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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 138, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/138\hfil Initial boundary-value problems]
{Unique solvability of initial boundary-value problems for
hyperbolic systems in cylinders whose base is a cusp domain}

\author[N. M. Hung,  V. T. Luong\hfil EJDE-2008/138\hfilneg]
{Nguyen Manh Hung,  Vu Trong Luong}  % in alphabetical order

\address{Nguyen Manh Hung \newline
Department of Mathematics, Hanoi National University of Education, Hanoi, Vietnam}
\email{hungnmmath@hnue.edu.vn}

\address{Vu Trong Luong \newline
Department of Mathematics, Taybac University,  Sonla, Vietnam}
\email{vutrongluong@gmail.com}

\thanks{Submitted May 8, 2008. Published October 14, 2008.}
\subjclass[2000]{35D05, 35D10, 35L55, 35M10}
\keywords{Initial boundary-value problems;  hyperbolic systems;
Cusp domain; \hfill\break\indent approximating boundary method;
generalized solution; existence; uniqueness; smoothness}


\begin{abstract}
 We study initial boundary-value problems for  hyperbolic systems
 of divergence form of arbitrary order in cylinders whose base
 is a cusp  domain. Our main results are to prove the  existence,
 uniqueness and the smoothness with respect to time variable of
 generalized solutions of these problems by using the method which
 we will denote as  ``approximating boundary method''.
\end{abstract}

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

\section{Introduction}\label{sec1}

The boundary problems for hyperbolic systems in smooth cylinders
have been well studied.  Fichera \cite{F}  established the
existence and  the uniqueness  of generalized solution, and he had
proved that if the right-hand parts, the coefficients and the
boundary are infinitely differentiable, so is the solution.  In
the case, when the non-smooth cylinders, the indicated methods can
not be applied since it is impossible to straighten the boundary
by a smooth transform.


In this paper, We consider the initial boundary-value problems for
hyperbolic systems in cylinders $Q_T=\Omega\times(0,T),
0<T<+\infty$, with base $\Omega$ is a cusp   domain. In section 2,
it is shown that the existence of a sequence of smooth domains
$\{\Omega^\varepsilon\}$ such that
$\Omega^\varepsilon\subset\Omega$ and $\lim_{\varepsilon\to
0}\Omega^\varepsilon=\Omega$;
 moreover, if $\Omega$ has a cusp point on boundary $\partial \Omega$,
then $\Gamma=\partial \Omega^\varepsilon\cap\partial\Omega$ is a
smooth $(n-1)$-dimensional manifold of the class $C^\infty$. In
section 3 we set up notation and state the initial boundary-value
problems. Section 4 establishes the existence, uniqueness  and the
smoothness with respect to time variable of generalized solutions
of these problems by the approximating boundary  method and
results in Fichera \cite{F}.

The main idea is to apply Theorem \ref{ABT} from Section 2 to
establish the unique solvability of the mentioned problem in
$V^{m,1}(Q_T)$,   where $V^{m,1}(Q_T)$ is a closed subspace of
$H^{m,1}(Q_T)$, and G\aa rding's inequality holds in
$V^{m,1}(Q_T)$.

 \section{Approximating boundary theorem}

\begin{theorem}\label{ABT}
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Then there
exists a sequence of smooth domains $\{\Omega^\varepsilon\}$ such
that $\Omega^\varepsilon\subset\Omega$ and $\lim_{\varepsilon\to
0} \Omega^\varepsilon  =\Omega$.
\end{theorem}

\begin{proof}
For $\varepsilon>0$ arbitrary, set $S^\varepsilon =\{x\in\Omega:
\text{dist}(x, \partial \Omega)\leq \varepsilon\},
\Omega^\varepsilon=\Omega\setminus S^\varepsilon$ and
$\partial\Omega^\varepsilon$ is the boundary of
$\Omega^\varepsilon$. Denote by $J(x)$ the characteristic function
of $\Omega^\varepsilon$ and by $J_h(x)$ the mollification of
$J(x)$, i.e,
$$
J_h(x)= \int_{\mathbb{R}^n}\theta_h(x-y)J(y)dy,
$$
where $\theta_h$ is a mollifier.
If $h<\frac{\varepsilon}{2}$, then $J_h(x)$ has following properties:
\begin{enumerate}
\item  $J_h(x)=0$ if $x\notin\Omega^{\varepsilon/2}$;
\item  $ 0\leq J_h(x)\leq1$;
\item  $J_h(x)=1$ in $\Omega^{2\varepsilon}$;
\item  $J_h\in C^\infty(\mathbb{R}^n)$.
\end{enumerate}
We now fix a constant $ c\in(0,1)$, and set
$\Omega_c^\varepsilon=\{x\in\Omega: J_h(x)>c\}$. It is obvious
that
$\Omega^{\varepsilon/2}\supset\Omega^\varepsilon_c\supset
\Omega^{2\varepsilon}$.
 Therefore, $\Omega_c^\varepsilon\subset\Omega$ and
$\lim_{\varepsilon\to 0}\Omega_c^\varepsilon =\Omega$,
$\partial\Omega_c^\varepsilon=\{x\in\Omega: J_h(x)=c\}$.

Assume that $K$ is a critical set of $J_h$.
By Sard's theorem  $ \mu(J_h(K))=0$, it implies that there exists a
constant $c_0 \in( 0,1)$
such that  $\{x\in \Omega : J_h(x)=c_0\}$ is not a critical set of  $J_h$.

