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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 151, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/151\hfil Hyperbolic equations in cusp domains]
{$L_p$-Regularity of solutions to first initial-boundary value
problem for hyperbolic equations in cusp domains}

\author[N. M. Hung, V. T. Luong\hfil EJDE-2009/151\hfilneg]
{Nguyen Manh Hung, Vu Trong Luong}  % in alphabetical order

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

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


\thanks{Submitted July 24, 2009. Published November 25, 2009.}
\subjclass[2000]{35D05, 35D10, 35L30, 35L35, 35G15}
\keywords{Hyperbolic equations; first initial-boundary value
problem; \hfill\break\indent $L_p$-regularity; cusp domain}

\begin{abstract}
 In this article, we establish  well-posedness and
 $L_p$-regularity of solutions to the first initial-boundary
 value problem for general higher order hyperbolic equations
 in cylinders whose base is a cusp domain.
\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}

Initial boundary-value problems for hyperbolic and parabolic type
equations in a cylinder with base containing conical point have
been studied by many authors \cite{H1, H2, H3, H4, KK}. The main
results are about the uniqueness and existence of the solutions,
and asymptotic expansions of the solution near a neighborhood of
conical point. Those results are mainly based on Galerkin's
approximate method and $L_2$-theory.

Boundary-value problems for elliptic type equations and systems
have also well studied. The main results, presented in \cite{K, M,
M1, S2}, established estimates in $L_p$ for solutions of elliptic
boundary value problems in domains with singular points on the
boundary.

The question is whether  similar results can be obtained based on
these results for initial boundary-value problems for
non-stationary equations. In this paper, we find the answer for
this question.

Firstly, we show the existence of a sequence of smooth domains
$\{\Omega^\epsilon\}_{\epsilon>0}$ such that
$\Omega^\epsilon\subset \Omega$ and
$\lim_{\epsilon\to0}\Omega^\epsilon=\Omega$. Furthermore, we
proved existence, uniqueness and smoothness, with respect to time
variable, of the generalized solution by approximating boundary
method, which can be applied for non-linear equations. Next, by
modifying the arguments in \cite{M1}, we take the term containing
the derivative in time of the unknown function to the right-hand
side of the equation, such that the problem can be considered as
an elliptic problem. With the help of some auxiliary results, we
apply the estimates in $L_p$ for solution of the elliptic boundary
value problem and our previous estimates to deal with the
$L_p$-regularity with respect to both of time and spatial
variables of the solution. Finally, in order to illustrate the
results above we show an example for the Cauchy-Dirichlet problem
for the beam equation in cylinder with base containing a cuspidal
point.

\section{Preliminaries}\label{sec2}

Let $\Omega$ be bounded domain in $\mathbb{R}^n$, $n\geq 2$, with
boundary $\partial\Omega$. Let $p, q$ be real numbers with
$1<p,q<+\infty$ and $\frac{1}{p}+\frac{1}{q}=1$.

We denote by $W^m_p(\Omega)$ the space of  all  $u=u(x),
x\in\Omega $ that have generalized derivatives   $D^\alpha u\in
L_p(\Omega)$, $|\alpha|\leq m$. The norm in this space is defined
as
\[
\big\Vert u \big\Vert_{ m;p}  = \Big(\int_{\Omega}
\sum_{|\alpha|=0}^m |D_x^\alpha u|^p \,dx \Big)^{1/p}.
\]
 In particular, ${\mathaccent"7017 W}^0_p(\Omega)\equiv L_p(\Omega)$.
The space ${\mathaccent"7017 W}^m_p(\Omega)$ is the completion  of
$ C^\infty_0(\Omega)$ in norm of the space $W^m_p(\Omega)$.

Setting $Q_T=\Omega\times(0,T)$, $0<T<+\infty$. We introduce  the
partial differential operator of order $2m$,
\begin{equation}\label{PDO}
 L=L(x,t; D_x)=\sum_{|\alpha|,|\beta|=0}^m D_x^\alpha
\Big( a_{\alpha\beta}(x,t)D_x^\beta\Big),
\end{equation}
where  $D_x^\alpha =i^\alpha\partial^\alpha_x$, $a_{\alpha\beta}$
are $s\times s-$ matrices of functions with complex values, and
$a_{\alpha\beta}$ are infinitely differentiable in
$\overline{Q}_T$ and  $ a_{\alpha\beta} = {a^*}_{\alpha\beta}$,
where ${a^*}_{\alpha\beta}$ denotes the transposed conjugate
matrix of  $a_{\alpha\beta}$. We have the following Green's
formula
$$
\int_{\Omega}Lu\,\overline{v}\,dx=B[u,v;t]
$$
which is valid for all $u, v\in C^\infty_0(\Omega)$ and  a.e.
$t\in[0,T)$, where
$$
B[u,v;t]= \sum_{|\alpha|,|\beta|=0}^m\int_{\Omega} a_{\alpha\beta
}(.,t)D_x^\beta u\,\overline{D_x^\alpha v}\,dx.
$$
We also assume the Garding's inequality,
\begin{equation}\label{GD}
B[u,u\,;t]\geq \gamma_0\|u\|^2_{W^m_2(\Omega)}
\end{equation}
which is valid for all $u\in {\mathaccent"7017 W}^m_2(\Omega)$ and
a.e. $t\in[0,T)$, where $\gamma_0$ is a positive constant
independent of $u$ and $t$.

