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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 20, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/20\hfil Burgers equations in a non-parabolic domain]
{Existence of solutions to Burgers equations in a non-parabolic domain }

\author[Y. Benia, B.-K. Sadallah \hfil EJDE-2018/20\hfilneg]
{Yassine Benia, Boubaker-Khaled Sadallah}

\address{Yassine Benia \newline
Dept of Mathematics and Informatics,
University of Benyoucef Benkhedda (Alger 1),
 16000, Algiers, Algeria}
\email{benia.yacine@yahoo.fr}

\address{Boubaker-Khaled Sadallah \newline
Lab. PDE \& Hist Maths; Dept of Mathematics, E.N.S.,
16050, Kouba, Algiers, Algeria}
\email{sadallah@ens-kouba.dz}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted December 14, 2017. Published January 15, 2018.}
\subjclass[2010]{35K58, 35Q35}
\keywords{Burgers equation; existence; uniqueness;
\hfill\break\indent non-parabolic domains; anisotropic Sobolev space}

\begin{abstract}
 In this article, we study the semilinear Burgers equation with time
 variable coefficients, subject to boundary condition in a non-parabolic
 domain. Some assumptions on the boundary of the domain and on the
 coefficients of the equation will be imposed. The right-hand side of
 the equation is taken in $L^2(\Omega)$. The method we used is based
 on the approximation of the non-parabolic domain by a sequence of subdomains
 which can be transformed into regular domains. This paper is an extension
 of the work \cite{benia}.
\end{abstract}

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

\section{Introduction}\indent

The Burgers equation is a fundamental partial differential equation in 
modeling many physical phenomena, such as fluid mechanics, acoustics, 
turbulence \cite{Burgers,Hopf}, traffic  flow, growth of interfaces, 
and financial mathematics \cite{Kevorkian,Whitham}.

In \cite{Sadallah},  the author studied a linear parabolic equation in 
a domain similar to the one considered in this work. Other references 
on the analysis of linear parabolic problems in non-regular domains are 
discussed for example in \cite{Aref,Grisvard,Kheloufi,Labbes}.

The work by Clark et al.\ \cite{Clark} is devoted to the homogeneous 
Burgers equation in non-parabolic domains which can be transformed 
into rectangle. In the same domains, we have established the existence, 
uniqueness and the optimal regularity of the solution to the non-homogeneous 
Burgers equation with time variable coefficients in an anisotropic Sobolev 
space (see \cite{benia}). The present paper is an extension of this last work 
to another type of non-regular domains.

Let $\Omega\subset\mathbb{R}^2$ be the ``triangular'' domain
\begin{equation*}
\Omega=\{(t,x)\in \mathbb{R}^2; \ 0<t<T, \ x\in I_t\},
\end{equation*}
where $T$ is a positive number and
\begin{equation*}
I_t=\{x\in \mathbb{R}; \ \varphi_1(t)<x<\varphi_2(t), \ t\in(0,T)\},
\end{equation*}
with
\begin{equation}\label{30}
 \varphi_1(0)=\varphi_2(0).
 \end{equation}
The functions  $\varphi_1$, $\varphi_2$ are defined on $[0,T]$, 
and belong to $\mathcal{C}^1(0,T)$.

The most interesting point of the problem studied here is the fact that  
$\varphi_1(0)=\varphi_2(0)$, because the domain is not rectangular and 
cannot be transformed into a regular domain without the appearance of some 
degenerate terms in the equation.

In $\Omega$, we consider the boundary-value problem for the non-homogeneous
 Burgers equation with variable coefficient
\begin{equation}\label{2}
\begin{gathered}
\partial_t u(t,x)+c(t)u(t,x)\partial_x u(t,x)-\partial^2_xu(t,x)=f(t,x) 
\quad  (t,x)\in\Omega,\\
u(t,\varphi_1(t))=u(t,\varphi_2(t))=0 \quad t\in (0,T),\\
\end{gathered}
\end{equation}
where $f\in L^2(\Omega)$ and $c(t)$ is given.

 We look for some conditions on the functions $c(t)$, $\varphi_1(t)$ and 
$\varphi_2(t)$ such that \eqref{2} admits a unique solution $u$ belonging 
to the anisotropic Sobolev space
\begin{equation*}
H^{1,2}(\Omega)=\{u\in L^2(\Omega); \partial_{t}u, \partial_{x}u, 
\partial^2_{x}u\in L^2(\Omega)\}.
\end{equation*}

In the sequel, we assume that there exist positive constants $c_1$ and $c_2$, 
such that
\begin{equation}\label{3}
\begin{aligned}
c_1\leq c(t)\leq c_2, \quad \text{for all } t\in(0,T),\\
\end{aligned}
\end{equation}
 and we note that
\begin{equation*}
\| u\|_{L^2(I_t)}
=\Big(\int_{\varphi_1(t)}^{\varphi_2(t)}| u(t,x)|^2
\,\mathrm{d}x\Big)^{1/2},
\end{equation*}
\begin{equation*}
\| u\|^2_{L^\infty(I_t)}=\underset{x\in I_t}{\sup}| u(t,x)|.
\end{equation*}
To establish the existence of a solution to \eqref{2}, we also assume that
\begin{equation}\label{4}
|\varphi'(t)|\leq \gamma \quad\text{for all} \ t\in [0,T],
\end{equation}
where $\gamma$ is a positive constant and $\varphi(t)=\varphi_2(t)-\varphi_1(t)$ 
for all $t\in [0,T]$.

\begin{remark} \rm
Once problem \eqref{2} is solved, we can deduce the solution of the problem
\begin{equation}\label{1}
\begin{gathered}
\partial_t u(t,x)+a(t)u(t,x)\partial_x u(t,x)-b(t)\partial^2_xu(t,x)
=f(t,x) \quad (t,x)\in\Omega,\\
u(t,\varphi_1(t))=u(t,\varphi_2(t))=0 \quad t\in (0,T).
\end{gathered}
\end{equation}
Indeed, consider the case where $a(t)$ and $b(t)$ are positive and bounded 
functions for all $t\in[0,T]$.  Let $h$ be defined by
$h:[0,T]\to [0,T']$
\[
 h(t)=\int^t_0 b(s)\mathrm{d}s,
\]
we put $\psi_i=\varphi_i\circ h^{-1}$ where $i=1, 2$. Using the change of 
variables $t'=h(t)$, $v(t',x)=u(t,x)$, \eqref{1} becomes equivalent to
 \eqref{2}, because it may be written as follows
\begin{gather*}
\partial_{t'} v(t',x)+c(t')v(t',x)\partial_x v(t',x)-\partial^2_xv(t',x)=g(t',x) 
\quad  (t',x)\in\Omega',\\
v(t',\psi_1(t'))=v(t',\psi_2(t'))=0, \quad t'\in (0,T'),\\
\end{gather*}
where $c(t')=\frac{a(t)}{b(t)}$, $g(t',x)=\frac{f(t,x)}{b(t)}$, $\Omega'=\{(t',x)\in \mathbb{R}^2; \ 0<t'<T', \ x\in I_{t'}\}$  and $T'=\int^T_0 b(s)\mathrm{d}s$.
\end{remark}