 Denote $\Omega^\varepsilon_{c_0}=\{x\in\Omega: J_h(x)>c_0\}$ and $F(x)=J_h(x)-c_0$.
If $x^0\in \partial\Omega_{c_0}^\varepsilon$, then
$F(x^0)=J_h(x^0)-c_0=0$ and $\text{grad} J_h(x^0)\ne 0$. This
implies there exists a $\frac{\partial J_h}{\partial x_i}(x^0)\ne
0 $, without loss generality we can suppose that $ \frac{\partial
J_h}{\partial x_n}(x^0)\ne 0 $. Using the implicit function
theorem, there exists a neighbourhood
 $W$ of $(x^0_1, \dots , x^0_{n-1})$ in $\mathbb{R}^{n-1}$ a neighbourhood $V$
of $x_n^0$ in $\mathbb{R}$ and an infinitely differentiable
function $z:W\longrightarrow \mathbb{R}$ such that $ x\in
U_{x^0}\cap \partial\Omega_{c_0}^\varepsilon, (\text{where }
U_{x^0}=W\times V)$ if and only if
 $x=(x_1, \dots , x_n)\in U_{x^0}$,  $x_n=z(x_1, \dots , x_{n-1})$.
Hence, $ \Omega_{c_0}^\varepsilon$ is smooth and
$\lim_{\varepsilon\to 0}\Omega_{c_0}^\varepsilon=\Omega$. The
theorem is proved.
\end{proof}
Suppose domain $\Omega$ is not smooth at one isolated point.
The definition is a formal description of domains with a  cusp point.

\begin{definition} \label{def2.1} \rm
We call a bounded domain $\Omega\subset \mathbb{R}^n$ a exterior
cusp domain if
\begin{enumerate}
  \item  $O\in \partial \Omega,  \partial \Omega\setminus\{O\}$ is a smooth $(n-1)$-dimensional manifold of the class $C^\infty$.
  \item Denote $x'=(x_1, x_2, \dots, x_{n-1})$, then
 $$
\{x\in\Omega:  0<x_n<1\}\equiv\{x=(x',x_n) \in\mathbb{R}^n:
|x'|<x_n^k \},\quad k\geq1.$$
\end{enumerate}
\end{definition}

\begin{definition} \label{def2.2}
We call a bounded domain $\Omega\subset \mathbb{R}^n$ a interior cusp
domain if
\begin{enumerate}
  \item  $O\in \partial \Omega,  \partial \Omega\setminus\{O\}$ is a smooth $(n-1)$-dimensional manifold of the class $C^\infty$.
  \item Denote $x'=(x_1, x_2, \dots, x_{n-1})$, then
 $$
\{x\in\Omega:  0<x_n<1\}\equiv\{x=(x',x_n) \in\mathbb{R}^n:
|x'|>x_n^k \}, \quad
k\geq1.
$$
\end{enumerate}
\end{definition}

Let $\varphi_\varepsilon\in C^\infty(\mathbb{R}_+)$,
$0<\varepsilon<1/4$, satisfying $0\leq\varphi_\varepsilon\leq 1$
and
$$
\varphi_\varepsilon(t)=1, \;\forall t<\varepsilon, \quad
\varphi_\varepsilon(t)=0, \; \forall t\geq 2\varepsilon.
$$
When $\Omega$ is a exterior cusp domain,  we set
$$
\Gamma_\varepsilon=\{x\in\mathbb{R}^n: x_n^k
=(1-\varphi_\varepsilon(|x'|))|x'|+2\varepsilon\varphi_\varepsilon(|x'|),
\,  0<x_n<1 \}.
$$
When $\Omega$ is a interior cusp domain,  we set
 $$
\Gamma_\varepsilon=\{x\in\mathbb{R}^n: x_n
=(1+\varphi_\varepsilon(|x'|))|x'|^{1/k}
-(2\varepsilon)^{1/k}\varphi_\varepsilon(|x'|), x_n<1\}
$$
Denote $\partial_0 \Omega=\{x\in \partial\Omega: x_n^k=|x'|,0<x_n<1\}$.
If  $|x'|\geq2\varepsilon$ then $\Gamma_\varepsilon \subset
\partial_0 \Omega$ else $\Gamma _\varepsilon\subset \Omega$.
We will denote by $\Omega^\varepsilon\subset\Omega$ a domain with boundary
$\partial \Omega^\varepsilon=\Gamma_\varepsilon
\cup (\partial \Omega\setminus\partial_0 \Omega)$ then
 $\{\Omega^\varepsilon\}$ is a smooth domain sequence,
 and $\lim_{\varepsilon\to0} \Omega^\varepsilon  =\Omega$.

\section{Statement of the problem}

 Set $Q_T=\Omega\times(0,T)$,
$Q^\varepsilon_T=\Omega^\varepsilon\times(0,T)$, $0<T<+\infty$,
$\Omega_\tau=\Omega\times\{t=\tau\} $.
For each multi index
$\alpha=(\alpha_1, \dots, \alpha_n)\in\mathbb{N}^n$,
set $|\alpha|=\alpha_1+\dots+\alpha_n$ and
$D^\alpha={\partial^{|\alpha|}}/{\partial x_1^{\alpha_1}\dots\partial x_n^{\alpha_n}}$.
Let us consider the partial differential operator of order $2m$
\begin{equation}\label{e1}
 L(x,t)= \sum_{|\alpha|,|\beta|=0}^m D^\alpha\left( a_{\alpha\beta}
(x,t)D^\beta\right),
\end{equation}
where  $a_{\alpha\beta}$ are $s\times s$ matrices whose entries are
complex valued functions, and
$ a_{\alpha\beta} = (-1)^{|\alpha|+|\beta|}{a^*}_{\alpha\beta}$.
${a^*}_{\alpha\beta}$ denotes the transposed conjugate matrix of
$a_{\alpha\beta}$, and $a_{\alpha\beta}$ are infinity differentiable
in $\overline{Q}_T$.
We assume that there exist a constant $c_0>0$ such that
\begin{equation} \label{e3.2}
\sum_{|\alpha|=|\beta|=m} a_{\alpha\beta}(x,t)\xi^\alpha
{\xi}^\beta\eta\overline{\eta}\geq c_0|\xi|^{2m}|\eta|^2,
\end{equation}
for all $\xi \in \mathbb{R}^n\backslash\{0\}, \eta \in
\mathbb{C}^s\backslash\{0\}$ and all $(x,t)\in \overline{Q}_T$.