Now we  introduce spaces on $Q_T$. Let $W_p^{m,1}(Q_T)$ be the
space consisting of  functions $u=u (x,t), (x,t)\in Q_T$ having
generalized derivatives $ D^\alpha u\in L_p(Q_T), |\alpha|\leq m$,
and $u_t\in L_p(Q_T)$,  with norm
\[
\big\Vert u \big\Vert_{ m,1;p}  =
\Big(\int_{Q_T}\sum_{|\alpha|=0}^m |D_x^\alpha u|^p\,dx\,dt
+\int_{Q_T}|u_t|^p\,dx\,dt \Big)^{1/p}.
\]
The space ${\mathaccent"7017 W}_p^{m,1 }(Q_T)$ is the closure in
$W_p^{m,1}(Q_T)$  of the set consisting of all functions in
$C^\infty(Q_T)$, which vanish near $S_T$ denoting by
$C_0^\infty(Q_T)$ for convenience.

We introduce the space $W_p^{-m,-1 }(Q_T)$ of generalized
functions on $Q_T$; it means that if $f\in W_p^{-m,-1 }(Q_T)$, the
$f$ admits the representation
\begin{equation}\label{eq.1}
f=\sum_{|\alpha|\leq m}D_x^\alpha f^{(\alpha)}+f^{(t)}_t
\end{equation}
where $f^{(\alpha)}, f^{(t)}\in L_p(Q_T), \,p\in(1,+\infty)$. The
norm in $W_p^{-m,-1 }(Q_T)$ can also be defined by
$$
\|f\|_{-m,-1;p}=\inf\sum_{|\alpha|\leq m}\|f^{(\alpha)}
\|_{L_p(Q_T)}+\|f^{(t)}_t\|_{L_p(Q_T)}.
$$
Here the infimum is taken over the set of all representations
\eqref{eq.1}. It is known that $W_p^{-m,-1 }(Q_T)$ and
${\mathaccent"7017 W}_q^{m,1 }(Q_T),q=\frac{p}{p-1}$, are dual to
one another. We also define
$$
\langle f,\eta \rangle=\int_{Q_T}f\overline{\eta} \,dx\,dt,
\quad f\in W_p^{-m,-1 }(Q_T),\;\eta\in {\mathaccent"7017 W}_q^{m,1
}(Q_T).
$$
It is clear that
\begin{gather*}
\|f\|_{-m,-1;p}=\sup\{|\langle f, \eta\rangle|:
\eta\in {\mathaccent"7017 W}_q^{m,1 }(Q_T),\|\eta\|_{m,1;q}=1\};\\
\|\eta\|_{m,1;q}=\sup\{|\langle f, \eta\rangle|:
f\in{W}_p^{-m,-1 }(Q_T),\|f\|_{-m,-1;p}=1\}.
\end{gather*}
In this paper, we consider the  problem
 \begin{gather}
 Lu-u_{tt}=f \text{ in } Q_T,\label{eq.2} \\
u =0, u_{t} =0\text{ on }\Omega,\label{eq.3}\\
\partial^j_\nu u =0\text{ on }S_T, j=0,1, \dots, m-1,\label{eq.4}
 \end{gather}
where $f:Q_T\to \mathbb{C}$ is a given function and $\partial^j_\nu u$
are derivatives with respect to the outer unit normal of
$S_T=\partial\Omega\times(0,T)$.
 Setting
$$
B_1[u,\eta]=\sum_{|\alpha|,|\beta|=0}^m
\int_{Q_T} a_{\alpha\beta }D^\beta u\,\overline{D^\alpha \eta}\,dx\,dt
+\int_{Q_T}u_t\overline{\eta_t}\,dx\,dt.
$$
for all $u\in {\mathaccent"7017 W}^{m,1}_p(Q_T),  \eta\in
{\mathaccent"7017 W}^{m,1}_q(Q_T)$.

\begin{definition} \label{def2.1} \rm
Let $f\in {W}_p^{-m,-1 }(Q_T)$; a function $u$ is called a
generalized $L_p$-solution of problem  \eqref{eq.2}--\eqref{eq.4}
if and only if $u$ belongs to ${\mathaccent"7017 W}_p^{m,1 }(Q_T),
u(x,0)=u_t(x,0)=0$, and the equality
\begin{equation}\label{eq.5}
 B_1[u,\eta]=\langle f, \eta\rangle
\end{equation}
holds for all $ \eta\in {\mathaccent"7017 W}_q^{m,1}(Q_T) $.
\end{definition}

 To prove uniqueness of the generalized $L_p$-solution of
\eqref{eq.2} -\eqref{eq.4}, we need to prove the  following lemma.

\begin{lemma} \label{lem2.1}
If $1<p\leq2$, then there exists a constant
$ \gamma_2= \gamma_2(p,n,m,|\Omega|,T)>0$, such that
\begin{equation} \label{GD.1}
\sup \{\big|B_1[u,\eta]\big|:\eta\in  {\mathaccent"7017 W}_q^{m,1
}(Q_T), \|\eta\|_{m,1;q}\leq  1\}\geq \gamma_2\|u\|_{m,1;p},\;
 \end{equation}
 for  all $ u\in {\mathaccent"7017 W}_p^{m,1}(Q_T)$.
\end{lemma}