For the study of problem \eqref{2} we will follow the method used  in \cite{Sadallah}, 
which consists in observing that this problem admits a unique solution in domains 
that can be transformed into rectangles, i.e., when $\varphi_1(0)\neq \varphi_2(0)$.

The paper is organized as follows. In the next section we study problem \eqref{2}
 in domain that can be transformed into a rectangle. When $\varphi_1$ and 
$\varphi_2$ are monotone on $(0,T)$,  we solve in Section \ref{s3} the problem 
in a triangular domain: We approximate this domain by a sequence of subdomains 
$(\Omega_n)_{n\in\mathbb{N}}$. Then we establish an a priori estimate of the type
\begin{equation*}
\| u_n \|^2_{H^{1,2}(\Omega_n)}\leq K \| f_n \|^2_{L^{2}(\Omega_n)}
\leq K \| f \|^2_{L^{2}(\Omega)},
\end{equation*}
where $u_n $ is the solution of \eqref{2} in $\Omega_n$ and  $K$ is a constant 
independent of $n$. This inequality allows us to pass to the limit in $n$. 
Finally, Section \ref{s4} is devoted to problem \eqref{2} in the case when 
$\varphi_1$ and $\varphi_2$ are monotone only near $0$.

Our main result is as follows.

\begin{theorem}\label{th1}
Assume that $c$ and $(\varphi_i(t))_{i=1,2}$ satisfy the conditions 
\eqref{30}, \eqref{3} and \eqref{4}.
Then, the problem
\begin{gather*}
\partial_t u(t,x)+c(t)u(t,x)\partial_x u(t,x)-\partial^2_xu(t,x)=f(t,x) \quad 
 (t,x)\in\Omega,\\
u(t,\varphi_1(t))=u(t,\varphi_2(t))=0 \quad t\in (0,T),\\
\end{gather*}
 admits in the triangular domain $\Omega$ a unique solution 
$u\in H^{1,2}(\Omega)$ in the following cases:

\noindent \textbf{Case 1.} $\varphi_1$ (resp $\varphi_2$) is a decreasing 
(resp increasing) function on $(0,T)$.

\noindent \textbf{Case 2.} $\varphi_1$ (resp $\varphi_2$) is a decreasing 
(resp increasing) function only near $0$.
\end{theorem}

Theses cases will be proved  in  Section \ref{s3}  
and Section \ref{s4}, respectively.

\section{Solution in a domain that can be transformed into a rectangle}\label{s2}

Let $\Omega\subset\mathbb{R}^2$ be the domain
\begin{gather*}
\Omega=\{(t,x)\in \mathbb{R}^2: 0<t<T, \; x\in I_t\},\\
I_t=\{x\in \mathbb{R}: \varphi_1(t)<x<\varphi_2(t), \; t\in(0,T)\}.
\end{gather*}
In this section, we assume that $\varphi_1(0)\neq \varphi_2(0)$.
 In other words
\begin{equation}\label{19}
\varphi_1(t)<\varphi_2(t)\quad \text{for all }  t\in[0,T].
\end{equation}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} %yas12.png
\end{center}
\caption{Domain that can be transformed into a rectangle.} \label{fig1}
\end{figure}


\begin{theorem}\label{th2}
If $f\in L^2(\Omega)$ and $c(t)$,  $(\varphi_i)_{i=1,2}$ satisfy 
the assumptions \eqref{3}, \eqref{4}  and \eqref{19}, then the problem
\begin{equation}\label{5}
\begin{gathered}
\partial_t u(t,x)+c(t)u(t,x)\partial_x u(t,x)-\partial^2_xu(t,x)=f(t,x) \quad 
 (t,x)\in\Omega,\\
u(0,x)=0 \quad x\in J=(\varphi_1(0),\varphi_2(0)),\\
u(t,\varphi_1(t))=u(t,\varphi_2(t))=0 \quad t\in (0,T),\\
\end{gathered}
\end{equation}
 admits a solution $u \in H^{1,2}(\Omega)$.
\end{theorem}

\begin{proof}
The change of variables: $\Omega \to R$
\[
(t,x)\mapsto (t,y)=\Big(t,\frac{x-\varphi_1(t)}{\varphi_2(t)-\varphi_1(t)}\Big)
\]
transforms $\Omega$ into the rectangle $R=(0,T)\times(0,1)$. 
Putting $u(t,x)=v(t,y)$ and $f(t,x)=g(t,y)$, problem \eqref{5} becomes
 \begin{equation}\label{7}
\begin{gathered}
\begin{aligned}
&\partial_tv(t,y)+p(t) v(t,y)\partial_yv(t,y)-q(t)\partial_y^2v(t,y)+r(t,y)\partial_yv(t,y)\\
&=g(t,y)  \quad(t,y)\in R,
\end{aligned}\\
v(0,y)=0  \quad  y\in(0,1),\\
v(t,0)=v(t,1)=0 \quad t\in (0,T),
\end{gathered}
\end{equation}
where
\begin{gather*}
\varphi(t)=\varphi_2(t)-\varphi_1(t), \quad 
p(t)=\frac{c(t)}{\varphi(t)}, \\
q(t)=\frac{1}{\varphi^2(t)}, \quad
r(t,y)=-\frac{y\varphi'(t)+\varphi_1'(t)}{\varphi(t)}.
\end{gather*}
This change of variables preserves 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*}
According to \eqref{3} and \eqref{4},  the functions $p, q$ and $r$ 
satisfy the following conditions
\begin{gather*}
\alpha<p(t)<\beta, \quad\forall t\in[0,T], \\
\alpha<q(t)<\beta,\quad\forall t\in[0,T], \\
| \partial_yr(t,y)|\leq \beta,\quad\forall(t,y)\in R,
\end{gather*}
where $\alpha$ and $\beta$ are positive constants.

