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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 116, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/116\hfil Resolutions of parabolic equations]
{Resolutions of parabolic equations in non-symmetric conical domains}

\author[A. Kheloufi\hfil EJDE-2012/116\hfilneg]
{Arezki Kheloufi}  % in alphabetical order

\address{Arezki Kheloufi \newline
Department of Technology, Faculty of Technology\\
B\'ejaia University, 6000 B\'ejaia, Algeria}
\email{arezkinet2000@yahoo.fr}

\thanks{Submitted May 31, 2012. Published July 9, 2012.}
\subjclass[2000]{35K05, 35K20}
\keywords{Parabolic equation; conical domain; anisotropic Sobolev space}

\begin{abstract}
 This article is devoted to the analysis of a two-space dimensional
 linear parabolic equation, subject to Cauchy-Dirichlet boundary
 conditions. The problem is set in a conical type domain and the
 right hand side term of the equation is taken in a Lebesgue space.
 One of the main issues of this work is that the domain can possibly
 be non regular. This work is an extension of the symmetric case
 studied in Sadallah \cite{Sadallah2011}.
\end{abstract}

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

\section{Introduction}

Let $Q$ be an open set of $ \mathbb{R}^3$ defined by
\[
Q=\{ (t,x_1,x_2) \in \mathbb{R} ^3:(x_1,x_2) \in \Omega_t,0<t<T\}
\]
where $T$ is a finite positive number and for a fixed $t$ in the
interval $] 0,T[ $, $\Omega_t$ is a bounded domain of
$ \mathbb{R} ^2$ defined by
\[
\Omega_t=\{ (x_1,x_2) \in \mathbb{R}^2:0
\leq \frac{x_1^2}{\varphi ^2(t) }+\frac{
x_2^2}{h^2(t)\varphi ^2(t) }<1\} .
\]
Here, $\varphi $ is a continuous real-valued function defined on
$[0,T ] $, Lipschitz continuous on $[0,T] $
and such that
\[
\varphi (0) =0,\quad \varphi (t) >0
\]
for every $t\in $ $] 0,T] $. $h$ is a Lipschitz
continuous real-valued function defined on $[0,T] $,
such that
\begin{equation}
0<\delta \leq h(t)\leq \beta   \label{HCondition}
\end{equation}
for every $t\in $ $[0,T] $, where $\delta $ and $\beta $
are positive constants.

In $Q$, we consider the boundary-value problem
\begin{equation}
\begin{gathered}
\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\in
L^2(Q) , \\
u\big|_{\partial Q-\Gamma_T}=0,
\end{gathered}   \label{leProbleme}
\end{equation}
where $L^2(Q)$ is the usual Lebesgue space on $Q$, $\partial Q$ is
the boundary of $Q$ and $\Gamma_T$ is the part of the boundary of
$Q$ where $t=T$.

The difficulty related to this kind of problems comes from this
singular situation for evolution problems; i.e., $\varphi $ is
allowed to vanish for $ t=0$, which prevents the domain $Q$  from being
transformed into a regular domain without the appearance of some
degenerate terms in the parabolic equation, see for example Sadallah
\cite{Sadallah1983}. In order to overcome this difficulty, we impose
a sufficient condition on the function $\varphi $; that is,
\begin{equation}
\varphi'(t) \varphi (t) 
\to  0 \quad \text{as }t\to 0,  \label{PhiCond1}
\end{equation}
and we obtain existence and regularity results for Problem
\eqref{leProbleme} by using the domain decomposition method. More
precisely, we will prove that Problem \eqref{leProbleme} has a
solution with optimal regularity, that is a solution $u$ belonging
to the anisotropic Sobolev space
\[
H_0^{1,2}(Q) :=\{ u\in H^{1,2}(Q)
:u\big|_{\partial Q-\Gamma_T}=0\} ,
\]
with
\[
H^{1,2}(Q) =\{ u\in L^2(Q) :\partial
_tu,\partial_{x_1}^{j}u,\partial_{x_2}^{j}u,\partial
_{x_1}\partial_{x_2} u\in L^2(Q) ,j=1,2\}.
\]
In Sadallah \cite{Sadallah2011} the same problem has been studied in
the case of a symmetric conical domain; i.e., in the case where
$h=1$. Further references on the analysis of parabolic problems in
non-cylindrical domains are:
 Alkhutov \cite{Alkhutov2002,Alkhutov2007}, Degtyarev \cite{Degtyarev2010},
Labbas, Medeghri and Sadallah \cite{Labbas2002,Labbas2005},
Sadallah \cite{Sadallah1983}. There are many other works concerning boundary-value problems in non-smooth domains (see, for example, Grisvard
\cite{Grisvard1985} and the references therein).

The organization of this article is as follows. In Section 2, first we
prove an uniqueness result for Problem \eqref{leProbleme}, then we
derive some technical lemmas which will allow us to prove an uniform
estimate (in a sense to be defined later). In Section 3, there are
two main steps. First, we prove that Problem \eqref{leProbleme}
admits a (unique) solution in the case of a domain which can be
transformed into a cylinder. Secondly, for $T$ small enough, we
prove that the result holds true in the case of a conical domain
under the above mentioned assumptions on functions $\varphi $ and
$h$. The method used here is based on the approximation of the
conical domain by a sequence of subdomains $(Q_n)_n$ which can be transformed into regular domains (cylinders). We
establish an uniform estimate of the type
\[
\| u_n\|_{H^{1,2}(Q_n) }\leq
K\| f\|_{L^2(Q_n) },
\]
where $u_n$ is the solution of Problem \eqref{leProbleme} in
$Q_n$ and $K $ is a constant independent of $n$. This allows us to
pass to the limit. Finally, in Section 4 we complete the proof of
our main result (Theorem \ref{thm3}).

\section{Preliminaries}

\begin{proposition}\label{Proposition4.1}
Problem \eqref{leProbleme} is uniquely solvable.
\end{proposition}

\begin{proof}
Let us consider $u\in H_0^{1,2}(\Omega ) $ a solution
of Problem \eqref{leProbleme} with a null right-hand side term. So,
\[
\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=0\quad\text{in }Q.
\]
In addition $u$ fulfils the boundary conditions
\[
u\big|_{\partial Q-\Gamma_T}=0.
\]
Using Green formula, we have
\begin{align*}
\int_Q(\partial_tu-\partial_{x_1}^2u-\partial
_{x_2}^2u) u\,dt\,dx_1\,dx_2 
&=\int_{\partial Q}(\frac{1}{2}| u| ^2\nu
_t-\partial_{x_1}u.u\nu_{x_1}-\partial_{x_2}u.u\nu
_{x_2}) d\sigma  \\
&\quad +\int_Q(| \partial_{x_1}u|^2+|\partial_{x_2}u| ^2) dt\,dx_1\,dx_2
\end{align*}
where $\nu_t$, $\nu_{x_1}$, $\nu_{x_2}$ are the components
of the unit outward normal vector at $\partial Q$. Taking into
account the boundary conditions, all the boundary integrals vanish
except $\int_{\partial Q}| u| ^2\nu_t\,d\sigma $. We have
\[
\int_{\partial Q}| u| ^2\nu_td\sigma
=\int_{\Gamma_T}| u| ^2\,dx_1\,dx_2.
\]
Then
\begin{align*}
&\int_Q(\partial_tu-\partial_{x_1}^2u-\partial
_{x_2}^2u) u\,dt\,dx_1\,dx_2 \\
&=\int_{\Gamma_T}\frac{1}{2}| u| ^2\,dx_1\,dx_2
+\int_Q(| \partial_{x_1}u|^2+|
\partial_{x_2}u| ^2) dt\,dx_1\,dx_2.
\end{align*}
Consequently,
\[
\int_Q(\partial_tu-\partial_{x_1}^2u-\partial
_{x_2}^2u) u\,dt\,dx_1\,dx_2=0
\]
yields
\[
\int_Q(| \partial_{x_1}u|
^2+|
\partial_{x_2}u| ^2) dt\,dx_1\,dx_2= 0,
\]
because
\[
\frac{1}{2}\int_{\Gamma_T}| u| ^2\,dx_1\,dx_2\geq 0 .
\]
This implies  $| \partial_{x_1}u|^2+|
\partial_{x_2}u| ^2=0$ and consequently
$\partial_{x_1}^2u=\partial_{x_2}^2u=0$. Then, the hypothesis
$\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=0$
gives $\partial_tu=0$. Thus, $u$ is constant. The boundary
conditions imply that $u=0$ in $Q$. This proves the uniqueness of
the solution of Problem \eqref{leProbleme}.
\end{proof}