\begin{proof}
We prove this result with $u\in C_0^\infty(Q_T)$.
Suppose that there is no $\gamma_2>0$ such that \eqref{GD.1} holds.
Then there is a sequence $\{u_k\}\subset C_0^\infty(Q_T)$ with
$\|u_k\|_{m,1;p}=1$ and
\begin{equation}\label{eq.6}
\sup \{\big|B_1[u_k,\eta]\big|:\eta\in  {\mathaccent"7017
W}_q^{m,1 }(Q_T), \|\eta\|_{m,1;q}\leq  1\}\leq \frac{1}{k},
\quad \text{for every } k\geq 1.
\end{equation}
Using Garding' inequality \eqref{GD},  we obtain
\begin{equation} \label{eq.7}
 \big|B_1[u_k,u_k]\big|\geq \gamma_0\|u_k\|^2_{m,0;2}
+\int_{Q_T}|u_{kt}|^2\,dx\,dt
\geq c_1 \|u\|^2_{m,1;2}.
 \end{equation}
On the other hand, by using H\"{o}lder's inequality with $1<p<2$,
$p^*=\frac{2}{p}$, $q^*=\frac{2}{2-p}$, we have
\begin{equation}\label{eq.8}
\|u_k\|^p_{m,1;p}= \sum_{|\alpha|=0}^m\int_{Q_T}
|D_x^\alpha u|^p \,dx\,dt +\int_{Q_T}|u_t|^p \,dx\,dt
\leq C_2\|u_k\|^p_{m,1;2},
 \end{equation}
where $C_2=C_2(p,|\Omega|,T)>0$.
Combining (\ref{eq.7}) and (\ref{eq.8}), we obtain
$$
\big|B_1[u_k,u_k]\big|\geq  C\|u_k\|^2_{m,1;p},
$$
where  $c$ is a constant independent of $k$.
 From the above inequality and (\ref{eq.6}), we have
$$
\|u_k\|^2_{m,1;p}\leq \frac{1}{k\,C},\quad \text{for } k= 1, 2, \dots.
$$
which  contradicts $\|u_k\|_{m,1;p}=1$. Therefore, there is a
constant $\gamma_2>0$ such that \eqref{GD.1} holds. Since
$u\in C_0^\infty(Q_T)$ which is dense in
${\mathaccent"7017 W}_p^{m,1}(Q_T) $,
this completes the proof.
\end{proof}

 Lemma \ref{lem2.1} implies the uniqueness of generalized
$L_p$-solution, according to the following theorem.

 \begin{theorem}\label{Theo1}
Assume that coefficients of operator \eqref{PDO} satisfy   \eqref{GD}
  and $f\in W^{-m,-1}_p(Q_T)$. Then  \eqref{eq.2} -\eqref{eq.4}
has at most one generalized $L_p$-solution.
\end{theorem}

\begin{proof}
Firstly, we  prove the theorem in the case $1<p\leq2$.
Suppose that  \eqref{eq.2}-\eqref{eq.4} has two generalized
$L_p$-solutions $u_1, u_2 $. Put $u=u_1- u_2 $, then
(\ref{eq.5}) implies that
\begin{equation*}
  B_1[u,\eta]=\sum_{|\alpha|,|\beta|=0}^m
\int_{Q_T} a_{\alpha\beta}(x,t)D_x^\beta\,
u\overline{D_x^\alpha \eta}\,dx\,dt
 +\int_{Q_T}u_t\,\overline{\eta}_t\,dx\,dt=0
\end{equation*}
holds for all $ \eta\in {\mathaccent"7017 W}_q^{m,1}(Q_T)$.
Combining inequality \eqref{GD.1} with the above equality,  we obtain
$$
\gamma_2\|u\|_{m,1;p}\leq\sup \{\big|B_1[u,\eta]\big|:
\eta\in  {\mathaccent"7017 W}_q^{m,1 }(Q_T), \|\eta\|_{m,1;q}\leq  1\}=0.
$$
Next, we  prove the theorem in the case $p>2$.
Since $p>2$, and $ Q_T $ is bounded, we have
${\mathaccent"7017 W}_p^{m,1 }(Q_T)\hookrightarrow {\mathaccent"7017 W}_2^{m,1 }
(Q_T)$.
Therefore, if $u$ is a generalized $L_p$-solution, and then $u$ is
a generalized $L_2$-solution.
We obtain the uniqueness of a generalized $L_p$-solution from
the uniqueness of a generalized $L_2$-solution.
Hence, $u\equiv 0$ in $Q_T$. This completes the proof of theorem.
\end{proof}

Next, we  prove the approximate boundary lemma, which is
the essential tool in solving \eqref{eq.2} -\eqref{eq.4}.

\begin{lemma}[\cite{L3}] \label{A}
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{lemma}

\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 $h<\varepsilon/2$, then $J_h(x)$ has following properties:
\begin{itemize}
\item [(1)] $J_h(x)=0$ if $x\notin\Omega^{\frac{\varepsilon}{2}}$;
\item [(2)] $ 0\leq J_h(x)\leq1, \forall x\in \Omega$;
\item [(3)] $J_h(x)=1$ in $\Omega^{2\varepsilon}$;
\item [(4)]  $J_h\in C_0^\infty(\mathbb{R}^n)$.
\end{itemize}

We now fix a constant $ c\in(0,1)$, set
$\Omega_c^\varepsilon=\{x\in\Omega: J_h(x)>c\}$.
It is obvious that
$\Omega^{\frac{\varepsilon}{2}}\supset\Omega^\varepsilon_c
\supset\Omega^{2\varepsilon}$.
 Therefore, $\Omega_c^\varepsilon\subset\Omega$ and
$\lim_{\varepsilon\to 0}\Omega_c^\varepsilon=\Omega$.

Assume that $K$ is the critical set of $J_h$, i.e. $K$ consisting
of all point $x$, such that the gradient of $J_h$ at $x$ vanishes.
A number $c\in\mathbb{R}$ such that $J_h^{-1}(c)$ contains at
least one $x\in K$ is called a critical value. By Sard's theorem
then the set of critical values of $J_h$ is of measure zero
(see\cite[Theorem 1.30]{TA}), it implies that there exists a
constant $c_0 \in( 0,1)$ such that  $c_0$  is not a critical value
of  $J_h$. Denote $\Omega^\varepsilon_{c_0}=\{x\in\Omega:
J_h(x)>c_0\}$ and $F(x)=J_h(x)-c_0$. For all $x^0\in
\partial\Omega_{c_0}^\varepsilon$, then $F(x^0)=J_h(x^0)-c_0=0$
and vector $ \text{grad}J_h(x^0)\ne 0$. This implies that there
exists a $\frac{\partial J_h}{\partial x_i}(x^0)\ne 0 $, without
loss of generality we can suppose that $ \frac{\partial
J_h}{\partial x_n}(x^0)\ne 0 $. Using the implicit function
theorem, we obtain that there exists a neighborhood $\mathcal{W}$
of $(x^0_1, \dots, x^0_{n-1})$ in $\mathbb{R}^{n-1}$  a
neighborhood $\mathcal{V}$ of $x_n^0$ in $\mathbb{R}$ and an
infinitely differentiable function $z:\mathcal{W}\to \mathbb{R}$
such that
$x\in  \mathcal{U}_{x^0}\cap \partial\Omega_{c_0}^\varepsilon$,
where
$\partial\Omega_c^\varepsilon=\{x\in\Omega: J_h(x)=c\}$,
$\mathcal{U}_{x^0}=\mathcal{W}\times \mathcal{V}$, 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 lemma proved.
\end{proof}