So, problem \eqref{5} is equivalent to problem \eqref{7}, and by \cite{benia}
problem \eqref{7} admits a solution $v\in H^{1,2}(R)$. 
Then, problem \eqref{5} in the domain $\Omega$ admits a solution  
$u\in H^{1,2}(\Omega)$.
\end{proof}

\section{Proof of Theorem \ref{th1}, Case 1}\label{s3}

Let
\begin{gather*}
\Omega=\{(t,x)\in \mathbb{R}^2: 0<t<T, \; x\in I_t\},\\
I_t=\{x\in \mathbb{R}: \varphi_1(t)<x<\varphi_2(t), \; t\in(0,T)\},
\end{gather*}
with
$\varphi_1(0)=\varphi_2(0)$ and $\varphi_1(T)<\varphi_2(T)$.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2} %yas11.png
\end{center}
\caption{Non-parabolic domain.} \label{fig2}
\end{figure}

For each $n\in \mathbb{N}^\star$, we define
\[
\Omega_n=\{(t,x)\in\mathbb{R}^2: \frac{1}{n}<t<T, \; x\in I_t\},
\]
and we set $f_n=f_{|\Omega_n}$, where $f$ is given in $L^2(\Omega)$. 
By Theorem \ref{th2} there exists a solution $u_n\in H^{1,2}(\Omega_n)$ 
of the problem
\begin{equation}\label{8}
\begin{gathered}
\partial_t u_n(t,x)+c(t)u_n(t,x)\partial_x u_n(t,x)-\partial^2_xu_n(t,x)\\
=f_n(t,x) \quad  (t,x)\in\Omega_n,
\\
u_n(\frac{1}{n},x)=0,  \quad \varphi_1(\frac{1}{n})<x<\varphi_2(\frac{1}{n}), \\
u_n(t,\varphi_1(t))=u_n(t,\varphi_2(t))=0 \quad t\in [\frac{1}{n},T],
\end{gathered}
\end{equation}
in $\Omega_n$.

To prove Case 1 of Theorem \ref{th1}, we have to pass to the limit in  
\eqref{8}. For this purpose we need the following result.

\begin{proposition}\label{p1}
There exists a positive constant $K$ independent of $n$ such that
\begin{equation*}
\| u_n \|^2_{H^{1,2}(\Omega_n)}\leq K \| f_n \|^2_{L^{2}(\Omega_n)}
\leq K \| f \|^2_{L^{2}(\Omega)}.
\end{equation*}
\end{proposition}

To prove this proposition we need some preliminary results.

\begin{lemma}\label{l1}
There exists a positive constant $K_1$ independent of $n$ such that
\begin{gather}\label{10}
\| u_n \|^2_{L^2(\Omega_n)}\leq K_1 \| \partial_x u_n \|^2_{L^2(\Omega_n)}, \\
\label{11}
\| \partial_x u_n \|^2_{L^2(\Omega_n)}\leq K_1 \| f_n\|^2_{L^2(\Omega_n)}.
\end{gather}
\end{lemma}

\begin{proof}
We have
\begin{equation*}
| u_n|^2=\Big| \int_{\varphi_1(t)}^{x}\partial_x u_n\,\mathrm{d}s\Big|^2
\leq (x-\varphi_1(t))\int_{\varphi_1(t)}^{x}| \partial_x u_n|^2\,\mathrm{d}s.
\end{equation*}
 integrating from $\varphi_1(t)$ to $\varphi_2(t)$, we obtain
\begin{equation*}
\int_{\varphi_1(t)}^{\varphi_2(t)}| u_n|^2\,\mathrm{d}x
\leq\int_{\varphi_1(t)}^{\varphi_2(t)}
\Big((x-\varphi_1(t))\int_{\varphi_1(t)}^{x}| \partial_x u_n|^2\,\mathrm{d}s\Big)
\,\mathrm{d}x,\\
\end{equation*}
hence
\begin{equation*}
\int_{\varphi_1(t)}^{\varphi_2(t)}| u_n|^2\,\mathrm{d}x
\leq(\varphi_2(t)-\varphi_1(t))\int_{\varphi_1(t)}^{\varphi_2(t)}
\int_{\varphi_1(t)}^{\varphi_2(t)}| \partial_x u_n|^2\,\mathrm{d}x\,\mathrm{d}x,
\end{equation*}
and
\begin{equation*}
\int_{\varphi_1(t)}^{\varphi_2(t)}| u_n|^2\,\mathrm{d}x
\leq(\varphi_2(t)-\varphi_1(t))^2\int_{\varphi_1(t)}^{\varphi_2(t)}
|\partial_xu_n|^2\,\mathrm{d}x.
\end{equation*}
Then, there exists a positive constant $K_1$ independent of $n$ such that
\begin{equation*}
\| u_n\|^2_{L^2(I_t)}\leq K_1 \|\partial_x u_n\|^2_{L^2(I_t)},
\end{equation*}
integrating between $\frac{1}{n}$ and $T$ we obtain  inequality \eqref{10}.