\begin{remark} \rm
In the sequel, we will be interested only by the question of the
existence of the solution of Problem \eqref{leProbleme}.
\end{remark}

The following result is well known (see, for example, \cite{Lions1968})

\begin{lemma}\label{Lemma4.1}
Let $D(0,1) $ be the unit disc of
$\mathbb{R} ^2$. Then, the Laplace operator $\Delta :H^2(
D(0,1) ) \cap H_0^{1}(D(0,1)) \to L^2(D(0,1) ) $ is
an isomorphism. Moreover, there exists a constant $C>0$ such that
\[
\| v\|_{H^2(D(0,1) )}\leq C\| \Delta v\|_{L^2(D(
0,1) ) } \text{, }\forall v\in H^2(D(0,1) ) .
\]
\end{lemma}

In the above lemma, $H^2$ and $H_0^{1}$ are the usual Sobolev
spaces defined, for instance, in Lions-Magenes \cite{Lions1968}. In
section 3, we will need the following result.

\begin{lemma}
\label{lem5}Let $t\in] \alpha_n,T[ $, where $(
\alpha_n)_n$ is a decreasing sequence to zero. Then,
there exists a constant $C>0$ independent of $n$ such that for each
$u_n\in H^2(\Omega_t) $, we have
\begin{itemize}
\item[(a)] $\| \partial_{x_1}u_n\|_{L^2(
\Omega_t) }^2\leq C\varphi ^2(t) \|
\Delta u_n\|_{L^2(\Omega_t) }^2$,

\item[(b)] $\| \partial_{x_2}u_n\|_{L^2(
\Omega_t) }^2\leq C\varphi ^2(t) \|
\Delta u_n\|_{L^2(\Omega_t) }^2$.
\end{itemize}
\end{lemma}

\begin{proof}
It is a direct consequence of Lemma \ref{Lemma4.1}. Indeed, let
$t\in] \alpha_n,T[ $ and define the following change
of variables
\[
\begin{array}{lll}
D(0,1) & \to & \Omega_t \\
(x_1,x_2) & \mapsto & (\varphi (
t) x_1,h(t) \varphi (t) x_2)
=(x_1',x_2').
\end{array}
\]
Set
\[
v(x_1,x_2) =u_n(\varphi (t) x_1,h(t) \varphi (t) x_2) ,
\]
then if $v\in H^2(D(0,1) ) $, $u_n$
belongs to $ H^2(\Omega_t) $.

(a) We have
\begin{align*}
\| \partial_{x_1}v\|_{L^2(D(
0,1) ) }^2 &=  \int_{D(0,1) }(
\partial_{x_1}v) ^2(x_1,x_2) \,dx_1\,dx_2 \\
&=  \int_{\Omega_t}(\partial_{x_1'}u_n) ^2(x_1',x_2')
\varphi ^2(t) \frac{1}{h(t)\varphi ^2(t) }dx_1'dx_2' \\
&=  \frac{1}{h(t)}\int_{\Omega_t}(\partial_{x_1'}u_n) ^2(x_1',x_2')
\,dx_1'\,dx_2' \\
&=  \frac{1}{h(t)}\| \partial_{x_1'}u_n\|_{L^2(\Omega_t) }^2.
\end{align*}
On the other hand,
\begin{align*}
\| \Delta v\|_{L^2(D(0,1)) }^2
&=  \int_{D(0,1) }[(\partial_{x_1}^2v+\partial_{x_2}^2v) (x_1,x_2) ]
^2\,dx_1\,dx_2 \\
&=  \int_{\Omega_t}(\varphi ^2(t) \partial
_{x_1'}^2u_n+(h\varphi)^2(t)
\partial_{x_2'}^2u_n) ^2(x_1',x_2') \text{ }\frac{dx_1'dx_2'}{(h\varphi^2)(
t) } \\
&=  \frac{\varphi ^2(t) }{h(t)}\int_{\Omega_t}(
\partial_{x_1'}^2u_n+h^2(t)\partial_{x_2'}^2u_n) ^2(x_1',x_2')
\,dx_1'dx_2' \\
& \leq   \frac{1}{\delta}\varphi ^2(t) \|
\Delta u_n\|_{L^2(\Omega_t)}^2,
\end{align*}
where $\delta$ is the constant which appears in \eqref{HCondition}.
Using Lemma \ref{Lemma4.1} and the condition \eqref{HCondition}, we
obtain the desired inequality.

(b) We have
\begin{align*}
\| \partial_{x_2}v\|_{L^2(D(0,1) ) }^2
&=  \int_{D(0,1) }(\partial_{x_2}v) ^2(x_1,x_2) \,dx_1\,dx_2 \\
&=  \int_{\Omega_t}(\partial_{x_2'}u_n) ^2(x_1',x_2')
h^2(t)\varphi ^2(t) \frac{1}{h(t)\varphi ^2(
t) }dx_1'dx_2' \\
&=  h(t)\int_{\Omega_t}(\partial_{x_2'}u_n) ^2(x_1',x_2')
\,dx_1'\,dx_2' \\
&=  h(t)\| \partial_{x_2'}u_n\|_{L^2(\Omega_t) }^2.
\end{align*}
On the other hand,
\[
\| \Delta v\|_{L^2(D(0,1)) }^2\leq \frac{1}{\delta}\varphi ^2(t)
\| \Delta u_n\|_{L^2(\Omega_t)}^2.
\]
Using the inequality
\[
\| \partial_{x_2}v\|_{L^2(D(0,1) ) }^2
 \leq   C\| \Delta v\|_{L^2(D(0,1) ) }^2
\]
of Lemma \ref{Lemma4.1} and  condition \eqref{HCondition}, we
obtain the desired inequality
\[
\| \partial_{x_2'}u_n\|_{L^2(\Omega_t) }^2
\leq   C\varphi ^2(t) \| \Delta u_n\|_{L^2(\Omega_t) }^2.
\]
\end{proof}