In this paper, we  use the usual functional spaces:
 $$
\overset{\circ}{C}^\infty(Q_T),  {C^\infty}(\overline{Q}_T), L_2(Q_T),
 L_2(\Omega ), H^m(\Omega ),   \overset{\circ}{H^m}(\Omega )
$$
(see \cite{H1, H2, F, S} for the precise definitions).
We introduce some functional spaces which will be used in this paper.

$H^{m,1}(Q_T)$ is the space consisting of all functions
$u = (u_1, \dots,u_s)$ from $L_2(Q_T)$ which have generalized
derivatives up to order
$m$ by $x$ and up to order $1$ by $t$ and belonging to $L_2(Q_T)$.
The norm in this space is defined as follows:
\[
 \Vert u \Vert_{m,1} =
\Big(\sum_{|\alpha|=0}^m\int_{Q_T} \Big(|D^\alpha u|^2 +
|u_t|^2\Big) \,dx\,dt \Big)^{1/2},
\]
where $| D^\alpha u| ^2 =  \sum_{i=1}^s |  D^\alpha u_i | ^2$,
$|u_t| ^2 = \sum_{i=1}^s \big| \partial  u_i / \partial t\big| ^2$.

$ \overset{\circ}{H}^{m,1}(Q_T)$ is the closure in $H^{m,1}(Q_T)$ of the
set consisting of all functions in  $ C^\infty (Q_T)$,
vanish near $S_T=\partial \Omega\times (0,T)$.

 $V^{m,1}(Q_T)$ is a closed subspace of $H^{m,1}(Q_T)$ having the following properties:
\begin{enumerate}
\item[(i)]  $V^{m,1}(Q_T)\supset \overset{\circ}{H}^{m,1}(Q_T)$;
\item[(ii)] Denote by
$$
B_T(u,v) =\sum_{|\alpha|,|\beta|=0}^m(-1)^{|\alpha|}\int_{Q_T}
 a_{\alpha\beta}(x,t)D^\beta u\overline{D^\alpha v}\,dx\,dt
$$
 and for $t \in [0,T)$,
$$
B(u,v)(t)=\sum_{|\alpha|,|\beta|=0}^m (-1)^{|\alpha|}
\int_{\Omega}a_{\alpha\beta}(x,t)D^\beta u\overline{D^\alpha v}dx,
\quad \forall u, v\in V^{m,1}(Q_T),
$$
then for all $u\in V^{m,1}(Q_T)$ satisfy
\begin{equation}\label{eq.1}
(-1)^mB(u,u)(t)\geq \gamma_0\|u(.,t)\|^2_{H^m(\Omega)}, \gamma_0 >0,
\quad \forall    t \text{ in }[0,T).
\end{equation}
\item[(iii)] Assume that $x_0\in \partial\Omega, U$ is a neighbourhood of $x_0$ in $\mathbb{R}^n$ and denote by
 $I=(\Omega\cap U)\times (0,T)$. Let
$\xi\in \overset{\circ}{C}^\infty(I)$ and $v\in V^{m,1}(Q_T)$,
then $\xi v\in V^{m,1}(I)$.

 In the case, $V^{m,1}(Q_T)= \overset{\circ}{H}^{m,1}(Q_T)$
 or $V^{m,1}(Q_T)=H^{m,1}(Q_T)$, condition  (iii)  is obvious.
\end{enumerate}
Suppose that $\{\Omega^\varepsilon\}$ is a sequence of smooth domains
as above, we set
$$
V_0^{m,1}(Q^\varepsilon_T)=\{u\in V^{m,1}(Q^\varepsilon_T):
\frac{\partial^j u}{\partial \nu^j}\Big|_{\Gamma^1_\varepsilon}=0,
j=0, 1, \dots, m-1\}
$$
where $\Gamma^1_\varepsilon=\{x\in\Gamma_\varepsilon:|x'|\leq 2\varepsilon\}$.
Then $V_0^{m,1}(Q^\varepsilon_T)$ is closed subspace of
$V^{m,1}(Q^\varepsilon_T)$.

By zero extension of $u\in V_0^{m,1}(Q^\varepsilon_T)$ out of
$Q^\varepsilon_T$, we regard that  $u\in V^{m,1}(Q_T)$; therefore,
from (\ref{eq.1}) we get the inequality
\begin{equation}\label{eq.2}
(-1)^mB(u,u)(t)\geq \gamma_0\|u(.,t)\|^2_{H^m(\Omega^\varepsilon)},
\gamma_0 >0, \text{ for all } t\in [0,T),
\end{equation}
 holds for all $u\in V_0^{m,1}(Q^\varepsilon_T)$.