Now, multiplying both sides of \eqref{8} by $u_n$ and integrating between
 $\varphi_1(t)$ and $\varphi_2(t)$, we obtain
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\int_{\varphi_1(t)}^{\varphi_2(t)} (u_n)^2\,\mathrm{d}x
+c(t)\int_{\varphi_1(t)}^{\varphi_2(t)}\partial_xu_nu^2_n\,\mathrm{d}x
-\int_{\varphi_1(t)}^{\varphi_2(t)}u_n\partial^2_xu_n\,\mathrm{d}x
=\int_{\varphi_1(t)}^{\varphi_2(t)} f_nu_n\,\mathrm{d}x.
\end{equation*}
Integration by parts gives
\begin{equation*}
c(t)\int_{\varphi_1(t)}^{\varphi_2(t)}\partial_xu_nu^2_n\,\mathrm{d}x
=\frac{c(t)}{3} \int_{\varphi_1(t)}^{\varphi_2(t)}\partial_x(u_n)^3\,\mathrm{d}x=0;
\end{equation*}
then
\begin{equation}\label{12}
\frac{1}{2}\frac{d}{dt}\int_{\varphi_1(t)}^{\varphi_2(t)}( u_n)^2\,\mathrm{d}x
+\int_{\varphi_1(t)}^{\varphi_2(t)}(\partial_xu_n)^2\,\mathrm{d}x
=\int_{\varphi_1(t)}^{\varphi_2(t)}f_nu_n\,\mathrm{d}x.
\end{equation}
 By integrating \eqref{12} from $1/n$ to $T$, we find that
\begin{align*}
&\frac{1}{2}\| u_n(T,x) \|^2_{L^2(I_T)}+
\int_{1/n}^T\| \partial_x u_n(s) \|^2_{L^2(I_t)}\,\mathrm{d}s\\
&\leq \int_{1/n}^T\| f_n(s)\|_{L^2(I_t)}\| u_n(s) \|_{L^2(I_t)}\,
\mathrm{d}s.
\end{align*}
Using the elementary inequality
\begin{equation}
|rs|\leq \frac{\varepsilon }{2}r^2+\frac{s^2}{2\varepsilon },\quad
\forall r,s\in R,\;  \forall \varepsilon >0,  \label{13}
\end{equation}
with $\varepsilon =K_1$, we obtain
\begin{align*}
&\frac{1}{2}\| u_n(T,x) \|^2_{L^2(I_T)}+ \int_{1/n}^T
\| \partial_x u_n(s) \|^2_{L^2(I_t)}\, \mathrm{d}s\\
&\leq \frac{K_1}{2} \int_{1/n}^T\| f_n(s)\|^2_{L^2(I_t)}
\, \mathrm{d}s+\frac{1}{2K_1}\int_{1/n}^T\| u_n(s) \|^2_{L^2(I_t)}\,
\mathrm{d}s.
\end{align*}
Thanks to \eqref{10}, we have
\begin{equation}
\| u_n(T,x)\|^2_{L^2(I_T)}+ \int_{1/n}^T\| \partial_x u_n(s) \|^2_{L^2(I_t)}\,
\mathrm{d}s\leq K_1 \int_{1/n}^T\| f_n(s)\|^2_{L^2(I_t)}\, \mathrm{d}s,
\end{equation}
so,
\begin{equation*}
\| \partial_x u_n \|^2_{L^2(\Omega_n)}\leq K_1\| f_n\|^2_{L^2(\Omega_n)}.
\end{equation*}
\end{proof}

\begin{corollary}\label{c1}
There exists a positive constant $K_2$ independent of $n$,
such that for all $t\in [1/n,T]$,
\begin{equation*}
\| \partial _xu_n\| _{L^2(I_t)}^2+\int_{1/n}^T\| \partial _x^2u_n(s)\| _{L^2(I_t)}^2\,
\mathrm{d}s\leq K_2.
\end{equation*}
\end{corollary}

\begin{proof}
Multiplying both sides of \eqref{8} by $\partial^2_xu_n$ and integrating between 
$\varphi_1(t)$ and $\varphi_2(t)$, we obtain
\begin{equation}\label{14}
\begin{aligned}
& \frac{1}{2}\frac{d}{dt} \int_{\varphi_1(t)}^{\varphi_2(t)}
(\partial_x u_n)^2\mathrm{d}x
+  \int_{\varphi_1(t)}^{\varphi_2(t)}(\partial^2_x u_n)^2\, \mathrm{d}x\\
& = -\int_{\varphi_1(t)}^{\varphi_2(t)}f_n\partial^2_x u_n\, \mathrm{d}x
+c(t) \int_{\varphi_1(t)}^{\varphi_2(t)}u_n\partial_x u_n\partial^2_x
u_n\, \mathrm{d}x.
\end{aligned}
\end{equation}
Using Cauchy-Schwartz inequality, \eqref{13} with
$\varepsilon =\frac{1}{2}$ leads to
\begin{equation}
\begin{aligned}
|  \int_{\varphi_1(t)}^{\varphi_2(t)}f_n\partial _x^2u_n\, \mathrm{d}x|
& \leq  \Big(\int_{\varphi_1(t)}^{\varphi_2(t)}| \partial _x^2u_n| ^2
\, \mathrm{d}x\Big) ^{1/2}
\Big(\int_{\varphi_1(t)}^{\varphi_2(t)}| f_n| ^2\, \mathrm{d}x\Big) ^{1/2} \\
& \leq  \frac{1}{4} \int_{\varphi_1(t)}^{\varphi_2(t)}|
\partial _x^2u_n| ^2\, \mathrm{d}x+
 \int_{\varphi_1(t)}^{\varphi_2(t)}| f_n| ^2\, \mathrm{d} x.
\end{aligned} \label{15}
\end{equation}
Now, we have to estimate the last term of \eqref{14}. An integration by parts
gives
\begin{equation*}
\int_{\varphi_1(t)}^{\varphi_2(t)}u_n\partial _xu_n\partial
_x^2u_n\, \mathrm{d}x
= \int_{\varphi_1(t)}^{\varphi_2(t)}u_n\partial _x
\Big(\frac{1}{2}(\partial _xu_n)^2\Big) \, \mathrm{d}x
=- \frac{1}{2} \int_{\varphi_1(t)}^{\varphi_2(t)}(\partial_xu_n) ^{3}\, \mathrm{d}x.
\end{equation*}

Since $\partial _xu_n$ satisfies $\int_{\varphi_1(t)}^{\varphi_2(t)}\partial _xu_n\,
\mathrm{d}x=0$ we deduce that the continuous function $\partial _xu_n$
is zero at some point $\xi(t)\in (\varphi_1(t),\varphi_2(t))$, and  by integrating
$2\partial_xu_n\partial _x^2u_n$ between $\xi(t)$ and $x$, we obtain
\begin{equation*}
2 \int _{\xi(t)}^{x}\partial _x u_n\partial _x^2u_n\, \mathrm{d}s \int _{\xi(t)}^{x}
= \partial _x(\partial _xu_n)^2\, \mathrm{d}s
=(\partial _xu_n)^2,
\end{equation*}
the Cauchy-Schwartz inequality gives
\begin{equation*}
\| \partial _xu_n\| _{L^{\infty }(I_t)}^2
\leq 2\| \partial _xu_n\| _{L^2(I_t)}\| \partial _x^2u_n\| _{L^2(I_t)},
\end{equation*}
but
\begin{equation*}
\| \partial _xu_n\| _{L^{3}(I_t)}^{3}\leq \| \partial
_xu_n\| _{L^2(I_t)}^2\| \partial _xu_n\| _{L^{\infty}(I_t)},
\end{equation*}
so, \eqref{3} yields
\begin{equation*}
|  \int_{\varphi_1(t)}^{\varphi_2(t)} c (t)u_n\partial
_xu_n\partial _x^2u_n\, \mathrm{d}x|
\leq \Big( \int_{\varphi_1(t)}^{\varphi_2(t)}| \partial _x^2u_n|
^2\, \mathrm{d}x\Big) ^{1/4}\Big(c_2^{4/5}
\int_{\varphi_1(t)}^{\varphi_2(t)}| \partial _xu_n| ^2\, \mathrm{d}x\Big) ^{5/4}.
\end{equation*}