\section{Local in time result}

\subsection{Case of a truncated domain $Q_{\alpha }$}

In this subsection, we replace $Q$ by $Q_{\alpha }$
\[
Q_{\alpha }=\{ (t,x_1,x_2) \in
\mathbb{R}^3:\frac{1}{\alpha} <t<T,0\leq \frac{x_1^2}{\varphi
^2(t) }+ \frac{x_2^2}{h^2(t)\varphi ^2(t) }<1\}
\]
with $\alpha >0$.

\begin{theorem}\label{thm1}
The problem
\begin{equation}
\begin{gathered}
\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\in
L^2(Q_{\alpha }) \text{,\medskip \medskip } \\
u\big|_{\partial Q_{\alpha }-\Gamma_T}=0,
\end{gathered}  \label{leProbleme2}
\end{equation}
admits a unique solution $u\in H^{1,2}(Q_{\alpha }) $.
\end{theorem}

\begin{proof}
The change of variables
\[
(t,x_1,x_2)  \mapsto  (t,y_1,y_2)
=(t,\frac{x_1}{\varphi (t) },\frac{x_2}{h(t)\varphi (t) })
\]
transforms $Q_{\alpha }$ into the cylinder
$P_{\alpha }=]\frac{1}{\alpha} ,T [ \times D(\frac{1}{\alpha}
,1) $, where $D(\frac{1}{\alpha} ,1) $ is the unit
disk centered on $(\frac{1}{\alpha} ,0,0) $. Putting
$u(t,x_1,x_2) =v(t,y_1,y_2) $ and
$f(t,x_1,x_2) =g(t,y_1,y_2) $, then
Problem \eqref{leProbleme2} is transformed, in $P_{\alpha }$ into
the  variable-coefficient parabolic problem
\begin{gather*}
\partial_tv-\frac{1}{\varphi ^2(t) }\partial_{y_1}^2v-
\frac{1}{h^2(t)\varphi ^2(t) }\partial
_{y_2}^2v-\frac{ \varphi'(t)
y_1}{\varphi (t) }\partial_{y_1}v-\frac{(
h\varphi )'(t) y_2}{
h(t)\varphi (t) }\partial_{y_2}v=g\medskip \\
v\big|_{\partial P_{\alpha }-\Gamma_T}=0.
\end{gather*}
This change of variables conserves the spaces $H^{1,2}$ and $L^2$.
In other words
\begin{gather*}
f\in L^2(Q_{\alpha }) \Rightarrow g\in L^2(P_{\alpha }) \\
u\in H^{1,2}(Q_{\alpha }) \Rightarrow v\in H^{1,2}(P_{\alpha }) .
\end{gather*}
\end{proof}

\begin{proposition} \label{Proposition2}
The operator
\[
-\big[\frac{\varphi'(t) y_1}{\varphi
(t) }\partial_{y_1}+\frac{(h\varphi )
'(t) y_2}{h(t)\varphi (t)
}\partial_{y_2}\big] :H_0^{1,2}(P_{\alpha })
\to L^2(P_{\alpha })
\]
is compact.
\end{proposition}

\begin{proof}
$P_{\alpha }$ has the horn property of Besov (see \cite{Besov1967}).
So, for $j=1,2$
\[
\begin{array}{llll}
\partial_{y_j} & H_0^{1,2}(P_{\alpha }) & \to &
H^{\frac{1}{2},1}(P_{\alpha }) \\
& v & \mapsto & \partial_{y_j}v,
\end{array}
\]
is continuous. Since $P_{\alpha }$ is bounded, the canonical
injection is compact from
$H^{\frac{1}{2},1}(P_{\alpha}) $ into $L^2(P_{\alpha }) $ (see for instance
\cite{Besov1967}), where
\[
H^{1/2,1}(P_{\alpha }) =L^2\Big(
\frac{1}{\alpha} ,T;H^{1}\big(D(\frac{1}{\alpha} ,1)
\big) \Big) \cap H^{1/2}\Big(\frac{1}{\alpha}
,T;L^2\big(D(\frac{1}{\alpha} ,1) \big) \Big).
\]
For the complete definitions of the $H^{r,s}$ Hilbertian Sobolev
spaces see for instance \cite{Lions1968}.

Consider the composition
\[
\begin{array}{cccccc}
\partial_{y_j}: & H_0^{1,2}(P_{\alpha })  & \to  &
H^{\frac{1}{2},1}(P_{\alpha })  & \to  &
L^2(P_{\alpha })  \\
& v & \mapsto  & \partial_{y_j}v & \mapsto  & \partial_{y_j}v,
\end{array}
\]
then $\partial_{y_j}$ is a compact operator from
$H_0^{1,2}(P_{\alpha }) $ into $L^2(P_{\alpha
}) $. Since $-\frac{ \varphi'(t)
}{\varphi (t) }$, $-\frac{ (h\varphi )
'(t) }{h(t)\varphi (t) }$ are
bounded functions, the operators
$-\frac{\varphi'(t) y_1}{\varphi (t) }\partial_{y_1}$,
$-\frac{(h\varphi )'(t) y_2}{h(t)\varphi (t) }\partial_{y_2}$ are also
compact from $H_0^{1,2}(P_{\alpha }) $ into
$L^2(P_{\alpha }) $. Consequently,
\[
-\big[\frac{\varphi'(t) y_1}{\varphi
(t) }\partial_{y_1}+\frac{(h\varphi )
'(t) y_2}{h(t)\varphi (t)
}\partial_{y_2}\big]
\]
is compact from $H_0^{1,2}(P_{\alpha }) $ to $L^2(P_{\alpha }) $.
\end{proof}

So, to complete the proof of Theorem \ref{thm1}, it is
sufficient to show that the operator
\[
\partial_t-\frac{1}{\varphi ^2(t) }\partial_{y_1}^2-
\frac{1}{h^2(t)\varphi ^2(t) }\partial_{y_2}^2
\]
is an isomorphism from $H_0^{1,2}(P_{\alpha }) $ into
$ L^2(P_{\alpha }) $.

\begin{lemma}
The operator
\[
\partial_t-\frac{1}{\varphi ^2(t) }\partial_{y_1}^2-
\frac{1}{h^2(t)\varphi ^2(t) }\partial_{y_2}^2
\]
is an isomorphism from $H_0^{1,2}(P_{\alpha }) $ to
$ L^2(P_{\alpha }) $.
\end{lemma}

\begin{proof}
Since the coefficients $\frac{1}{\varphi ^2(t) }$ and
$\frac{ 1}{h^2(t)\varphi ^2(t) }$ are bounded in
$\overline{ P_{\alpha }}$, the optimal regularity is given by
Ladyzhenskaya-Solonnikov-Ural'tseva \cite{Ladyzhenskaya}.
\end{proof}

We shall need the following result  to justify the calculus
of this section.

\begin{lemma}\label{lem4}
The space
\[
\{ u\in H^{4}(P_{\alpha }) :u\big|_{\partial_p P_{\alpha }}=0\}
\]
is dense in the space
\[
\{ u\in H^{1,2}(P_{\alpha }) :u\big|_{\partial_p P_{\alpha }}=0\} .
\]
Here, $\partial_{p}P_{\alpha }$ is the parabolic boundary of
$P_{\alpha }$ and $H^{4}$ stands for the usual Sobolev space
defined, for instance, in Lions-Magenes \cite{Lions1968}.
\end{lemma}

The proof of the above lemma can be found in \cite{Kheloufi2010}.