We have the following results obtained  in \cite{F}.

 \begin{theorem}[\cite{F}] \label{G.F}
If $f\in C^\infty(\overline{Q}^\varepsilon_T)$ and
$\frac{\partial^k f}{\partial t^k}\Big|_{t=0}=0$, for
$k=0,1,\dots $, then there is the unique function $u\in
C^\infty(\overline{Q}^\varepsilon_T) $ such that
 \begin{equation}\label{eq.2b}
(-1)^{m-1}L(x,t)u-u_{tt}=f(x,t)\quad \text{in } Q^\varepsilon_T
\end{equation}
satisfies initial conditions $u(x,0)=u_t(x,0)=0$; moreover,
$u\in V_0^{m,1}(Q^\varepsilon_T)$ and   boundary conditions
 \begin{equation}\label{eq.3}
\langle L(x,t)u,v\rangle_{L_2(\Omega^\varepsilon)} =B(u,v)(t)
\end{equation}
 holds for all $v\in  V_0^{m,1}(Q^\varepsilon_T)$ and all $t\in [0,T]$,
where the scalar product is
$$
\langle u , v\rangle_{L_2(\Omega^\varepsilon)}
=\int_{ \Omega^\varepsilon}u\overline{v}dx.
$$
\end{theorem}

In the cylinder $Q_T$, we consider  systems
\begin{equation}\label{eq.4}
(-1)^{m-1}L(x,t)u-u_{tt}=f(x,t),
\end{equation}
where $f\in L_2(Q_T)$.

 \begin{definition} \label{def3.1} \rm
A function  $u\in V^{m,1}(Q_T)$ is a generalized solution of {\it
initial boundary-value problems} for systems (\ref{eq.4}) if  it
satisfies following equalities:
\begin{equation}\label{eq.5}
(-1)^{m-1}B_T(u,\eta) +\langle u_t,\eta_t
\rangle_{{L_2(Q_T)}}=\langle f,\eta\rangle_{{L_2(Q_T)}}
\end{equation}
for all test function $ \eta \in V^{m,1}(Q_T), \eta(x,T)=0$,
and  initial conditions holds
 \begin{equation}\label{eq.5.1}
u(x,0)=u_t(x,0)=0.
\end{equation}
 \end{definition}

In particular, $V^{m,1}(Q_T)=\overset{\circ}{H}^{m,1}(Q_T)$ or
$V^{m,1}(Q_T)=H^{m,1}(Q_T)$, then we have definition of
generalized solutions of {\it the fist initial boundary-value
problem } or {\it  second initial boundary-value problems }for
system (\ref{eq.4}).

\section{The uniqueness and existence theorems}

In this section, we investigate the unique solvability of  initial
boundary-value problems for the systems (\ref{eq.4}). We start
with studying the uniqueness theorem.

 \begin{theorem} \label{thm4.1}
Assume that for a positive constant $\mu$,
$$
\sup\big\{\big| \frac{\partial a_{\alpha\beta}}{\partial t}\big|,
 \left|a_{\alpha\beta}\right|:(x,t)\in Q_T, 0\leq|\alpha|,|\beta|
\leq m\big\}\leq \mu.
$$
Then   the initial boundary-value problems (\ref{eq.5}),
(\ref{eq.5.1}) for  systems (\ref{eq.4}) has no more than one
generalized solution in $V^{m,1}(Q_T)$.
\end{theorem}