Suppose that $\{\Omega^\epsilon\}$ is a smooth domain subsequence and
$\lim_{\varepsilon\to 0}\Omega^\varepsilon=\Omega$.
Set $Q^\epsilon_T=\Omega^\epsilon\times(0,T),
S^\epsilon_T=\partial\Omega^\epsilon\times(0,T)$.
It is known that the problem
\begin{gather*}
 Lu-u_{tt}=f \text{ in } Q^\epsilon_T,  \\
u =0, u_{t} =0\text{ on }\Omega^\epsilon, \\
\partial^j_\nu u =0\text{ on }S^\epsilon_T, j=0,1, \dots, m-1,
 \end{gather*}
has a unique function
$u^\epsilon(x,t)\in C^\infty(\overline{Q^\epsilon_T})$;
if  $ f\in C^\infty(\overline{Q^\epsilon_T}),  f_{ t^k}\big|_{t=0}=0$,
for $ k=0,1,\dots$.
Moreover,
$ u^\epsilon(.,t) \in {\mathaccent"7017 W}_2^{m}(\Omega^\epsilon)$,
 for all $t\in [0,T]$, (see\cite{F, LD, KR}).


 \section{Main results}\label{sec3}

\subsection{Existence of generalized $L_p$-solutions}
In this subsection, we  prove the existence  of generalized $L_p$-solution.
Firstly, we prove the needed following propositions:

  \begin{proposition}\label{pro.1a}
 Suppose that   $1<p\leq 2$ and $f\in C^\infty(\overline{Q_T})$, and
$ f_{ t^k}\big|_{t=0}=0$, for $ k=0,1,\dots$ ;
then $u^\epsilon $ is a generalized $L_p$-solution of
 \eqref{eq.2}-\eqref{eq.3} in $Q_T^\epsilon$ satisfying
$$
\|u^\epsilon\|_ {m,1;p} \leq C \|f\|_{-m,-1;p}
$$
where the constant $C$  is independent of $\epsilon, u$ and $f$.
\end{proposition}

\begin{proof}
 From $ u ^\epsilon$ satisfying system \eqref{eq.2} in
$Q^\epsilon_T$; i. e.,
$$
f=Lu^\epsilon - u^\epsilon_{tt}, \text{ in } Q^\epsilon_T,
$$
 we have
 $$
\langle f,\eta \rangle=\int_{Q^\epsilon_T}Lu^\epsilon\,\overline{\eta}
 \,dx\,dt-\int_{Q^\epsilon_T}u^\epsilon_{tt} \,\overline{\eta}\,dx\,dt
$$
valid for all $\eta\in {\mathaccent"7017 W}_q^{m,1}(Q_T^\epsilon)$.

By using Green's formula and integrating by parts with respect to $t$,
we obtain from the equality above that
\begin{equation}\label{eq.5b}
 B_1[u^\epsilon,\eta] =\langle f,\eta \rangle
\end{equation}
valid for all $\eta\in {\mathaccent"7017
W}_q^{m,1}(Q_T^\epsilon)$. This clearly shows that  $ u^\epsilon
$ is a generalized $L_p$-solutions of problem \eqref{eq.2}
-\eqref{eq.3} in $Q_T^\epsilon$;
  otherwise, using   inequality \eqref{GD.1},  we conclude
from (\ref{eq.5b}) that
$$
\|u^\epsilon\|_ {m,1;p} \leq C \|f\|_{-m,-1;p}.
$$
 \end{proof}

Now we prove the existence of the generalized $L_p$-solution of
\eqref{eq.2}-\eqref{eq.4} in  $Q_T$,  when the assumptions of
Proposition \ref{pro.1a} are satisfied.

\begin{proposition}\label{pro.1b}
Let the  following hypothesis  be satisfied:
\begin{itemize}
\item[(i)] $1<p\leq 2$,
\item[(ii)] $ f\in C^\infty(\overline{Q_T})$, and
$ f_{ t^k}\big|_{t=0}=0$, for $ k=0,1,\dots$
\end{itemize}
Then  \eqref{eq.2}-\eqref{eq.4}
in cylinder $Q_T$ has a generalized $L_p$-solution
$u\in  \overset{\circ}{W_p^{m,1 }}(Q_T)$ which  satisfies
\begin{equation}
 \| u\|_ {m,1;p} \leq C \|f \|_{-m,-1;p}
\end{equation}
where $C$ is a constant independent of $u$ and $f$.
\end{proposition}

\begin{proof}
By Proposition \ref{pro.1a} we have
\begin{equation}
 \|u^\epsilon\|_ {m,1;p} \leq C \|f \|_{-m,-1;p}
\end{equation}
where the constant $C$  does not depend on $\epsilon$.
Setting
$\widetilde{u}^\epsilon=u^\epsilon$ in $Q^\epsilon_T$, and vanishes
 outside $Q^\epsilon_T$.
 From the inequality above we obtain
\begin{equation}\label{eq.5c}
\|\widetilde{u^\epsilon}\|_ {m,1;p} \leq C \|f \|_{-m,-1;p}
 \end{equation}
where the constant $C$  does not depend on $\epsilon$.