\begin{remark} \label{remq1}\rm
 In Lemma \ref{lem4}, we can replace $P_{\alpha }$ by
$Q_{\alpha }$ with the help of the change of variables defined above.
\end{remark}

\subsection{Case of a conical type domain}

In this case, we define $Q$ by
\[
Q=\{ (t,x_1,x_2) \in \mathbb{R}
^3:0<t<T,0\leq \frac{x_1^2}{\varphi ^2(t)
}+\frac{ x_2^2}{h^2(t)\varphi ^2(t) }<1\}
\]
with
\begin{equation}
\varphi (0) =0,\quad \varphi (t) >0,\quad t\in]0,T] .  \label{RaccordCondition}
\end{equation}
We assume that the functions $h$ and $\varphi $ satisfy conditions
\eqref{HCondition} and \eqref{PhiCond1}. For each $n\in \mathbb{N}^*$,
we define $Q_n$ by
\[
Q_n=\{ (t,x_1,x_2) \in \mathbb{R}^3:\frac{1}{n}<t<T,
0\leq \frac{x_1^2}{\varphi ^2(t) }+\frac{x_2^2}{h^2(t)\varphi ^2(t)
}<1\}
\]
and we denote $f_n=f_{/Q_n}$ and $ u_n\in H^{1,2}(
Q_n) $ the solution of Problem \eqref{leProbleme} in
$Q_n$. Such a solution exists by Theorem \ref{thm1}.

\begin{proposition}\label{Prop}
There exists a constant $K_1$ independent of $n$ such that
\[
\| u_n\|_{H^{1,2}(Q_n) }\leq
K_1\| f_n\|_{L^2(Q_n) }\leq
K_1\| f\|_{L^2(Q) },
\]
where $\|u_n\|_{H^{1,2}(
Q_n)}=\big(\|u_n\|^2_{H^{1}(
Q_n)}+\sum\limits_{i,j=1}^2\|\partial_{x_i}\,\partial_{x_j}u_n\|^2_{L^2(
Q_n)}\big)^{1/2}$.
\end{proposition}

To prove Proposition \ref{Prop}, we need the following
result which is a consequence of Lemma \ref{lem5} and Grisvard-Looss
\cite{Grisvard1976} (see Theorem 2.2).

\begin{lemma}\label{lem7} 
There exists a constant $C>0$ independent of $n$ such that
\[
\| \partial_{x_1}^2u_n\|_{L^2(Q_n) }^2+\| \partial_{x_2}^2u_n\|
_{L^2(Q_n) }^2+\| \partial_{x_1x_2}^2u_n\|_{L^2(Q_n)}^2
\leq C\| \Delta u_n\|_{L^2(Q_n) }^2.
\]
\end{lemma}

\subsection*{Proof of Proposition \ref{Prop}}
Let us denote the inner product in $L^2(Q_n) $ by
 $\langle \cdot,\cdot\rangle $, then we have
\begin{align*}
\| f_n\|_{L^2(Q_n) }^2 
&= \langle \partial_tu_n-\Delta u_n,\partial_tu_n-\Delta u_n\rangle \\
&=  \| \partial_tu_n\|_{L^2(Q_n) }^2+\| \Delta u_n\|_{L^2(Q_n)
}^2-2\langle \partial_tu_n,\Delta u_n\rangle
\end{align*}

\noindent\textbf{Estimation of $-2\langle \partial_tu_n,\Delta
u_n\rangle$:}
We have
\[
\partial_tu_n.\Delta u_n=\partial_{x_1}(\partial
_tu_n\partial_{x_1}u_n) +\partial_{x_2}(
\partial_tu_n\partial_{x_2}u_n) -\frac{1}{2}\partial
_t[(\partial_{x_1}u_n) ^2+(\partial
_{x_2}u_n) ^2] .
\]
Then
\begin{align*}
-2\langle \partial_tu_n,\Delta u_n\rangle &=
-2\int_{Q_n}
\partial_tu_n.\Delta u_ndt\,dx_1\,dx_2 \\
&=  -2\int_{Q_n}[\partial_{x_1}(\partial
_tu_n\partial_{x_1}u_n) +\partial_{x_2}(
\partial
_tu_n\partial_{x_2}u_n) ] dt\,dx_1\,dx_2 \\
&\quad +\int_{Q_n}\partial_t[(\partial_{x_1}u_n)
^2+(\partial_{x_2}u_n) ^2] dt\,dx_1\,dx_2 \\
&=  \int_{\partial Q_n}[| \nabla u_n|
^2\nu_t-2\partial_tu_n(\partial_{x_1}u_n\nu
_{x_1}+\partial_{x_2}u_n\nu_{x_2}) ] d\sigma
\end{align*}
where $\nu_t,\nu_{x_1},\nu_{x_2}$ are the components of the
unit outward normal vector at $\partial Q_n$. We shall rewrite the
boundary integral making use of the boundary conditions. On the part
of the boundary of $Q_n$\ where $t=\frac{1}{n}$, we have $u_n=0$
and consequently $
\partial_{x_1}u_n=\partial_{x_2}u_n=0$. The corresponding boundary
integral vanishes. On the part of the boundary\ where $t=T$, we have
$\nu_{x_1}=0$, $\nu_{x_2}=0$ and $\nu_t=1$. Accordingly the
corresponding boundary integral
\[
A=\int_{\Gamma_T}| \nabla u_n|^2\,dx_1\,dx_2
\]
is nonnegative. On the part of the boundary\ where
$\frac{x_1^2}{ \varphi ^2(t)
}+\frac{x_2^2}{h^2(t)\varphi ^2(t) }=1$, we have
\[
\nu_{x_1}=\frac{h(t)\cos \theta }{\sqrt{(\varphi'(t) h(t)\cos ^2\theta +(h\varphi )'(
t) \sin^2 \theta ) ^2+(h(t)\cos \theta )
^2+\sin ^2\theta }} ,
\]
\[
\nu_{x_2}=\frac{\sin \theta }{\sqrt{(\varphi'(t) h(t)\cos ^2\theta +(h\varphi )'(
t) \sin^2 \theta ) ^2+(h(t)\cos \theta )
^2+\sin ^2\theta }} ,
\]
\[
\nu_t=\frac{-(\varphi'(t) h(t)\cos
^2\theta +(h\varphi )'(t) \sin^2 \theta
) }{\sqrt{(\varphi'(t) h(t)\cos
^2\theta +(h\varphi )'(t) \sin^2 \theta
) ^2+(h(t)\cos \theta ) ^2+\sin ^2\theta }}
\]
and $u_n(t,\varphi (t) \cos \theta ,h(t)\varphi
(t) \sin \theta ) =0$. Differentiating with
respect to $t$ then with respect\ to $\theta $ we obtain
\begin{gather*}
\partial_tu_n=-\varphi'(t) \cos \theta .\partial
_{x_1}u_n-(h\varphi )'(t) \sin \theta.\partial_{x_2}u_n,
\\
\sin \theta .\partial_{x_1}u_n=h(t)\cos \theta .\partial_{x_2}u_n .
\end{gather*}
Consequently the corresponding boundary integral is
\begin{align*}
J_n &=  -2\int_0^{2\pi }\int_{1/n}^T\partial
_tu_n.(h\varphi \cos \theta .\partial
_{x_1}u_n+h\varphi \sin \theta .\partial
_{x_2}u_n) \,dt\,d\theta   \\
&\quad -\int_0^{2\pi }\int_{1/n}^T| \nabla
u_n| ^2((h\varphi )'\varphi \sin
^2\theta +\varphi'(h\varphi )\cos ^2\theta )
\,dt\,d\theta
\\
&=  2\int_0^{2\pi }\int_{1/n}^T\{(\varphi
'\cos \theta .\partial_{x_1}u_n+(h\varphi )'\sin \theta
.\partial_{x_2}u_n)\\
&\quad  \times  (h\varphi \cos \theta .\partial_{x_1}u_n+h\varphi
\sin
\theta .\partial_{x_2}u_n) \}\,dt\,d\theta  \\
& \quad -\int_0^{2\pi }\int_{1/n}^T| \nabla
u_n| ^2((h\varphi )'\varphi \sin
^2\theta
+\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta   \\
&=  2\int_0^{2\pi }\int_{1/n}^T| \nabla
u_n| ^2((h\varphi )'\varphi \sin
^2\theta
+\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta   \\
&\quad -\int_0^{2\pi }\int_{1/n}^T| \nabla
u_n| ^2((h\varphi )'\varphi \sin
^2\theta +\varphi'h\varphi \cos ^2\theta )
\,dt\,d\theta
  \\
&=  \int_0^{2\pi }\int_{1/n}^T| \nabla
u_n| ^2((h\varphi )'\varphi \sin
^2\theta
+\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta .
\end{align*}
Finally,
\begin{equation}
\begin{split}
-2\langle \partial_tu_n,\Delta u_n\rangle
&= \int_0^{2\pi }\int_{1/n}^T| \nabla
u_n| ^2((h\varphi )'\varphi \sin^2\theta +\varphi'h\varphi
\cos ^2\theta ) \,dt\,d\theta \\
&\quad +\int_{\Gamma_T}| \nabla u_n|^2(T,x_1,x_2) \,dx_1\,dx_2.
\end{split}\label{3}
\end{equation}