\begin{proof}
Suppose that problems (\ref{eq.5}), (\ref{eq.5.1}) has two solutions
$u_1,u_2$ in $V^{m,1} (Q_T)$.  Put $u=u_1-u_2$,   (\ref{eq.5}) implies
\begin{equation} \label{e.1}
(-1)^{m-1}B_T(u,\eta) +\langle u_t,\eta_t\rangle_{{L_2(Q_T)}}=0
\end{equation}
for all  $\eta\in V^{m,1}(Q_T), \eta(x,T)=0$.
For $b\in (0,T)$, we set
$$
\eta(x,t)=
\begin{cases}
0 &\text{if  }t\in(b, T]\\
\int_b^tu(x,\tau)d\tau & \text{if  }  t\in[0,b].
\end{cases}
$$
It is easy to check that $\eta(x,t)\in V^{m,1}(Q_T)$,
$\eta_t(x,t)=u(x,t), \eta(x,T)=0$. Put $\eta(x,t)$ in
(\ref{e.1}), we have
$$
(-1)^{m-1}\int_0^bB(\eta_t,\eta)(t)dt+\int_{Q_b}\eta_{tt}
\overline{\eta}_t\,dx\,dt=0.
$$
 Adding this equality with its complex conjugate, we obtain
\begin{equation} \label{e.2}
2\mathop{\rm Re}(-1)^{m-1}\int_0^bB(\eta_t,\eta)(t)dt
+\int_{Q_b}\frac{\partial}{\partial t} |\eta_t|^2\,dx\,dt=0.
\end{equation}
We have
\begin{align*}
&2\mathop{\rm Re}\int_0^b (-1)^{m -1}B(\eta_t, \eta)(t)dt \\
&= \int_0^\tau (-1)^{m -1}\frac{\partial }{\partial t
}\Big(B(\eta, \eta)(t) \Big)dt
+\mathop{\rm Re}\sum_{|\alpha|,|\beta|=0}^m\int_{Q_b}(-1)^{|\alpha|+m}
\frac{\partial a_{\alpha\beta}}{\partial t}D^\beta \eta
\overline{D^\alpha \eta}\,dx\,dt \\
& = (-1)^{m-1 } \big[B(\eta, \eta )(b) -B(\eta, \eta )(0) \big]\\
&\quad +\mathop{\rm Re}\sum_{|\alpha|,|\beta|=0}^m\int_{Q_b}(-1)^{|\alpha|+m}
\frac{\partial a_{\alpha\beta}}{\partial t}D^\beta \eta \overline{D^\alpha
\eta}\,dx\,dt.
\end{align*}
Since $B(\eta, \eta)(b)=0$, it implies
\begin{equation} \label{e.3}
\begin{aligned}
 &2\mathop{\rm Re}\int_0^b (-1)^{m -1}B(\eta_t,\eta)(t)dt\\
& = (-1)^{m} B(\eta, \eta)(0)+\mathop{\rm Re}\sum_{|\alpha|,|\beta|=0}^m\int_{Q
_b}(-1)^{|\alpha|+m}\frac{\partial a_{\alpha\beta}}{\partial
t}D^\beta \eta \overline{D^\alpha \eta}\,dx\,dt.
\end{aligned}
\end{equation}
and since $\eta_t(x,0)=u(x,0)=0$,
\begin{equation}\label{e.4}
\int_{\Omega }\int_0^b \frac{\partial }{\partial t }|\eta_t|^2
dtdx =\int_{\Omega }|\eta_t(x,b)|^2dx .
\end{equation}
Put (\ref{e.3}) and(\ref{e.4}) in (\ref{e.2}), we obtain
\begin{equation}\label{e.5}
\begin{aligned}
&(-1)^{m} B(\eta, \eta)(0)+\int_{\Omega }|\eta_t(x,b)|^2dx\;\;\;\\
&=\mathop{\rm Re}\sum_{|\alpha|,|\beta|=0}^m\int_{Q
_b}(-1)^{|\alpha|+m-1}\frac{\partial a_{\alpha\beta}}{\partial
t}D^\beta \eta \overline{D^\alpha \eta}\,dx\,dt.
\end{aligned}
\end{equation}
Set
$$
v^\alpha(x,t)=\int_t^0D^\alpha u(x,\tau)d\tau,\quad  t\in [0,b]
$$
then
$$
v^\alpha(x,b) =\int_b^0D^\alpha u(x,\tau)d\tau=\int_b^0D^\alpha
\eta_t(x,\tau)d\tau=D^\alpha \eta(x,0).
$$
Using  (\ref{eq.1}) we have
\begin{equation}\label{e.6}
 (-1)^mB(\eta,\eta)(0)\geq\gamma_0\int_{\Omega}\sum_{|\alpha|=0}^m|
D^\alpha \eta(x,0)|^2dx=\gamma_0\int_{\Omega}\sum_{|\alpha|=0}^m
|v^\alpha(x,b)|^2dx.
\end{equation}
 From equality (\ref{e.5}), we use (\ref{e.6}) and Cauchy inequality,
we will obtain
$$
\gamma_0\int_{\Omega}\sum_{|\alpha|=0}^m|v^\alpha(x,b)|^2dx
 +\int_{\Omega }|v(x,b)|^2dx\leq K_1 \int_{Q _b}\sum_{|\alpha|=0}^m
 |v^\alpha(x,t)|^2\,dx\,dt$$ set
$K=K_1/ \gamma_0$ is a constant independent of $b$, then
$$
 \int_{\Omega}\sum_{|\alpha|=0}^m|v^\alpha(x,b)|^2dx\leq
 K  \int_{Q _b} \sum_{|\alpha|=0}^m|v^\alpha(x,t)|^2 xdt
$$
By setting
$$
y(b)=\int_{\Omega}\sum_{|\alpha|=0}^m|v^\alpha(x,b)|^2dx
$$
we have
$$
y(b)\leq K\int_0^by(t)dt,
$$
The Gronwall-Bellmann inequality  implies
$y(b)=u(x,b) =0$, for all $b\in (0,T)$ and all $x\in \Omega$;
 hence, $u\equiv0$ in $Q_T$.
\end{proof}

Now, we establish the existence of generalized solutions of mentioned
problems by {\it the approximating boundary method}.
Firstly, we will prove some following needed propositions.

\begin{proposition}\label{bd.1}
If $f\in C^\infty(\overline{Q }_T),  \frac{\partial^k f}{\partial
t^k}\Big|_{t=0}=0$, for $k=0,1,\dots $ and
$$
\sup\big\{\big| \frac{\partial a_{\alpha\beta}}{\partial t}\big|,
|a_{\alpha\beta}|:(x,t)\in Q_T, 0\leq|\alpha|,|\beta|\leq m\big\}\leq \mu,
$$
then the generalized solutions  $u^\varepsilon\in{V_0^{m,1}(Q^\varepsilon_T)}$
of  problem (\ref{eq.5}), (\ref{eq.5.1}),  in  smooth cylinders
$Q^\varepsilon_T$,  satisfies the  estimate
$$
\|u^\varepsilon\|^2_ {m,1} \leq C\|f\|^2_{L_2(Q^\varepsilon_T)}$$
 where $C$  is a constant independent of $\varepsilon$.
\end{proposition}

\begin{proof}
By Theorem  \ref{G.F}, we have
$u^\varepsilon\in C^\infty(\overline{Q}^\varepsilon_T)
\cap V_0^{m,1}(Q^\varepsilon_T) $,
satisfying systems  \eqref{eq.2b}, boundary conditions (\ref{eq.3}) and
initial conditions
 $u^\varepsilon(x,0)=u^\varepsilon_t(x,0)=0$, it is clear that
$u^\varepsilon$ is the generalized solution of problem
(\ref{eq.5}), (\ref{eq.5.1})  in $Q_T^\varepsilon$.