Finally, by Young's inequality
$|AB|\leq \frac{|A|^{p}}{p}+\frac{|B|^{p'}}{p'}$, with $1<p<\infty $ and
$p'=\frac{p}{p-1}$.
Choosing $p=4$  (then $p'=\frac{4}{3}$)
\[
A=\Big(\int_{\varphi_1(t)}^{\varphi_2(t)}| \partial _x^2u_n| ^2\, \mathrm{d}x\Big)^{1/4},\quad
 B=\Big(c_2^{4/5} \int_{\varphi_1(t)}^{\varphi_2(t)}| \partial
_xu_n| ^2\, \mathrm{d}x\Big)^{5/4},
\]
the estimate of the last term of \eqref{14} becomes
\begin{equation}
\begin{aligned}
&\big|  \int_{\varphi_1(t)}^{\varphi_2(t)}c (t)u_n\partial
_xu_n\partial _x^2u_n\, \mathrm{d}x\big|\\
&\leq \frac{1}{4} \int_{\varphi_1(t)}^{\varphi_2(t)}| \partial _x^2u_n| ^2\, \mathrm{d}x+\frac{3}{4}c_2^{4/3}
\Big(\int_{\varphi_1(t)}^{\varphi_2(t)}| \partial _xu_n| ^2\, \mathrm{d}x\Big) ^{5/3}.
\end{aligned}
\label{16}
\end{equation}
Let us return to \eqref{14}:
 By integrating between $\frac{1}{n}$ and $t$,
 from the estimates \eqref{15} and \eqref{16}, we obtain
\begin{align*}
&\| \partial_xu_n\|^2_{L^2(I_t)}+
\int_{1/n}^{t}\| \partial^2_xu_n(s)\|^2_{L^2(I_t)}\, \mathrm{d}s \\
&\leq 2
\int_{1/n}^{t} \| f_n(s)\|^2_{L^2(I_t)}\, \mathrm{d}s
 +\frac{3}{2}c_2^{4/3} \int_{1/n}^{t}\Big(\|
\partial_xu_n(s)\|^2_{L^2(I_t)}\Big)^{5/3}\, \mathrm{d}s.
\end{align*}
$f_n\in L^2(\Omega_n)$, then there exists a constant $c_3$ such that
\begin{align*}
&\| \partial_xu_n\|^2_{L^2(I_t)}+
\int_{1/n}^{t}\| \partial^2_xu_n(s)\|^2_{L^2(I_t)}\,\mathrm{d}s\\
&\leq c_3+ \frac{3}{2}c_2^{4/3} \int_{1/n}^{t}\Big(\|
\partial_xu_n(s)\|^2_{L^2(I_t)}\Big)^{2/3}
\| \partial_xu_n(s)\|^2_{L^2(I_t)}\, \mathrm{d}s.
\end{align*}
Consequently, the function
\[
\varphi (t)=\| \partial _xu_n\|
_{L^2(I_t)}^2+\int_{1/n}^{t}\| \partial
_x^2u_n(s)\| _{L^2(I_t)}^2\, \mathrm{d}s
\]
 satisfies the inequality
\begin{equation*}
\varphi (t)\leq c_3+\int_{1/n}^{t}\Big(\frac{3}{2}c_2^{4/3} \| \partial
_xu_n(s)\| _{L^2(I_t)}^{4/3}\Big)\varphi (s)\mathrm{d}s,
\end{equation*}
Gronwall's inequality shows that
\begin{equation*}
\varphi (t)\leq c_3\exp \Big(\int_{1/n}^{t}(\frac{3}{2}c_2^{4/3} \| \partial
_xu_n(s)\| _{L^2(I_t)}^{4/3})\mathrm{d}s\Big) .
\end{equation*}
According to Lemma \ref{l1} the integral
$\int_{1/n}^{t}\| \partial _xu_n\| _{L^2(I_t)}^{4/3}\mathrm{d}s$ 
is bounded by a constant
independent of $n$. So there exists a positive constant $K_2$
such that
\begin{equation*}
\| \partial _xu_n\| _{L^2(I_t)}^2+
\int_{1/n}^T\| \partial _x^2u_n(s)\| _{L^2(I_t)}^2\,\mathrm{d}s\leq K_2.
\end{equation*}
\end{proof}

\begin{lemma}\label{l2}
There exists a constant $K_3$ independent of $n$ such that
\begin{equation*}
\| \partial_t u_n \|^2_{L^2(\Omega_n)}
+\| \partial^2_x u_n \|^2_{L^2(\Omega_n)}\leq K_3\| f_n\|^2_{L^2(\Omega_n)}.
\end{equation*}
\end{lemma}

Then Theorem \ref{p1} is a direct consequence of  Lemmas \ref{l1} and \ref{l2}.