\begin{lemma} \label{lem6}
One has
\begin{align*}
-2\langle \partial_tu_n,\Delta u_n\rangle 
&= 2\int_{Q_n}(\frac{\varphi'}{\varphi
}x_1\partial_{x_1}u_n+\frac{ (h\varphi )'}{h\varphi
}x_2\partial_{x_2}u_n) \Delta
u_ndt\,dx_1\,dx_2 \\
& \quad +\int_{\Gamma_T}| \nabla u_n|
^2(T,x_1,x_2) \,dx_1\,dx_2.
\end{align*}
\end{lemma}

\begin{proof}
For $\frac{1}{n}<t<T$, consider the following parametrization of the
domain $ \Omega_t$
\[
\begin{array}{lll}
(0,2\pi ) & \to & \Omega_t \\
\theta & \to & (\varphi (t) \cos \theta
,h(t)\varphi (t) \sin \theta ) =(
x_1,x_2).
\end{array}
\]
Let us denote the inner product in $L^2(\Omega_t) $
by $ \langle \cdot,\cdot\rangle $, and set
\[
I_n=\big\langle \Delta u_n,\frac{\varphi'}{\varphi }
x_1\partial_{x_1}u_n+\frac{(h\varphi )'}{h\varphi }
x_2\partial_{x_2}u_n\big\rangle
\]
then we have
\begin{align*}
I_n &= \int_{\Omega_t}(\partial
_{x_1}^2u_n+\partial_{x_2}^2u_n) (
\frac{\varphi'}{\varphi }x_1\partial_{x_1}u_n+\frac{(h\varphi )'
}{h\varphi }x_2\partial_{x_2}u_n)
\,dx_1\,dx_2
  \\
&=  \int_{\Omega_t}(\frac{\varphi'}{\varphi }
x_1\partial_{x_1}^2u_n\partial
_{x_1}u_n+\frac{(h\varphi )'}{h\varphi }x_2\partial
_{x_2}^2u_n\partial
_{x_2}u_n) \,dx_1\,dx_2   \\
& \quad +\int_{\Omega_t}(\frac{\varphi'}{\varphi }
x_1\partial_{x_2}^2u_n\partial
_{x_1}u_n+\frac{(h\varphi )'}{h\varphi }x_2\partial
_{x_1}^2u_n\partial
_{x_2}u_n) \,dx_1\,dx_2  .
\end{align*}
Using Green formula, we obtain
\begin{align*}
I_n &=  \frac{1}{2}\int_{\Omega_t}(\frac{\varphi
'}{\varphi }x_1\partial_{x_1}(
\partial
_{x_1}u_n) ^2+\frac{(h\varphi )'}{h\varphi }
x_2\partial_{x_2}(\partial_{x_2}u_n)
^2)\,dx_1\,dx_2   \\
&\quad +\int_{\Omega_t}(\frac{\varphi'}{\varphi }
x_1\partial_{x_2}(\partial_{x_2}u_n) \partial
_{x_1}u_n+\frac{(h\varphi )'}{h\varphi }x_2\partial
_{x_1}(\partial_{x_1}u_n) \partial
_{x_2}u_n)
\,dx_1\,dx_2 \\
&=  \frac{1}{2}\int_{\partial \Omega_t}(\frac{\varphi
'}{\varphi }x_1\nu_{x_1}(\partial_{x_1}u_n) ^2+
\frac{(h\varphi )'}{h\varphi }x_2\nu_{x_2}(
\partial
_{x_2}u_n) ^2) d\sigma  \\
&\quad -\frac{1}{2}\int_{\Omega_t}(\frac{\varphi'}{
\varphi }(\partial_{x_1}u_n) ^2+\frac{(h\varphi
)'}{h\varphi }(\partial_{x_2}u_n)
^2)
\,dx_1\,dx_2   \\
&\quad +\int_{\partial \Omega_t}(\frac{\varphi'}{\varphi }
x_1\nu_{x_2}+\frac{(h\varphi )'}{h\varphi }x_2\nu
_{x_1}) \partial_{x_1}u_n\partial_{x_2}u_nd\sigma
   \\
&\quad -\int_{\Omega_t}(\frac{\varphi'}{\varphi }
x_1\partial_{x_2}u_n\partial
_{x_1x_2}^2u_n+\frac{(h\varphi )'}{h\varphi
}x_2\partial_{x_1}u_n\partial
_{x_1x_2}^2u_n) \,dx_1\,dx_2
\end{align*}
where $\nu_{x_1},\nu_{x_2}$ are the components of the unit
outward normal vector at $\partial \Omega_t$. Then
\begin{align*}
I_n &=  \frac{1}{2}\int_{\partial \Omega_t}(
\frac{\varphi'}{\varphi }x_1\nu_{x_1}(\partial
_{x_1}u_n) ^2+\frac{(h\varphi )'}{h\varphi
}x_2\nu_{x_2}(
\partial_{x_2}u_n) ^2) d\sigma    \\
&\quad -\frac{1}{2}\int_{\Omega_t}(\frac{\varphi'}{
\varphi }(\partial_{x_1}u_n) ^2+\frac{(h\varphi
)'}{h\varphi }(\partial_{x_2}u_n)
^2)
\,dx_1\,dx_2   \\
&\quad +\int_{\partial \Omega_t}(\frac{\varphi'}{\varphi }
x_1\nu_{x_2}+\frac{(h\varphi )'}{h\varphi }x_2\nu
_{x_1}) \partial_{x_1}u_n\partial_{x_2}u_nd\sigma
   \\
&\quad -\frac{1}{2}\int_{\Omega_t}(\frac{\varphi'}{
\varphi }x_1\partial_{x_1}(\partial_{x_2}u_n) ^2+
\frac{(h\varphi )'}{h\varphi }x_2\partial_{x_2}(
\partial_{x_1}u_n) ^2) \,dx_1\,dx_2.
\end{align*}
Thus,
\begin{align*}
I_n &=  \frac{1}{2}\int_{\partial \Omega_t}\Big(
\frac{\varphi'}{\varphi }x_1\nu_{x_1}(\partial
_{x_1}u_n) ^2+\frac{(h\varphi )'}{h\varphi
}x_2\nu_{x_2}(
\partial_{x_2}u_n) ^2\Big) d\sigma    \\
&\quad  -\frac{1}{2}\int_{\Omega_t}\Big(\frac{\varphi'}{
\varphi }(\partial_{x_1}u_n) ^2+\frac{(h\varphi
)'}{h\varphi }(\partial_{x_2}u_n)
^2\Big)
\,dx_1\,dx_2   \\
&\quad  +\int_{\partial \Omega_t}\Big(\frac{\varphi'}{\varphi }
x_1\nu_{x_2}+\frac{(h\varphi )'}{h\varphi }x_2\nu
_{x_1}\Big) \partial_{x_1}u_n\partial_{x_2}u_nd\sigma
   \\
&\quad  -\frac{1}{2}\int_{\partial \Omega_t}(\frac{\varphi
'}{\varphi }x_1\nu_{x_1}(\partial_{x_2}u_n) ^2+
\frac{(h\varphi )'}{h\varphi }x_2\nu_{x_2}(
\partial
_{x_1}u_n) ^2) \,dx_1\,dx_2   \\
&\quad  +\frac{1}{2}\int_{\Omega_t}\Big(\frac{\varphi'}{
\varphi }(\partial_{x_1}u_n) ^2+\frac{(h\varphi