After multiplying  \eqref{eq.2b} by $\overline{u}^\varepsilon_t$,
integrating on $Q^\varepsilon_\tau, (\tau<T)$, we obtain
$$
(-1)^{m-1}\int_0^\tau \int_{\Omega^\varepsilon} Lu^k
\overline{u^\varepsilon_t}\,dx\,dt
-\int_0^\tau\int_{\Omega^k}u^\varepsilon_{tt}\overline{u^\varepsilon_t}\,dx\,dt
=\int_{Q^\varepsilon_\tau}f\overline{u^\varepsilon_t}\,dx\,dt.
$$
From that, and using equality (\ref{eq.3}), we get
$$
(-1)^{m }\int_0^\tau B(u^\varepsilon, u^\varepsilon_t)(t)dt
+\int_0^\tau\int_{\Omega^\varepsilon}u^k_{tt}\overline{u^\varepsilon_t}\,dx\,dt
=-\int_{Q^\varepsilon_\tau}f\overline{u^\varepsilon_t}\,dx\,dt
$$
 Adding this equality with its complex conjugate we obtain
\begin{equation}\label{eq.6}
  (-1)^{m }2\mathop{\rm Re}\int_0^\tau B(u^\varepsilon, u^\varepsilon_t)(t)dt
+\int_{\Omega^\varepsilon}\int_0^\tau \frac{\partial }{\partial t
}|u^\varepsilon_t|^2 dtdx
=-2\mathop{\rm Re}\int_{Q^\varepsilon_\tau}f\overline{u^\varepsilon_t}\,dx\,dt.
\end{equation}
We now change the left terms of (\ref{eq.6})
\begin{align*}
&2(-1)^{m }\mathop{\rm Re}\int_0^\tau B(u^\varepsilon, u^\varepsilon_t)(t)dt\\
&= (-1)^{m }\mathop{\rm Re}\int_0^\tau \frac{\partial }{\partial t }
\Big(B(u^\varepsilon, u^\varepsilon )(t) \Big)dt\\
&\quad  -\mathop{\rm Re}\int_{Q^\varepsilon_\tau}\sum_{\alpha,\beta=0}^m(-1)
^{|\alpha|+m}\frac{\partial a_{\alpha\beta}}{\partial t}D^\beta u^\varepsilon
\overline{D^\alpha u^\varepsilon}\,dx\,dt \\
&= (-1)^{m } \mathop{\rm Re}B(u^\varepsilon, u^\varepsilon )(\tau)
-\mathop{\rm Re}\int_{Q^\varepsilon_\tau}
 \sum_{\alpha,\beta=0}^m(-1)^{|\alpha|+m}
 \frac{\partial a_{\alpha\beta}}{\partial t}D^\beta u^\varepsilon
\overline{D^\alpha u^\varepsilon}\,dx\,dt.
\end{align*}
Using the initial conditions, we get
 $$
\int_{\Omega^\varepsilon}\int_0^\tau \frac{\partial }{\partial t
}|u^\varepsilon_t|^2 dtdx
=\int_{\Omega^\varepsilon_\tau}|u_t^\varepsilon|^2dx
=\|u^\varepsilon_t(x,t)\|_{L_2(\Omega^\varepsilon_\tau)}
 $$
Therefore, basing on what has been discussed above, equality (\ref{eq.6})
can be rewritten as the form
\begin{equation}\label{eq.7}
\begin{aligned}
&(-1)^{m } B(u^\varepsilon, u^\varepsilon
)(\tau)+\|u_t(x,t)\|_{L_2(\Omega^\varepsilon_\tau)}\\
&=\mathop{\rm Re}\int_{Q^\varepsilon_\tau}(-1)^{|\alpha|+m}\sum_{|\alpha|,|\beta|=0}^m\frac{\partial
a_{\alpha\beta}}{\partial t}D^\beta u^\varepsilon
\overline{D^\alpha
u^\varepsilon}\,dx\,dt-2\mathop{\rm Re}\int_{Q^\varepsilon_\tau}
f\overline{u^\varepsilon_t}\,dx\,dt.
\end{aligned}
\end{equation}
 From (\ref{eq.7}), by using inequality (\ref{eq.2}),
and Cauchy inequality, we obtain
\begin{align*}
 &\gamma_0\|u^\varepsilon(x,t) \|^2_{H^m(\Omega_\tau^\varepsilon)}
 +\|u^\varepsilon_t(x,t)\|_{L_2(\Omega^\varepsilon_\tau)} \\
 &\leq C_1\Big( \sum_{|\alpha| =0}^m\int_{Q^\varepsilon_\tau}
\big(|D^\alpha u^\varepsilon|^2+|u^\varepsilon_t|^2\big)\,dx\,dt
+\|f\|^2_{L_2(Q^\varepsilon_T)}\Big).
\end{align*}
Therefore,
\begin{align*}
 &\|u^\varepsilon(x,t) \|^2_{H^m(\Omega_\tau^\varepsilon)}
 +\|u^\varepsilon_t(x,t)\|_{L_2(\Omega^\varepsilon_\tau)} \\
&\leq C_2\Big(\int_0^\tau\Big(\|u^\varepsilon(.,t)
\|^2_{H^m(\Omega_t^\varepsilon)}+\|u^\varepsilon_t(.,t)
 \|_{L_2(\Omega_t^\varepsilon)}\Big)
dt+\|f\|^2_{L_2(Q^\varepsilon_T)}\Big)
\end{align*}
where  $C_2= C_1/ \text{min}\{\gamma_0;1\} $.
Denote
$$
Z(\tau)=\|u^\varepsilon(x,t)\|^2_{H^m(\Omega_\tau^\varepsilon)}
 +\|u^\varepsilon_t(x,t)\|_{L_2(\Omega^\varepsilon_\tau)}
$$
 We get
$$
Z(\tau)\leq C_2 \Big(\int_0^\tau
Z(t)dt+\|f\|^2_{L_2(Q^\varepsilon_T)}\Big).
$$
The Gronwall - Bellmann inequality implies
$$
Z(\tau)\leq C_2e^{C_2\tau}\|f\|^2_{L_2(Q^\varepsilon_T)},\quad
 \forall \tau\in(0,T)
$$
By integrating with respect to $\tau$ from $0$ to $T$ this inequality,
we obtain
$$
\|u^\varepsilon\|^2_{m,1}\leq C\|f\|^2_{L_2(Q^\varepsilon_T)},\quad
C=C_2(e^{C_2T}-1),
$$
where $C$ is a absolute constant.
\end{proof}

In the next proposition, we prove result of proposition \ref{bd.1}
without conditions $\frac{\partial^k f}{\partial t^k}\Big|_{t=0}=0$,
for $k=0,1,\dots$.

