dans un domaine
non cylindrique}, C.R.A.S, Paris, 335
(2002), 1017--1022.

\bibitem{Lab1} R. Labbas, A. Medeghri, B.-K. Sadallah;
\emph{On a parabolic equation in a triangular domain},
Applied Mathematics and Computation
130 (2002), 511--523.

\bibitem{Lab2} R. Labbas, A. Medeghri, B.-K. Sadallah;
\emph{An $L^{p}$ approach for the study of degenerate parabolic equation},
Electron. J. Diff. Equ., vol. 2005,  no. 36, (2005), 1--20.

\bibitem{Lions} J. L. Lions, E. Magenes;
\emph{Probl\`{e}mes aux limites non homog\`{e}nes et applications},
$1,2$, Dunod, Paris (1968).

\bibitem{Sad1} B.-K. Sadallah;
\emph{Etude d'un probl\`{e}me $2m-$parabolique dans
des domaines plan non rectangulaires}, Boll. Un. Mat. Ital.,
(5), 2-B (1983), 51--112.

\bibitem{Sad2} B. K. Sadallah;
\emph{Regularity of a parabolic equation solution
in a nonsmooth and unbounded domain}, J. Aust. Math. Soc., 84 (2)
(2008),  265--276.

\bibitem{Sad3} B. K. Sadallah;
\emph{A remark on a parabolic problem in a
sectorial domain}, Applied Mathematics E-Notes, 8 (2008),
263-270.

\bibitem{Sav} G. Savar\'{e};
\emph{Parabolic problems with mixed variable lateral
conditions: an abstract approach}, J. Math. Pures et Appl. 76
(1997), 321-351.

\end{thebibliography}

\end{document}