\begin{proof}
To prove Lemma  \ref{l2}, we develop the inner product in $L^2(\Omega_n)$,
\begin{align*}
\| f_n\|^2_{L^2(\Omega_n)}
&=(\partial_tu_n+c(t) u_n\partial_xu_n-\partial^2_xu_n,
 \partial_tu_n+c(t)u_n\partial_xu_n-\partial^2_xu_n)_{L^2(\Omega_n)}\\
&=\| \partial_tu_n \|^2_{L^2(\Omega_n)}+\| \partial^2_xu_n\|^2_{L^2(\Omega_n)}
 +\| c(t)u_n\partial_xu_n\|^2_{L^2(\Omega_n)}\\
&\quad -2(\partial_tu_n,\partial^2_xu_n)_{L^2(\Omega_n)}
 +2(\partial_tu_n,c(t)u_n\partial_xu_n)_{L^2(\Omega_n)}\\
&\quad -2(c(t)u_n\partial_xu_n,\partial^2_xu_n)_{L^2(\Omega_n)},
\end{align*}
so,
\begin{equation}\label{17}
\begin{aligned}
&\| \partial_tu_n \|^2_{L^2(\Omega_n)}+\|  \partial^2_xu_n\|^2_{L^2(\Omega_n)}\\
&=\| f_n\|^2_{L^2(\Omega_n)}-\| c(t)u_n\partial_xu_n\|^2_{L^2(\Omega_n)}
 +2(c(t)u_n\partial_xu_n,\partial^2_xu_n)_{L^2(\Omega_n)}
\\
&\quad -2(\partial_tu_n,c(t)u_n\partial_xu_n)_{L^2(\Omega_n)}
 +2(\partial_tu_n,\partial^2_xu_n)_{L^2(\Omega_n)}.
\end{aligned}
\end{equation}
Using \eqref{3} and \eqref{13} with $\varepsilon=1/2$, we obtain
\begin{equation}\label{21}
\left| -2(\partial_tu_n,c(t)u_n\partial_xu_n)_{L^2(\Omega_n)}\right|
\leq \frac{1}{2}\| \partial_tu_n \|^2_{L^2(\Omega_n)}+2c_2^2\| u_n\partial_xu_n\|^2_{L^2(\Omega_n)},
\end{equation}
and
\begin{equation}\label{20}
\left| 2(c(t)u_n\partial_xu_n,\partial^2_xu_n)_{L^2(\Omega_n)}\right|
\leq 2c_2^2\| u_n\partial_xu_n\|^2_{L^2(\Omega_n)}+\frac{1}{2}
\| \partial^2_xu_n \|^2_{L^2(\Omega_n)}.
\end{equation}

Now calculating the last term of \eqref{17},
\begin{align*}
(\partial_tu_n,\partial^2_xu_n)_{L^2(\Omega_n)}
&=-\int_{1/n}^T\int_{\varphi_1(t)}^{\varphi_2(t)}\partial_t(\partial_xu_n)
  \partial_xu_n \,\mathrm{d}x\mathrm{d}t
 +\int_{1/n}^T\left[\partial_tu_n\partial_xu_n\right]_{\varphi_1(t)}^{\varphi_2(t)} 
 \,\mathrm{d}t\\
&=-\frac{1}{2}\int_{1/n}^T\int_{\varphi_1(t)}^{\varphi_2(t)}
 \partial_t(\partial_xu_n)^2  \,\mathrm{d}x\mathrm{d}t
 +\int_{1/n}^T\left[\partial_tu_n\partial_xu_n\right]_{\varphi_1(t)}^{\varphi_2(t)} 
 \,\mathrm{d}t\\
&=-\frac{1}{2}\Big[\int_{\varphi_1(t)}^{\varphi_2(t)}(\partial_xu_n)^2  
 \,\mathrm{d}x\Big]_{1/n}^T
 +\int_{1/n}^T\left[\partial_tu_n\partial_xu_n\right]_{\varphi_1(t)}^{\varphi_2(t)} 
 \,\mathrm{d}t\\
&=-\frac{1}{2}\int_{\varphi_1(T)}^{\varphi_2(T)}(\partial_xu_n)^2(T,x)  \,\mathrm{d}x
 +\frac{1}{2}\int_{\varphi_1(\frac{1}{n})}^{\varphi_2(\frac{1}{n})}
 (\partial_xu_n)^2(\frac{1}{n},x)\,\mathrm{d}x\\
&\quad +\int_{1/n}^T\partial_tu_n(t,\varphi_2(t))\partial_xu_n(t,\varphi_2(t)) 
 \,\mathrm{d}t\\
&\quad -\int_{1/n}^T\partial_tu_n(t,\varphi_1(t))\partial_xu_n(t,\varphi_1(t)) 
 \,\mathrm{d}t.
\end{align*}
According to the boundary conditions, we have
\begin{equation*}
 \partial_t u_n(t,\varphi_i(t))+\varphi'_i(t)\partial_x u_n(t,\varphi_i(t))=0, \quad
 i=1,2,
\end{equation*}
so
\begin{align*}
(\partial_tu_n,\partial^2_xu_n)_{L^2(\Omega_n)} 
&=-\frac{1}{2}\int_{\varphi_1(T)}^{\varphi_2(T)}(\partial_xu_n)^2(T,x) 
 \,\mathrm{d}x
-\int_{1/n}^T\varphi'_2(t)(\partial_xu_n(t,\varphi_2(t)))^2 \,\mathrm{d}t\\
&\quad +\int_{1/n}^T\varphi'_1(t)(\partial_xu_n(t,\varphi_1(t)))^2 \,\mathrm{d}t,
\end{align*}
it follows that
\begin{equation} \label{22}
(\partial_tu_n,\partial^2_xu_n)\leq 0.
\end{equation}
From \eqref{21}, \eqref{20} and \eqref{22},  \eqref{17} becomes
\begin{equation}\label{18}
\| \partial_tu_n \|^2_{L^2(\Omega_n)}+\| \partial^2_xu_n\|^2_{L^2(\Omega_n)}\leq
2\| f_n\|^2_{L^2(\Omega_n)}+10c_2^2\| u_n\partial_xu_n\|^2_{L^2(\Omega_n)}.
\end{equation}

On the other hand, using the injection of $H^{1}_{0}(I_t)$
in $L^{\infty}(I_t)$, we obtain
\begin{align*}
\big|  \int _{1/n}^T \int _{\varphi_1(t)}^{\varphi_2(t)}(u_{n}
\partial_{x}u_{n})^{2}\, \mathrm{d}x\,\mathrm{d}t\big|  
& \leq   \int _{1/n}^T\Big( \| u_{n}\|^{2}_{L^{\infty}(I_t)} 
\int_{\varphi_1(t)}^{\varphi_2(t)}| \partial_{x}u_{n}|^{2}\, \mathrm{d}x\big)
\, \mathrm{d}t\\
& \leq  \int _{1/n}^T\| u_{n}\|^{2}_{H^{1}_{0}(I_t)}
 \| \partial_{x}u_{n}\|^{2}_{L^{2}(I_t)}\, \mathrm{d}t\\
& \leq \| u_{n}\|^{2}_{L^{\infty}(\frac{1}{n},T;H^{1}_{0}(I_t))}
 \| \partial_{x}u_{n}\|^{2}_{L^{2}(\Omega_n)},
\end{align*}
According to Corollary \ref{c1}, $\| u_{n}\|^{2}_{L^{\infty
}(\frac{1}{n},T;H^{1}_{0}(I_t))}$ is bounded, then
by \eqref{11} and \eqref{18}, there exists a constant $K_3$  independent of $n$,
 such that
\begin{equation*}
\| \partial_t u_n \|^2_{L^2(\Omega_n)}
 +\| \partial^2_x u_n \|^2_{L^2(\Omega_n)}\leq K_3\| f_n\|^2_{L^2(\Omega_n)}.
\end{equation*}
However,
\begin{equation*}
\| f_n\|^2_{L^2(\Omega_n)}\leq \| f\|^2_{L^2(\Omega)},
\end{equation*}
 then, from lemmas \ref{l1} and \ref{l2} , there exists a constant $K$  
independent of $n$, such that
\begin{equation*}
\| u_n \|^2_{H^{1,2}(\Omega_n)}\leq K \| f_n\|^2_{L^2(\Omega_n)}
\leq K \| f\|^2_{L^2(\Omega)}.
\end{equation*}
This completes the proof.
\end{proof}