 It implies that the set $\{\widetilde{u}^\varepsilon\}_{\varepsilon>0}$
is uniformly bounded in the space
${\mathaccent"7017 W}_p^{m,1}(Q_T)$. So we can take a subsequence,
denoted also by $\widetilde{u}^{\varepsilon} $
for convenience, which converges weakly to a function
$ u \in {\mathaccent"7017 W}_p^{m,1 }(Q_T)$.
We will show that $ u $ is a generalized $L_p$-solution of
 \eqref{eq.2}-\eqref{eq.4} in  cylinder $Q_T$.
 In fact  for all $\eta\in {\mathaccent"7017 W}_q^{m,1 }(Q_T),
$ there exists $\eta_\delta\in C_0^\infty(Q_T)$  such that $
  \eta_\delta\equiv 0$ in $Q_T\setminus Q^\varepsilon_T,
$ and $ \|\eta_\delta-\eta\|_{m,1;q}\to 0$ when
$\delta\to 0$. Since $\widetilde{u}^{\varepsilon}$  is a
generalized solution of  \eqref{eq.2}-\eqref{eq.4} in the
smooth cylinder $Q^\varepsilon_T$, we have
$$
 B_1[\widetilde{u}^\varepsilon, \eta_{\delta}]
=\langle f,\eta_{\delta} \rangle
$$
 Passing to the limit when $\varepsilon\to 0,\delta\to0$ for the
weakly convergent sequence, we get
$$
 B_1[u, \eta ]=\langle f,\eta \rangle
$$
Since ${\mathaccent"7017 W}_p^{m,1 }(Q_T)$ is imbedded
continuously into $L_p(\Omega)$, the trace sequence
$\{\widetilde{u}^{\varepsilon}(x,0)\}$ of
$\{\widetilde{u}^\varepsilon(x,t)\}$ converges weakly to the trace
$ u(x,0)$ of $u(x,t)$ in $L_p(\Omega )$. On the other hand,
$\widetilde{u}^{\varepsilon}(x,0)=0$, so that $ u(x,0)=0$;by
analogous arguments, we have $u_t(x,0)=0$. Hence, $ u(x,t)$ is a
generalized $L_p$-solution of  \eqref{eq.2}-\eqref{eq.4}.
Moreover, from (\ref{eq.5c}) we have
$$
\|u\|_{m,1;p} \leq\varliminf_{\varepsilon\to0}
\|\widetilde{u}^{\varepsilon}\|_{m,1;p}\leq C \|f \|_{-m,-1;p}.
$$
\end{proof}

Proposition \ref{pro.1b} stated the existence of generalized
$L_p$-solutions of  \eqref{eq.2}-\eqref{eq.4}   in
${\mathaccent"7017 W}_p^{m,1 }(Q_T)$ when $f\in
C^\infty(\overline{Q}_T)$ and $ f_{ t^k}\big|_{t=0}=0$,
for $k=0,1,\dots$. We now establish the problem when
$f\in W^{-m,-1}_p(Q_T )$.

\begin{theorem}\label{Theo2}
Suppose that  $f\in W^{-m,-1}_p(Q_T ), p\in(1,+\infty) $,
 then  \eqref{eq.2}-\eqref{eq.4}
has a generalized $L_p$-solution  $u\in{\mathaccent"7017 W}_p^{m,1
}(Q_T) $, and
\begin{equation}
 \| u\|^p_ {m,1;p} \leq C \|f\|^p_{0,p},
\end{equation}
where $C$ is a constant independent of $u$ and $f$.
\end{theorem}

\begin{proof}
 We start by studying the case $1<p\leq 2$.
Denote
$$
f_h(x,t)= \begin{cases}
0, &  \text{outside }  Q^\epsilon_T\\
f(x,t), &   t>h \\
0, &    t\leq h
\end{cases}
$$
for all $h>0$. We denote by $g_{\frac{h}{2}}$ the mollification
of $f_h$. Then $g_{\frac{h}{2}}\in C_0^\infty(Q_T),
 g_{\frac{h}{2}}\equiv 0, t<\frac{h}{2}$ and
$g_{\frac{h}{2}} \to f$ in $W^{-m,-1}_p(Q_T ) $.
 By Proposition \ref{pro.1b}, problem \eqref{eq.2}-\eqref{eq.4}
has a generalized $L_p$-solution
$u_h\in\overset{\circ}{W}_p^{m,1 }(Q_T) $ with replacing
$f$ by $g_{\frac{h}{2}}$,  and the following estimates holds
\begin{equation} \label{eq.6a}
 \| u_h\|_ {m,1;p} \leq C \| g_{\frac{h}{2}} \|_{-m,-1;p}
\end{equation}
where $C$ is a constant independent of $h, u$ and $f$.
Since $\{g_{\frac{h}{2}}  \}$ is a Cauchy sequence in
$L_p(Q_T)$ and inequality (\ref{eq.6a}), it follows that
$\{u_h\}$ is a Cauchy sequence in $\overset{\circ}{W}_p^{m,1 }(Q_T)$.
Hence, $u_h\to u\in \overset{\circ}{W}_p^{m,1 }(Q_T)$, then $u$ is
a generalized $L_p$-solutions of \eqref{eq.2}-\eqref{eq.4}
and satisfies
$$
\| u\|_ {m,1;p} \leq C \|f\|_{-m,-1;p}\,.
$$
Thus, the theorem is proved in the case $1<p\leq2$.