\begin{proposition}\label{bd.2}
If $f\in C^\infty(\overline{Q}_T)$ and
 $$
\sup\big\{\big| \frac{\partial a_{\alpha\beta}}{\partial t}\big|,
|a_{\alpha\beta}|:(x,t)\in Q_T, 0\leq|\alpha|,|\beta|\leq m\big\}\leq \mu,
$$
 then generalized solution $u^\varepsilon\in{V_0^{m,1}(Q^\varepsilon_T)}$
of  problem (\ref{eq.5}), (\ref{eq.5.1})   in  smooth cylinders
$Q^\varepsilon_T$  satisfies the following estimates
$$
\|u^\varepsilon\|^2_ {m,1} \leq C\|f\|^2_{L_2(Q^\varepsilon_T)}
$$
 where $C$  is a constant independent of $\varepsilon$.
\end{proposition}

\begin{proof}
Denote
$$
f_h(x,t)=
\begin{cases}
0 &  \text{if } (x,t)\ne Q_T^\varepsilon \\
f(x,t) &   \text{if } (x,t)\in Q_T^\varepsilon, t>h \\
0, &  \text{if } (x,t)\in Q_T^\varepsilon, t\leq h
\end{cases}
$$
for all $h>0$. We will denote by $g_{\frac{h}{2}}$ the
mollification of $f_h$;  i.e.,
$$
g_{\frac{h}{2}}(x,t)=
\int_{\mathbb{R}^{n+ 1}}\theta_{\frac{h}{2}}(x-y, t-\tau)f_h(y,\tau)dyd\tau,
$$
where $\theta_h $ is a mollifier.
Then $g_{\frac{h}{2}}\in C^\infty(\overline{Q^\varepsilon_T}),
g_{\frac{h}{2}}\equiv 0, t<\frac{h}{2}$ and
$g_{\frac{h}{2}}\to  f$ in $L_2(Q_T^\varepsilon)$. Applying
proposition (\ref{bd.1}) to replace $f$ by $g_{\frac{h}{2}}$, we
get $u^\varepsilon_h$ as a generalized solution of the problem
(\ref{eq.5}), (\ref{eq.5.1}) in $Q_T^\varepsilon$ and the
following estimate holds
$$
\|u_h^\varepsilon\|^2_ {m,1} \leq C\|g_{\frac{h}{2}}\|^2
_{L_2(Q^\varepsilon_T)}
$$
where $C$ is a absolute constant. As $g_{\frac{h}{2}}\to  f$
in $L_2(Q_T^\varepsilon)$, $\{u^\varepsilon_h\}$ is a Cauchy
sequence in $V_0^{m,1}(Q_T^\varepsilon)$. Therefore,
$u^\varepsilon_h \to  u^\varepsilon, (h\to  0)$.
It is easy to see that $u^\varepsilon$ is a generalized solution of
the problem and satisfies the following estimate
$$
\|u^\varepsilon\|^2_ {m,1} \leq C\|f\|^2_{L_2(Q^\varepsilon_T)}.
$$
\end{proof}

We now prove the existence of a generalized solution to problem
(\ref{eq.5}), (\ref{eq.5.1})  in  $ Q_T $  when
$f\in C^\infty(\overline{Q}_T)$.

\begin{proposition}\label{bd.3}
Assume that $f\in C^\infty(\overline{Q}_T)$ and
$$
\sup\big\{\big| \frac{\partial a_{\alpha\beta}}{\partial t}\big|,
|a_{\alpha\beta}|:(x,t)\in Q_T, 0\leq|\alpha|,|\beta|\leq m\big\}
\leq \mu.
$$
Then problem (\ref{eq.5}), (\ref{eq.5.1})  in cylinder $Q_T$ has
the generalized solution  $u\in V^{m,1}(Q_T) $ and
\begin{equation}
\|u\|^2_{m,1} \leq C\|f\|^2_{L_2(Q_T)}
\end{equation}
where $C$ is a constant independent of $u$ and $f$.
\end{proposition}

\begin{proof}
By proposition \ref{bd.2},  the generalized solution
$u^\varepsilon\in V_0^{m,1}(Q^\varepsilon_T) $ of
problem (\ref{eq.5}), (\ref{eq.5.1})  in the smooth cylinder
$Q^\varepsilon_T$ satisfies the following inequality
\begin{equation}
\|u^\varepsilon\|^2_{m,1}\leq C\|f\|^2_{L_2(Q^\varepsilon_T)}
\end{equation}
where $C$ is a constant independent of $\varepsilon$.

Since $ \|f\|^2_{L_2(Q^\varepsilon_T)}\leq \|f\|^2_{L_2(Q_T)}$, we have
\begin{equation*}
\|u_\varepsilon\|^2_{m,1}\leq C\|f\|^2_{L_2(Q_T)}.
\end{equation*}
Set
$$
\widetilde{u}_\varepsilon(x,t)=
\begin{cases}
u^\varepsilon(x,t)& \text{if }(x,t)\in Q^\varepsilon_T\\
0 & \text{if }(x,t)\in Q_T\setminus Q^\varepsilon_T
\end{cases}
$$
then
\begin{equation}\label{eq.8}
\|\widetilde{u}_\varepsilon\|^2_{m,1}=\|u^\varepsilon\|^2_{m,1}
\leq  C\|f\|^2_{L_2(Q_T)}.
\end{equation}