)'}{h\varphi }(\partial_{x_2}u_n)
^2\Big) \,dx_1\,dx_2
\end{align*}
and then
\begin{align*}
I_n &=  \frac{1}{2}\int_{\partial \Omega_t}\Big(
\frac{\varphi'}{\varphi }x_1\nu_{x_1}(\partial
_{x_1}u_n) ^2+\frac{(h\varphi )'}{h\varphi
}x_2\nu_{x_2}(
\partial_{x_2}u_n) ^2\Big) d\sigma    \\
&\quad  +\int_{\partial \Omega_t}(\frac{\varphi'}{\varphi }
x_1\nu_{x_2}+\frac{(h\varphi )'}{h\varphi }x_2\nu
_{x_1}) \partial_{x_1}u_n\partial_{x_2}u_nd\sigma
   \\
&\quad  -\frac{1}{2}\int_{\partial \Omega_t}\Big(\frac{\varphi
'}{\varphi }x_1\nu_{x_1}(\partial_{x_2}u_n) ^2+
\frac{(h\varphi )'}{h\varphi }x_2\nu_{x_2}(
\partial
_{x_1}u_n) ^2\Big) \,dx_1\,dx_2.
\end{align*}
Consequently,
\begin{align*}
I_n&=\frac{1}{2}\int_0^{2\pi }\Big(\frac{\varphi'}{\varphi }
\varphi h\varphi (\cos \theta .\partial_{x_1}u_n) ^2+
\frac{(h\varphi )'}{h\varphi }\varphi h\varphi (\sin
\theta
.\partial_{x_2}u_n) ^2\Big) d\theta   \\
&\quad +\int_0^{2\pi }(\frac{\varphi'}{\varphi }\varphi ^2+
\frac{(h\varphi )'}{h\varphi }(h\varphi )^2) \sin
\theta \cos \theta .\partial_{x_1}u_n\partial
_{x_2}u_nd\theta
\\
&\quad -\frac{1}{2}\int_0^{2\pi }\Big(\frac{\varphi'}{\varphi }
\varphi h\varphi (\cos \theta .\partial_{x_2}u_n) ^2+
\frac{(h\varphi )'}{h\varphi }\varphi h\varphi (\sin
\theta .\partial_{x_1}u_n) ^2\Big) d\theta   \\
&=\frac{1}{2}\int_0^{2\pi }\Big(\varphi'h\varphi
\Big(\cos \theta .\partial_{x_1}u_n\Big) ^2+\varphi
(h\varphi )'(\sin \theta .\partial
_{x_2}u_n) ^2\Big) d\theta
  \\
&\quad +\int_0^{2\pi }\Big(\varphi'\varphi +(h\varphi
)'h\varphi \Big) \sin \theta \cos \theta .\partial
_{x_1}u_n\partial
_{x_2}u_nd\theta   \\
&\quad -\frac{1}{2}\int_0^{2\pi }\Big(\varphi'h\varphi
(\cos \theta .\partial_{x_2}u_n) ^2+\varphi
(h\varphi )'(\sin \theta .\partial_{x_1}u_n) ^2\Big) d\theta .
\end{align*}
The boundary condition $u_n(t,\varphi (t) \cos
\theta ,h(t)\varphi (t) \sin \theta ) =0$ leads to
\[
\sin \theta .\partial_{x_1}u_n=h(t)\cos \theta .\partial
_{x_2}u_n;
\]
then
\[
\sin \theta \cos \theta .\partial_{x_1}u_n\partial
_{x_2}u_n=h(t)(\cos \theta .\partial_{x_2}u_n)
^2
\]
and
\[
h(t)\sin \theta \cos \theta .\partial_{x_1}u_n\partial
_{x_2}u_n=(\sin \theta .\partial_{x_1}u_n)^2.
\]
Consequently,
\begin{align*}
I_n&= \frac{1}{2}\int_0^{2\pi }\Big(\varphi'h\varphi (\cos \theta .\partial_{x_1}u_n)
^2+\varphi (h\varphi )'(\sin \theta .\partial
_{x_2}u_n) ^2\Big)
d\theta   \\
&\quad +\int_0^{2\pi }\Big(\varphi'h\varphi (\cos
\theta .\partial_{x_2}u_n) ^2+\varphi (h\varphi
)'(\sin
\theta .\partial_{x_1}u_n) ^2\Big) d\theta   \\
&\quad -\frac{1}{2}\int_0^{2\pi }(\varphi'h\varphi
(\cos \theta .\partial_{x_2}u_n) ^2+\varphi
(h\varphi )'(\sin \theta .\partial
_{x_1}u_n) ^2) d\theta
  \\
&=  \frac{1}{2}\int_0^{2\pi }\Big(\varphi'h\varphi
(\cos \theta .\partial_{x_1}u_n) ^2+\varphi
(h\varphi )'(\sin \theta .\partial
_{x_2}u_n) ^2\Big) d\theta
  \\
&\quad +\frac{1}{2}\int_0^{2\pi }\Big(\varphi'h\varphi
(\cos \theta .\partial_{x_2}u_n) ^2+\varphi
(h\varphi )'(\sin \theta .\partial
_{x_1}u_n) ^2\Big) d\theta
  \\
&=  \frac{1}{2}\int_0^{2\pi }\Big\{\varphi'h\varphi (
\cos \theta .\partial_{x_1}u_n) ^2+\varphi (h\varphi
)'(\sin \theta .\partial_{x_2}u_n) ^2 \\
& +\varphi'h\varphi (\cos \theta .\partial
_{x_2}u_n) ^2+\varphi (h\varphi )'(\sin
\theta
.\partial_{x_1}u_n) ^2\Big\}d\theta   \\
&=  \frac{1}{2}\int_0^{2\pi }[(\partial
_{x_1}u_n) ^2+(\partial_{x_2}u_n)
^2] (\varphi (h\varphi )'\sin ^2\theta
+\varphi'h\varphi \cos
^2\theta ) d\theta .
\end{align*}
So
\[
I_n=\frac{1}{2}\int_0^{2\pi }| \nabla u_n|
^2(\varphi (h\varphi )'\sin ^2\theta +\varphi
'h\varphi \cos ^2\theta ) d\theta
\]
and
\begin{align*}
&\int_{1/n}^T\int_0^{2\pi }| \nabla
u_n| ^2(\varphi (h\varphi )'\sin
^2\theta +\varphi'h\varphi \cos ^2\theta ) \,dt\,d\theta   \\
&=2\int_{Q_n}(\frac{\varphi'}{\varphi
}x_1\partial_{x_1}u_n+\frac{(h\varphi )'}{h\varphi
}x_2\partial
_{x_2}u_n) \Delta u_ndt\,dx_1\,dx_2.
\end{align*}
Finally, by  \eqref{3}, it follows that
\begin{align*}
-2\langle \partial_tu_n,\Delta u_n\rangle
 &= 2\int_{Q_n}\Big(\frac{\varphi'}{\varphi }x_1\partial_{x_1}u_n+\frac{
(h\varphi )'}{h\varphi }x_2\partial_{x_2}u_n\Big)
\Delta u_ndt\,dx_1\,dx_2 \\
&\quad  +\int_{\Gamma_T}| \nabla u_n|^2(T,x_1,x_2) \,dx_1\,dx_2.
\end{align*}
\end{proof}