Now we study the case $p>2$. It is clear that
$q=\frac{p}{p-1}\in(1,2)$; by the proof above, for any
$g\in W_q^{-m,-1 }(Q_T)$ there exists a solution
$v\in {\mathaccent"7017 W}_q^{m,1 }(Q_T)$ of the adjoint problem
\begin{equation}\label{eq.6b}
B_1[v,u]=\langle g,u\rangle
\end{equation}
for all $u\in {\mathaccent"7017 W}_p^{m,1 }(Q_T)$, and
$$
\| v\|_ {m,1;q} \leq C \| g  \|_{-m,-1;q}.
$$
 We suppose that $f\in C^\infty(\overline{Q_T})$, $f_{t^k}(x,0)=0$,
$k=0,1, \dots$ and for $u=u^\epsilon$ in (\ref{eq.6b}). Then, by
(\ref{eq.6b}), we have
\begin{align*}
|\langle g, u^\epsilon\rangle|
&=|B_1[v,u^\epsilon]|=|\overline{B_1[u^\epsilon,v]}|
=|\langle f, v\rangle|\\
&\leq \| f  \|_{-m,-1;p}\| v\|_ {m,1;q}\\
&\leq C\| f  \|_{-m,-1;p}\| g  \|_{-m,-1;q}
\end{align*}
for any $g\in W_q^{-m,-1 }(Q_T)$. This implies
$$
\|u^\epsilon\|_{m,1;p}
=\sup\big\{\frac{|\langle g, u^\epsilon\rangle|}{\| g  \|_{-m,-1;q}}:
\;0\ne g\in W_q^{-m,-1 }(Q_T)\big\}\leq C\| f  \|_{-m,-1;p}.
$$
 From this inequality and arguments analogous to proofs above,
we get the proof of the theorem in this case.
The proof is complete.
\end{proof}

We should remark that by replacing the condition
$f\in W_p^{-m,-1 }(Q_T)$ by condition $f\in L_p(Q_T)$,
and noting that
$$
\|f\|_{W_p^{-m,-1 }(Q_T)}\leq  \|f\|_{L_p(Q_T)},
$$
we obtain the following theorem.

\begin{theorem}\label{Theo3}
If $f\in L_p(Q_T), p\in(1,+\infty)$, then \eqref{eq.2}-\eqref{eq.4},
in the cylinder $Q_T$, has a generalized
$L_p$-solution $u\in  \overset{\circ}{W_p^{m,1 }}(Q_T)$ which  satisfies
\begin{equation*}
 \| u\|_ {m,1;p} \leq C \|f \|_{L_p(Q_T)}
 \end{equation*}
where $C$ is a constant independent of $u$ and $f$.
\end{theorem}

\subsection{Smoothness of the generalized $L_p$-solution with
respect to time}

The following theorem shows that the generalized $L_p$-solution
 $u\in {\mathaccent"7017 W}_p^{m,1}(Q_T)$ of problem
\eqref{eq.2}-\eqref{eq.4}  is smooth with respect to time variable
 $t$ if right hand-side $f$ and
coefficients of operator \eqref{PDO} are smooth enough
with respect to $t$.

\begin{theorem}\label{Theo4}
 Let $h$ be a positive integer, and  assume that
\begin{itemize}
\item[(1)] $  f_{t^k} \in L_p(Q_T)$, $k\leq h$,
\item[(2)] $ f_{t^k}\big|_{t=0}=0,x\in \Omega$, $k\leq h-1$,
\item[(3)] $\sup\big\{\big|
\frac{\partial^k a_{\alpha\beta}}{\partial t^k}\big|, k<h+1:(x,t)
\in Q_T, 0\leq|\alpha|,|\beta|\leq m\big\}\leq\mu $.
 \end{itemize}
Then the generalized solution $u\in {\mathaccent"7017
W}_p^{m,1}(Q_T)$   of  \eqref{eq.2}-\eqref{eq.4}  has
generalized derivatives with respect to $t$ up to order $h$ in
${\mathaccent"7017 W}_p^{m,1}(Q_T)$ and satisfies the estimate
\begin{equation}
\|u_{t^h}\|_{m,1;p}\leq c\sum_{k=0}^{h} \|f_{t^k}\|_{L_p(Q_T)},
\end{equation}
where $c$ is a constant independent of $u$ and $f$.
\end{theorem}

\begin{proof}
 In the case $1<p\leq 2$, Clearly, we needed only to show that
\begin{equation}\label{eq.6c}
\|u^\epsilon_{t^h}\|_{m,1;p}\leq \sum_{k=0}^h\|f_{t^k}\|_{L_p(Q_T)}
\end{equation}
where $f\in C^\infty (\overline{Q_T}), f_{t^k}(x,0)=0, x\in \Omega$.
It is proved by induction on $h$.
According to Proposition \ref{pro.1a}, inequality (\ref{eq.6c})
is valid for $h=0$. Now let it be true for $h-1$; we will
prove that this also holds for $h$.

 From the fact that $u ^\epsilon$ satisfies  \eqref{eq.2}
in $Q^\epsilon_T$, we have
\begin{equation}
f=Lu^\epsilon-u^\epsilon_{tt}.\label{eq.7b}
\end{equation}
  Differentiating equality (\ref{eq.7b}), $h$ times with
respect to $t$,  it follows that
$$
f_{t^h}=Lu^\epsilon_{t^h}+
\sum_{k=0}^{h-1} \binom{h-1}{k}  D_x^\alpha(a_{\alpha\beta t^{h-k}}
D_x^\beta u^\epsilon_{t^k}) -u^\epsilon_{t^{h+2}}.
$$
Therefore,
\begin{align*}
\langle f_{t^h}, v\rangle
&=\int_{Q_T}Lu_{t^h}v\,dx\,dt+\sum_{k=0}^{h-1} \binom{h-1}{k}
 \int_{Q_T}\sum_{|\alpha|,|\beta|=0}^m  D_x^\alpha
 (a_{\alpha\beta t^{h-k}} D_x^\beta u^\epsilon_{t^k})\overline{v}\,dx\,dt\\
&\quad - \int_{Q_T}u^\epsilon_{t^{h+1}}\overline{v}\,dx\,dt
\end{align*}
for all $v\in  {\mathaccent"7017 W}_q^{m,1}(Q_T)$.