 This means that the set $\{\widetilde{u}_\varepsilon\}_{\varepsilon>0}$
is uniform bounded in the space $V^{m,1}(Q_T) $. So we can take a
subsequence, denote also by $\widetilde{u}_{\varepsilon} $
for convenience, which converges weakly to a function
$\widetilde{u}_0\in V^{m,1}(Q_T) $.
 We will show that $\widetilde{u}_{0}$ is a generalized solution of
problem (\ref{eq.5}), (\ref{eq.5.1})  in  cylinder $Q_T$. In fact,
for all $\eta\in V^{m,1}(Q_T), \eta(x,T)=0$ there exists
$\eta_\delta\in C^\infty(\overline{Q_T})$  such that
$\eta_\delta\equiv 0$ in $Q_T\setminus Q^\varepsilon_T, \eta_\delta(x,T)=0$,
and $ \|\eta_\delta-\eta\|_{m,1}\longrightarrow 0$ when
$\delta\to  0$.
Since $u^{\varepsilon}$  is a generalized solution of problem
(\ref{eq.5}), (\ref{eq.5.1})  in smooth cylinder $Q^\varepsilon_T$, we have
$$
(-1)^{m-1}B_T(u^\varepsilon,\eta_\delta) +\langle u^\varepsilon_t,\eta_{\delta t}\rangle_{{L_2(Q^\varepsilon_T)}}=\langle f,\eta_\delta\rangle_{{L_2(Q^\varepsilon_T)}}
$$
 Rewrite this equality in the form
$$
(-1)^{m-1}B_T(\widetilde{u}_{\varepsilon},\eta_\delta) +\langle\widetilde{u}_{\varepsilon t},\eta_{\delta t}\rangle_{{L_2(Q_T)}}=\langle f,\eta_\delta\rangle_{{L_2(Q^\varepsilon_T)}}
$$
 Passing to the limit when $\varepsilon\to  0,\delta\to 0$
for the weakly convergent sequence, we get
$$
(-1)^{m-1}B_T(\widetilde{u}_0,\eta) +\langle\widetilde{u}_{0 t},
\eta_{t}\rangle_{{L_2(Q_T)}}=\langle f,\eta\rangle_{{L_2(Q_T)}}
$$
Since $V^{m,1}(Q_T)$ is embedded continuously into $L_2(\Omega)$,
the trace sequence $\{\widetilde{u}_{\varepsilon}(x,0)\}$ of
$\{\widetilde{u}_\varepsilon(x,t)\}$ converges weakly to the trace
$\widetilde{u}_0(x,0)$ of $\widetilde{u}_0(x,t)$ in $L_2(\Omega )$.
On the other hand, $\widetilde{u}_{\varepsilon}(x,0)=0$, so that
$ \widetilde{u}_0(x,0)=0$.
Hence, $\widetilde{u}_0(x,t)$ is a generalized solution of
problem (\ref{eq.5}), (\ref{eq.5.1}). Moreover, from (\ref{eq.8}) we have
$$
\|\widetilde{u}_{0}\|^2_{m,1} \leq\varliminf_{\varepsilon\to 0}
\|\widetilde{u}_{\varepsilon}\|^2_{m,1}\leq C\|f\|^2_{L_2(Q_T)}.
$$
\end{proof}

Proposition \ref{bd.3} states the existence of generalized solutions of
problem  (\ref{eq.5}), (\ref{eq.5.1})  in $V^{m,1}(Q_T)$ when
$f\in C^\infty(\overline{Q}_T)$. Using this proposition and
properties of mollification of $f\in L_2(Q_T )$, we obtain the
following theorem.

\begin{theorem} \label{thm4.2}
If $f\in L_2(Q_T), $ and
$$
\sup\big\{\big| \frac{\partial a_{\alpha\beta}}{\partial t}\big|,
 |a_{\alpha\beta}|:(x,t)\in Q_T, 0\leq|\alpha|,|\beta|\leq m\big\}
 \leq \mu,
$$
then problem (\ref{eq.5}), (\ref{eq.5.1}) in the cylinder $Q_T$
has generalized solutions  $u\in V^{m,1}(Q_T)$ and
\[
\|u\|^2_{m,1}\leq C\|f\|^2_{L_2(Q_T)},
\]
 where $C$ is a constant independent of $u$ and $f$.
\end{theorem}

 The following theorem shows that the generalized solution
$u\in V^{m,1}(Q_T)$ of  problem (\ref{eq.5}), (\ref{eq.5.1}) is smooth with
respect to time variable $t$, if the right-hand  side $f$ and
coefficients of the operator (\ref{e1}) are smooth enough with
respect to $t$.

\begin{theorem} Let
\begin{itemize}
\item[(i)] $\frac{\partial^k f}{\partial t^k} \in L_2(Q_T),k\leq h$;
\item[(ii)] $\frac{\partial^k f}{\partial t^k}\Big|_{t=0}=0$,
 $x\in \Omega$, $k\leq h-1$;
\item[(iii)] $\sup\big\{\big| \frac{\partial^k a_{\alpha\beta}}{\partial t^k}
\big|, k<h:(x,t)\in Q_T, 0\leq|\alpha|,|\beta|\leq m\big\}\leq\mu $.
\end{itemize}
Then the generalized solution $u\in V^{m,1}(Q_T)$  of problem
 \eqref{eq.5}, \eqref{eq.5.1} has generalized derivatives with
respect to $t$ up to order $h$ in $ V^{m,1}(Q_T)$ and
\[
\big\|\frac{\partial^h u}{\partial t^h}\big\|^2_{m,1}
\leq C\sum_{k=0}^h \big\|\frac{\partial^k f}{\partial t^k}\big\|^2_{L_2(Q_T)}
\]
where $C$ is a constant independent of $u$ and $f$.
\end{theorem}

This theorem is proved by arguments analogous to those in the proof
of propositions \ref{bd.1}, \ref{bd.2}, \ref{bd.3} and by induction
on $h$.

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee for his/her
helpful comments and suggestions.

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\end{document}