\subsubsection*{Existence and uniqueness}
Choose a sequence $(\Omega_n)_{n\in\mathbb{N}}$ of the domains 
defined previously, such that $\Omega_n\subseteq\Omega$, as 
$n\longrightarrow+\infty$ then $\Omega_n\longrightarrow\Omega$.

Consider $u_n\in H^{1,2}(\Omega_n)$ the solution of
\begin{gather*}
\partial_t u_n(t,x)+c(t)u_n(t,x)\partial_x u_n(t,x)-\partial^2_xu_n(t,x)
=f_n(t,x) \quad  (t,x)\in\Omega_n,\\
u_n(\frac{1}{n},x)=0 \quad \varphi_1(\frac{1}{n})<x<\varphi_2(\frac{1}{n}), \\
u_n(t,\varphi_1(t))=u_n(t,\varphi_2(t))=0 \quad t\in ]\frac{1}{n},T[.
\end{gather*}
We know that a solution $u_n$ exists by the Theorem \ref{th2}. 
Let $\widetilde{u_n}$ be the  extension by zero of $u_n$ outside $\Omega_n$.
From the proposition \ref{p1} results the inequality
\begin{equation*}
\| \widetilde{u_n} \|^2_{L^2(\Omega_n)}
+\| \partial_t\widetilde{ u_n} \|^2_{L^2(\Omega_n)}
+\| \partial_x\widetilde{ u_n} \|^2_{L^2(\Omega_n)}
+\| \partial^2_x\widetilde{ u_n} \|^2_{L^2(\Omega_n)}\leq C\| f\|^2_{L^2(\Omega)}.
\end{equation*}
This implies that $\widetilde{u_n}$, $\partial_t\widetilde{ u_n}$ and
 $\partial^j_x \widetilde{u_n}$, $j=1,2$ are bounded in $L^2(\Omega_n)$, 
from Corollary \ref{c1} $\widetilde{u_n\partial_x u_n}$  is bounded in  $L^2(\Omega_n)$. So, it is possible to extract a subsequence from $u_n$, still denoted $u_n$ such that
\begin{equation*}
\partial_t\widetilde{ u_n}\to {\partial_t u} \quad \text{weakly in} \quad L^2(\Omega_n),
\end{equation*}
\begin{equation*}
\widetilde{\partial^2_x u_n}\to {\partial^2_x u} \quad \text{weakly in} \quad L^2(\Omega_n),
\end{equation*}
 \begin{equation*}
\widetilde{u_n}\partial_x \widetilde{u_n}u_n \to u\partial_x u \quad
 \text{weakly in } L^2(\Omega_n).
\end{equation*}
Then $u\in H^{1,2}(\Omega)$ is solution to problem \eqref{2}.

For the uniqueness, let us observe that any solution $u\in H^{1,2}(\Omega)$ 
of problem \eqref{2}  is in $L^{\infty }(0,T,H^1_0(I_t))$. 
Indeed, by the same way as in Corollary \ref{c1}, we prove that there exists 
a positive constant $K_2$ such that for all $t\in [0,T]$
\begin{equation*}
\| \partial _xu\| _{L^2(I_t)}^2+\int_{0}^T\| \partial _x^2u(s)\| _{L^2(I_t)}^2
\,\mathrm{d}s\leq K_2.
\end{equation*}

Let $u_1,u_2\in H^{1,2}(\Omega)$ be two solutions of \eqref{2}.
 We put $u=u_1-u_2$. It is clear that $u\in L^{\infty }(0,T,H^1_0(I_t))$.
The equations satisfied by $u_1$ and $u_2$ leads to
\begin{equation*}
 \int _{\varphi_1(t)}^{\varphi_2(t)}[ \partial _{t}uw+c
(t)uw\partial _xu_1+c (t)u_2w\partial _xu+\partial
_xu\partial _xw] \, \mathrm{d}x=0.
\end{equation*}
Taking, for $t\in [ 0,T]$, $w=u$ as a test function, we deduce that
\begin{equation}  \label{122}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}\| u\| _{L^2(I_t)}^2+\| \partial _xu\| _{L^2(I_t)}^2\\
&=-c(t) \int _{\varphi_1(t)}^{\varphi_2(t)}u^2\partial
_xu_1\, \mathrm{d}x -c(t) \int_{\varphi_1(t)}^{\varphi_2(t)}u_2u\partial _xu
\, \mathrm{d}x.
\end{aligned}
\end{equation}
An integration by parts gives
\begin{equation*}
c (t) \int _{\varphi_1(t)}^{\varphi_2(t)}u^2\partial _xu_1\,
\mathrm{d}x=-2c (t) \int _{\varphi_1(t)}^{\varphi_2(t)}u\partial_xuu_1\, \mathrm{d}x,
\end{equation*}
then \eqref{122} becomes
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\| u\| _{L^2(I_t)}^2+\|
\partial _xu\| _{L^2(I_t)}^2=
\int _{\varphi_1(t)}^{\varphi_2(t)}c (t)(2u_1-u_2) u\partial _xu\,
\mathrm{d}x.
\end{equation*}
By \eqref{3} and inequality \eqref{13} with
$\varepsilon =2$, we obtain
\begin{align*}
&\big|  \int _{\varphi_1(t)}^{\varphi_2(t)}c (t)(2u_1-u_2) u\partial _xu
\, \mathrm{d}x\big| \\
&\leq \frac{1}{4}c _2^2(2\| u_1\| _{L^{\infty
}(0,T,H^1_0(I_t))}+\| u_2\| _{L^{\infty }(0,T,H^1_0(I_t))}) ^2
\| u\| _{L^2(I_t)}^2+\| \partial _xu\| _{L^2(I_t)}^2.
\end{align*}
So, we deduce that there exists a non-negative constant $D$, such as
\begin{equation*}
\frac{1}{2}\frac{d}{dt}\| u\| _{L^2(I_t)}^2
\leq D\| u\|_{L^2(I_t)}^2,
\end{equation*}
and Gronwall's lemma leads to $u=0$. This completes the proof of 
Theorem \ref{th1}, Case 1.