Now, we continue the proof of Proposition \ref{Prop}. We have
\begin{align*}
&| \int_{Q_n}\Big(\frac{\varphi'}{\varphi }
x_1\partial_{x_1}u_n+\frac{(h\varphi )'}{h\varphi }
x_2\partial_{x_2}u_n\Big) \Delta u_ndt\,dx_1\,dx_2|  \\
&\leq \| \Delta u_n\|_{L^2(Q_n)}\| \frac{\varphi'}{\varphi }x_1\partial
_{x_1}u_n\|_{L^2(Q_n) } 
+\| \Delta u_n\|_{L^2(Q_n)}\| \frac{(h\varphi )'}{h\varphi }x_2\partial
_{x_2}u_n\|_{L^2(Q_n) },
\end{align*}
but Lemma \ref{lem5} yields
\begin{align*}
\| \frac{\varphi'}{\varphi }x_1\partial_{x_1}u_n\|_{L^2(Q_n) }^2 
&=\int_{1/n}^T\varphi ^{\prime 2}(t)
\int_{\Omega_t}(\frac{x_1}{\varphi (t)
}) ^2(\partial_{x_1}u_n) ^2dt\,dx_1\,dx_2 \\
& \leq  \int_{1/n}^T\varphi ^{\prime 2}(t)
\int_{\Omega_t}(\partial_{x_1}u_n)
^2dt\,dx_1\,dx_2 \\
&\leq  C^2\int_{1/n}^T(\varphi (t) \varphi'(t) ) ^2\int_{\Omega
_t}(\Delta u_n) ^2dt\,dx_1\,dx_2 \\
&\leq  C^2\epsilon ^2\| \Delta u_n\|
_{L^2(Q_n) }^2,
\end{align*}
since $(\varphi (t) \varphi'(t) ) \leq \epsilon $.
Similarly, we have 
\[
\| \frac{(h\varphi )'}{h\varphi }x_2\partial
_{x_2}u_n\|_{L^2(Q_n) }^2 \leq
C^2\epsilon ^2\| \Delta u_n\|_{L^2(Q_n) }^2.
\]
Then
\[
| \int_{Q_n}\Big(\frac{\varphi'}{\varphi }
x_1\partial_{x_1}u_n+\frac{(h\varphi )'}{h\varphi }
x_2\partial_{x_2}u_n\Big) \Delta
u_ndt\,dx_1\,dx_2| \leq 2C\epsilon \| \Delta
u_n\|_{L^2(Q_n) }^2.
\]
Therefore, Lemma \ref{lem6} shows that
\begin{align*}
| 2\langle \partial_tu_n,\Delta u_n\rangle|
& \geq  -2\big| \int_{Q_n}\Big(\frac{\varphi
'}{\varphi } x_1\partial_{x_1}u_n+\frac{(h\varphi
)'}{h\varphi } x_2\partial_{x_2}u_n\Big) \Delta
u_ndt\,dx_1\,dx_2\big|
 \\
&\quad  +\int_{\Gamma_T}| \nabla u_n|^2(T,x_1,x_2) \,dx_1\,dx_2 \\
& \geq  -4C\epsilon \| \Delta u_n\|_{L^2(Q_n) }^2.
\end{align*}
Hence
\begin{align*}
\| f_n\|_{L^2(Q_n) }^2
&= \| \partial_tu_n\|_{L^2(Q_n)}^2+\| \Delta u_n\|_{L^2(Q_n)
}^2-2\langle \partial_tu_n,\Delta u_n\rangle  \\
& \geq  \| \partial_tu_n\|_{L^2(Q_n) }^2+(1-4C\epsilon ) \| \Delta
u_n\|_{L^2(Q_n) }^2.
\end{align*}
Then, it is sufficient to choose $\epsilon $ such that
$1-4C\epsilon >0$ to
get a constant $K_0>0$ independent of $n$ such that
\[
\| f_n\|_{L^2(Q_n) }\geq K_0\| u_n\|_{H^{1,2}(Q_n)},
\]
and since
\[
\| f_n\|_{L^2(Q_n) }\leq \| f\|_{L^2(Q_n) },
\]
there exists a constant $K_1>0$, independent of $n$ satisfying
\[
\| u_n\|_{H^{1,2}(Q_n) }\leq
K_1\| f_n\|_{L^2(Q_n) }\leq K_1\| f\|_{L^2(Q) }.
\]
This completes the proof of Proposition \ref{Prop}.

\subsection*{Passage to the limit}
We are now in position to prove the main result of this work.

\begin{theorem} \label{thm2}
Assume that the functions $h$ and $\varphi $ verify the
conditions \eqref{HCondition}, \eqref{PhiCond1} and
\eqref{RaccordCondition}. Then, for $T$ small enough, Problem
\eqref{leProbleme} admits a unique solution $u\in H^{1,2}(Q) $.
\end{theorem}

\begin{proof}
Choose a sequence $Q_n$ $n=1,2, \dots$, of truncated conical domains
(see subsection 3.2) such that $Q_n\subseteq Q$. Then we have
$Q_n\to Q$, as $n\to \infty $.