\section{Proof of Theorem \ref{th1}, Case 2}\label{s4}

 In this case  we set $\Omega=Q_1\cup Q_2\cup \Gamma_{T_1}$ where
\begin{gather*}
Q_1=\{(t,x)\in \mathbb{R}^2:  0<t<T_1, \; x\in I_t\},\\
Q_2=\{(t,x)\in \mathbb{R}^2:  T_1<t<T, \; x\in I_t\},\\
\Gamma_{T_1}=\{(T_1,x)\in \mathbb{R}^2:  x\in I_{T_1}\},
\end{gather*}
with $T_1$ small enough. $f\in L^2(\Omega)$ and $f_i=f_{| Q_i}$, $i=1,2$.

Theorem \ref{th1}, Case 1, applied to the domain $Q_1$, shows that there 
exists a unique solution $u_1\in H^{1,2}(Q_1)$ of the problem
\begin{gather*}
\partial_t u_1(t,x)+c(t)u_1(t,x)\partial_x u_1(t,x)-\partial^2_xu_1(t,x)\\
=f_1(t,x) \quad (t,x)\in Q_1, \\
u_1(t,\varphi_1(t))=u_1(t,\varphi_2(t))=0 \quad t\in (0,T_1).
\end{gather*}

\begin{lemma}\label{l4}
If $u\in H^{1,2}\left((T_1,T)\times(0,1)\right)$, then 
$u_{| t=T_1}\in H^1(\{T_1\}\times(0,1))$.
\end{lemma}

The above lemma is a special case of \cite[Theorem 2.1, Vol. 2]{Lions}.
Using the transformation $[T_1,T]\times[0,1]\to Q_2$,
\[
(t,x) \mapsto (t,y)= (t,(\varphi_2(t)-\varphi_1(t))x+\varphi_1(t))
\]
we deduce from Lemma \ref{l4} the following result.

\begin{lemma}
If $u\in H^{1,2}(Q_2)$, then $u_{| \Gamma_{T_1}}\in H^1(\Gamma_{T_1})$.
\end{lemma}

We denote the trace ${u_1}_{|\Gamma_{T_1}}$ by $u_0$ which is in the 
Sobolev space $H^1(\Gamma_{T_1})$ because $u_1\in H^{1,2}(Q_1)$.

Theorem \ref{th2} applied to the domain $Q_2$,  shows that there exists 
a unique solution $u_2\in H^{1,2}(Q_2)$ of the problem
\begin{gather*}
\partial_t u_2(t,x)+c(t)u_2(t,x)\partial_x u_2(t,x)-\partial^2_xu_2(t,x)
=f_2(t,x) \quad  (t,x)\in Q_2,\\
u_2(0,x)=u_0(x)  \quad \varphi_1(T_1)<x<\varphi_2(T_1), \\
u_2(t,\varphi_1(t))=u_2(t,\varphi_2(t))=0 \quad t\in [T_1,T],
\end{gather*}
putting
\[
u=\begin{cases}
u_1 \quad \text{in } \quad Q_1,\\
u_2 \quad \text{in } \quad Q_2,
\end{cases}
\]
we observe that $u\in H^{1,2}(\Omega)$ because 
${u_1}_{|\Gamma_{T_1}}={u_2}_{|\Gamma_{T_1}}$ and is a solution of the problem
\begin{gather*}
\partial_t u(t,x)+c(t)u(t,x)\partial_x u(t,x)-\partial^2_xu(t,x)=f(t,x) 
\quad  (t,x)\in\Omega,\\
u(t,\varphi_1(t))=u(t,\varphi_2(t))=0 \quad t\in (0,T).
\end{gather*}
We prove the uniqueness of the solution by the same way as in Case 1.

\begin{thebibliography}{99}

\bibitem{Aref} V. N. Aref'ev; L. A. Bagurov;
  \emph{Solutions of the heat equation in domains with singularities}, Mathematical
Notes, 64, 2 (1998), 139-153.

\bibitem{benia} Y. Benia, B.-K. Sadallah;  
\emph{Existence of solutions to Burgers equations in domains that can be transformed 
into rectangles}, Electron. J. Diff. Equ., 157 (2016), 1-13.

\bibitem{Burgers} J. M. Burgers;
 \emph{A mathematical model illustrating the theory of turbulence}, Adv. Appl. Mech.,
1 (1948), 171-199.

\bibitem{Clark} H. R. Clark, M. A. Rincon; 
 \emph{A. Silva; Analysis and numerical simulation of viscous Burgers
equation}, Numerical Functional Analysis and Optimization, 32:7 (2011), 695-716.

\bibitem{Grisvard} P. Grisvard; 
 \emph{Elliptic Problems in Nonsmooth Domains}, Monographs and Studies in Mathematics
24, Pitman, Boston, 1985.

\bibitem{Hopf} E. Hopf; 
The partial differential equation $u_t + uu_x = \nu u_{xx}$,
 Comm. Pure Appl. Math. 3 (1950), 201-230.

\bibitem{Kevorkian}  J. Kevorkian;  
\emph{Partial differential equations: Analytical solution techniquess}, 
Brooks=Cole Pub, Company, California, 1990.

\bibitem{Kheloufi} A. Kheloufi, B.-K. Sadallah;  
\emph{Parabolic equations with Robin type boundary conditions in a 
non-rectangular domain}, Electron. J. Diff. Equ., 25 (2010), 511-523.

\bibitem{Labbes} R. Labbas, A. Medeghri, B.-K. Sadallah; 
\emph{On a parabolic equation in a triangular domain}, Applied Mathematics 
and Computation, 230 (2002), 511-523.

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

\bibitem{Sadallah} B.-K. Sadallah;  
\emph{\'{E}tude d'un probl\`{e}me $2m$-parabolique dans des domaines plan
 non rectangulaires}, Boll. U. M. I., (6) 2-B (1983), 51-112.

\bibitem{Whitham} G. B. Whitham;  
\emph{Lectures on wave propagation}, Narosa Pub. House, New Delhi, 1979.

\end{thebibliography}

\end{document}