Consider the solution $u_n\in H^{1,2}(Q_n) $ of the
Cauchy-Dirichlet problem
\begin{gather*}
\partial_tu_n-\partial_{x_1}^2u_n-\partial_{x_2}^2u_n=f\quad
\text{in }Q_n \\
u_n\big|_{\partial Q_n-\Gamma_T}=0,
\end{gather*}
where $\Gamma_T$ is the part of the boundary of $Q_n$ where
$t=T$. Such a solution $u_n$ exists by Theorem \ref{thm1}. Let
$\widetilde{u}_n$ the 0-extension of $u_n$ to $Q$. By 
Proposition \ref{Prop}, we know that there exists a constant $C$
such that
\[
\| \widetilde{u}_n\|_{L^2(Q)}+\| \partial_t\widetilde{u}_n\|_{L^2(
Q) }+\sum_{ i,j=0,\,  1\leq i+j\leq2} ^2
\| \partial_{x_1}^{j}\partial_{x_2}^{j}\widetilde{u}_n\|_{L^2(Q) }\leq
C\| f\|_{L^2(Q) }.
\]
This means that $\widetilde{u}_n$, $\partial_t\widetilde{u}_n$,
$ \partial_{x_1}^{j}\partial_{x_2}^{j}\widetilde{u}_n$ for
 $1\leq i+j\leq 2$ are bounded functions in
$L^2(Q) $. So for a suitable increasing sequence of
integers $n_k$, $k=1,2,\dots$, there exist functions
$u, v, v_{i,j}$, $1\leq i+j\leq 2$ in $L^2(Q) $ such that
\begin{gather*}
\widetilde{u}_{n_k}  \rightharpoonup  u \quad
\text{weakly in $L^2(Q)$ as $k\to \infty$} \\
\partial_t\widetilde{u}_{n_k}  \rightharpoonup  v \quad
\text{weakly in $L^2(Q)$ as $k\to \infty$} \\
\partial_{x_1}^{j}\partial_{x_2}^{j}\widetilde{u}_{n_k}
\rightharpoonup  v_{i,j} \quad
\text{weakly in $L^2(Q)$ as $ k\to \infty$, $1\leq i+j\leq 2$.}
\end{gather*}
Clearly,
\[
v=\partial_tu,\quad v_{i,j}=\partial_{x_1}^{i}\partial
_{x_2}^{j}u , \quad 1\leq i+j\leq 2
\]
in the sense of distributions in $Q$ and so in $L^2(Q)$.
So, $ u\in H^{1,2}(Q) $ and
\[
\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\quad \text{in }Q.
\]
On the other  hand, the solution $u$ satisfies the boundary conditions
$u\big|_{\partial Q-\Gamma_T}=0$ since
$u\big|_{Q_n}=u_n$ for all $n\in \mathbb{N}^*$.
This proves the existence of a solution to Problem
\eqref{leProbleme}.
\end{proof}

\section{Global in time result}

Assume that $Q$ satisfies \eqref{RaccordCondition}. In the case
where $T$ is not in the neighborhood of zero, we set $Q=D_1\cup
D_2\cup \Gamma_{T_1}$ where
\begin{gather*}
D_1=\{ (t,x_1,x_2) \in \mathbb{R}
^3:0<t<T_1,\text{ }0\leq \frac{x_1^2}{\varphi ^2(
t) }+\frac{x_2^2}{(h\varphi ) ^2(t) } <1\}
\\
D_2=\{ (t,x_1,x_2) \in \mathbb{R}^3:
T_1<t<T,\text{ }0\leq \frac{x_1^2}{\varphi ^2(t) }+\frac{x_2^2}{(h\varphi ) ^2(
t) } <1\}
\\
\Gamma_{T_1}=\{ (T_1,x_1,x_2) \in
\mathbb{R}^3:\text{ }0\leq \frac{x_1^2}{\varphi ^2(T_1) }
+ \frac{x_2^2}{(h\varphi ) ^2(T_1) }<1\}
\end{gather*}
with $T_1$ small enough.

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

Theorem \ref{thm2} applied to the conical domain $D_1$, shows that
there exists a unique solution $u_1\in H^{1,2}(D_1)$ of the
problem
\begin{equation}
\begin{gathered}
\partial_tu_1-\partial_{x_1}^2u_1-\partial
_{x_2}^2u_1=f_1,\quad f_1\in L^2(D_1) \\
u_1\big|_{\partial D_1-\Gamma_{T_1}}=0.
\end{gathered} \label{PbmGlobal1}
\end{equation}

Hereafter, we denote the trace $u_{1/\Gamma_{T_1}}$ by $\psi $
which is in the Sobolev space $H^{1}(\Gamma_{T_1}) $
because $u_1\in H^{1,2}(D_1) $ (see \cite{Lions1968}).

Now, consider the following problem in $D_2$,
\begin{equation}
\begin{gathered}
\partial_tu_2-\partial_{x_1}^2u_2-\partial
_{x_2}^2u_2=f_2\quad f_2 \in L^2(D_2) \\
u_{2/\Gamma_{T_1}}=\psi \\
u_2\big|_{\partial D_2-(\Gamma_{T_1}\cup \Gamma_T)}=0
\end{gathered}
  \label{PbmGlobal2}
\end{equation}

We use the following result, which is a consequence of 
\cite[Theorem 4.3, Vol. 2]{Lions1968}, to solve Problem \eqref{PbmGlobal2}.

\begin{proposition}\label{Proposition4.3}
Let $Q$ be the cylinder $] 0,T[\times D(0,1) $, $f\in L^2(Q) $ and
 $\psi \in H^{1}(\gamma_0) $. Then, the problem
\begin{gather*}
\partial_tu-\partial_{x_1}^2u-\partial_{x_2}^2u=f\text{ in }Q
\\
u\big|_{\gamma_0}=\psi \\
u\big|_{\gamma_0\cup \gamma_1}=0
\end{gather*}
where $\gamma_0=\{ 0\} \times D(0,1) $,
$\gamma_1=] 0,T[ \times \partial D(0,1) $,
admits a (unique) solution $u\in H^{1,2}(Q) $.
\end{proposition}

\begin{remark}\label{Remark4.2} \rm
In the application of \cite[Theorem 4.3, Vol.2]{Lions1968}, 
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
\[
(t,x_1,x_2) \mapsto (t,y_1,y_2)
=(t,\varphi (t) x_1,(h\varphi )(t) x_2) ,
\]
we deduce the following result.

\begin{proposition}\label{Propositio4.4}
Problem \eqref{PbmGlobal2} admits a (unique)
solution $ u_2\in H^{1,2}(D_2) $.
\end{proposition}

So, the function $u$ defined by
\[
u=\begin{cases}
u_1 &\text{in }D_1 \\
u_2 &\text{in }D_2
\end{cases}
\]
is the (unique) solution of Problem \eqref{leProbleme} for an
arbitrary $T$. Our second main result is as follows.

\begin{theorem} \label{thm3}
Assume that the functions $h$ and $\varphi $ verify  
conditions \eqref{HCondition}, \eqref{PhiCond1} and \eqref{RaccordCondition}. 
Then, Problem \eqref{leProbleme} admits a unique solution $u\in H^{1,2}(Q) $.
\end{theorem}

\subsection*{Acknowledgments}
The author is thankful to Professor B. K.
Sadallah (Ecole Normale Sup\'{e}rieure de Kouba, Algeria) for his
help during the preparation of this work, and to the anonymous
referees for their careful reading of a previous version of the
manuscript, which led to a substantial improvement of this
manuscript.

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