By using Green's formula and integrating by parts,
\begin{equation*}
B_1[u^\epsilon_{t^h},v]=\langle f_{t^h}, v\rangle-\sum_{k=0}^{h-1}
 \binom{h-1}{k}\int_{Q_T}\sum_{|\alpha|,|\beta|=0}^m
a_{\alpha\beta t^{h-k}} D_x^\beta
 u^\epsilon_{t^k}\overline{ D_x^\alpha v}\,dx\,dt.
\end{equation*}
for all $v\in  {\mathaccent"7017 W}_q^{m,1}(Q_T)$.

 From the inequality above and H\"{o}lder's inequality, we have
\begin{equation}\label{eq.7c}
|B_1[u^\epsilon_{t^h},v]|\leq C(\|f_{t^h}\|_{L_p(Q_T)}
+\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p})\|v\|_{m,1;q}
\end{equation}
 for all $v\in{\mathaccent"7017 W}_q^{m,1}(Q_T) $.
By using \eqref{GD.1}, (\ref{eq.7c}) and the induction assumption,
we obtain
$$
\|u^\epsilon_{t^h}\|_{m,1;p}
\leq C\sum_{k=0}^{h}\|f_{t^k}\|_{L_p(Q_T)}
$$
where $C$ is a constant independent of $\epsilon, u$.
The proof is completed in this case.

In the case $p>2$. It is easy to recognize that
$q=\frac{p}{p-1}\in(1,2)$; by Theorem \ref{Theo2}, for any
$g\in W_q^{-m,-1 }(Q_T)$ there exists a solution
$v\in {\mathaccent"7017 W}_q^{m,1 }(Q_T)$ of the adjoint problem
\begin{equation}\label{eq.7d}
B_1[v,u]=\langle g,u\rangle
\end{equation}
which for all $u\in {\mathaccent"7017 W}_p^{m,1 }(Q_T)$, and
$$
\| v\|_ {m,1;q} \leq C \| g  \|_{-m,-1;q}.
$$
We assume that $f\in C^\infty(\overline{Q_T})$, $f_{t^k}(x,0)=0$,
$k=0,1, \dots$ and for $u=u^\epsilon_{t^h}$ in (\ref{eq.7d}). Then, by
(\ref{eq.7d}) and (\ref{eq.7c}), we have
\begin{align*}
|\langle g, u^\epsilon_{t^h}\rangle|
&=|B_1[v,u^\epsilon_{t^h}]|=|\overline{B_1[u^\epsilon_{t^h},v]}|\\
&\leq C(\|f_{t^h}\|_{L_p(Q_T)} +\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p})
\|v\|_{m,1;q}\\
&\leq C(\|f_{t^h}\|_{L_p(Q_T)} +\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p})
\| g  \|_{-m,-1;q}
\end{align*}
for any $g\in W_q^{-m,-1 }(Q_T)$. Hence,
\begin{align*}
\|u^\epsilon_{t^h}\|_{m,1;p}
&=\sup\big\{\frac{|\langle g, u^\epsilon_{t^h}\rangle|}
{\| g  \|_{-m,-1;q}}: \;0\ne g\in W_q^{-m,-1 }(Q_T)\big\}\\
&\leq  C(\|f_{t^h}\|_{L_p(Q_T)} +\sum_{k=0}^{h-1}\|u^\epsilon_{t^k}\|_{m,1;p}).
 \end{align*}
From this inequality and induction assumption, we have the proof
of this case, and complete the proof.
\end{proof}

 \subsection{Regularity of the generalized  $L_p$-solution}\label{sec4}

In this section, we consider problem \eqref{eq.2}-\eqref{eq.4}
in cylinders $Q_T=\Omega\times (0,T)$,
where its base $\Omega$ is  described as follows:

 Let $\varphi$ be an infinitely differentiable positive function
on the interval $(0,1]$ satisfying the conditions
\begin{itemize}
  \item[(i)] $\lim_{\tau\to0}\varphi(\tau)^{k-1}\varphi(\tau)^{(k)}<\infty$
   for  $k=1,2, \dots$;
  \item[(ii)] $\int_0^1\frac{d\tau}{\varphi(\tau)}=+\infty$
\end{itemize}
 These conditions are satisfied, for example, by the function
$\varphi(\tau)=\tau^\alpha$ if $\alpha\geq1$.
Obviously, conditions (i) and (ii) imply $\varphi(0)=0$.
Suppose that $\Omega$ is a bounded domain in
$\mathbb{R}^n (n\geq 2),\partial \Omega\backslash\{\mathcal{O}\}$
is smooth, and
$$
\{x\in\Omega:\;0< x_n<1\}=\{x\in\mathbb{R}^n: x_n<1,\;
 x'\in\varphi(x_n)\omega\},
$$
where $x'=(x_1,\dots,x_{n-1}), \omega$ is a smooth domain in
$\mathbb{R}^{n-1}$. Then the mapping
\begin{equation}\label{map}
y_j=\frac{x_j}{\varphi(x_n)}, \text{ if } j=1,\dots,n-1,\quad
\text{and}\quad
 y_n= \int_{x_n}^1\frac{d\tau}{\varphi(\tau)}
\end{equation}
takes the set $\{x\in\Omega:\; 0<x_n<1\}$ onto the half-cylinder
$\mathcal{C_+}=\{y\in\mathbb{R}^n: y'\in \omega
,\;y_n>0\}=\omega\times (0,+\infty)$.
Moreover, it follows that
$$
\det \Big(\frac{\partial y_j}{\partial
x_k}\Big)_{j,k=1,\dots,n}=\varphi(x_n)^{-n}.
$$
It is known that the function $\varphi$ can be extended to an
infinitely differentiable positive function on the interval
$(0,+\infty)$.
  To consider the problem, we need to introduce some
weighted Sobolev spaces.
The space $W^l_{p,\beta,\gamma}(\Omega)$ can be defined as the
closure of the set
$C_0^\infty(\overline{\Omega}\backslash\{\mathcal{O}\})$
with respect to the norm
$$
\|u\|_{W^l_{p,\beta,\gamma}(\Omega)}
=\Big(\int_{\Omega}
\sum_{|\alpha|\leq l}e^{p\beta y_n(x_n)}\varphi(x_n)^{p(\gamma-l+|\alpha|)}
|D^\alpha u|^p\,dx\Big)^{1/p}.
$$
Let $X, Y$ be Banach spaces, we denote by $L_p(0,T;X)$ the spaces
consisting of all measurable functions
$u:(0,T)\to X$ with norm
$$
\|u\|_{L_p(0,T;X)}=\Big(\int^T_0\|u(t)\|^p_X\,dt\Big)^{1/p},
$$
and by $W_p^k(0,T;X,Y), k=1,2$, the spaces consisting of
functions $u\in L_p(0,T;X)$
such that generalized derivatives $u_{t^k}=u^{(k)}$ exist and
belong to $L_p(0,T;Y)$, (see \cite{EV}), with norm
$$
\|u\|_{W_p^k(0,T;X,Y)}=\Big( \|u\|^2_{L_p(0,T;X)}
+\sum_{j=1}^k\|u_{t^j}\|^p_{L_p(0,T;Y)}\Big)^{1/p}.
$$
For short notation, we set
\begin{gather*}
V_p^l(\Omega)=W_{p,0,0}^l(\Omega), \quad
V_p^{l,k}(Q_T)=W_p^k(0,T; V_p^l(\Omega) ,L_p(\Omega)),\\
W_{p,\beta,\gamma}^{l,k}(Q_T)= W_p^k(0,T; W_{p,\beta,\gamma}^l(\Omega),
L_p(\Omega)).
\end{gather*}
Finally, we define the weighted Sobolev space
$W_{p,\beta,\gamma}^l(Q_T)$ as the set of  functions defined
in $Q_T$ such that
$$
\|u\|_{W_{p,\beta,\gamma}^l(Q_T)}
=\Big(\int_{Q_T}\sum_{|\alpha|+k\leq l}
e^{2\beta y(x_n)}\varphi(x_n)^{p(|\gamma-l+\alpha|+k)}
|D^\alpha u_{t^k}|^p\,dx\,dt\Big)^{1/p}<+\infty.
$$
To simplify notation, we continue to write $V_p^l(Q_T)$
instead of $W_{p,0,0}^l(Q_T)$.

  Moreover, we assume that the functions
\begin{equation}
\widehat{a}_{\alpha\beta}(y,.)=\varphi(x(y))^{2m-|\alpha|
-|\beta|}a_{\alpha\beta}(x(y),.)
\end{equation}
satisfy the condition of stabilization for $y_n\to +\infty$ for a.e.
 $t$ in $(0,T)$(see\cite[Sec.9]{M1}).
Then the coefficients of the operators
$\widehat{L}(y,t;\,D_y)$, which arises from the operators
 $ \varphi(x_n)^{2m}L(x,t;D_x)$ via the coordinate change $x\to y$,
stabilize for $y_n\to +\infty$. If we replace the coefficients of
the differential operator $\widehat{L}(y,t;\,D_y)$
by their limits for $y_n\to +\infty$, we get differential
operator (denote also by $\widehat{L}(y',t;\,D_{y'}, D_{y_n})$
for convenience)
which has coefficients depending only on $y'$ and $t$.


 By the following proposition, we can apply the results
of the Dirichlet problem to elliptic equations in domains
with cuspidal points on boundary.

\begin{proposition} \label{pro.1}
Suppose that $u=u(x,t)$ is a generalized  solution of  problem
\eqref{eq.2} -\eqref{eq.4} and $u_{tt}\in L_p(Q_T)$. Then for a.e.
$t\in(0,T),\,u(t)=u(.,t)$ is a generalized solution in
${\mathaccent"7017 W}_p^m(\Omega)$ of the Dirichlet problem for
elliptic equation
\begin{equation}
 L(.,t;D_x)u=f_1(.,t)   \label{eq4.1}
\end{equation}
where $f_1=u_{tt}+f$.
\end{proposition}

\begin{proof}
For any $ \psi \in {\mathaccent"7017 W}_q^m(\Omega)$,
$\theta\in C^\infty_0(0,T)$ and setting $v(x,t)=\psi(x)\theta(t)$, we
substitute the function $v(x,t)$  into (\ref{eq.5}), we conclude
that
\begin{equation}\label{eq4.2}
  \int_{Q_T} \Big[\sum_{|\alpha|,|\beta|=0}^m a_{\alpha\beta}D_x^\beta\,
u\overline{D_x^\alpha \psi}
-(u_{tt}\,\overline{\psi}+f\overline{\psi})\Big]\overline{\theta(t)}
\,dx\,dt= 0.
\end{equation}
We will denote by
$$
\xi(t)=  \int_{\Omega} \Big[\sum_{|\alpha|,|\beta|=0}^m
a_{\alpha\beta}D_x^\beta\, u\overline{D_x^\alpha \psi }
- (u_{tt}\,\overline{\psi}+f\overline{\psi })\Big]\,dx,
$$
then $\xi(t)\in L_p(0,T)$. Noting that
$\theta\in C^\infty_0(0,T)$, which dense in $L_q(0,T)$ and
 using Fubini's theorem, we obtain from (\ref{eq4.2}) that
\begin{equation}\label{eq4.3}
  \int_0^T\xi(t)\overline{\theta}(t)\,dt= 0, \quad\text{for any }
\theta\in L_q(0,T), (1/p+1/q=1).
\end{equation}
Therefore,
$$
\|\xi\|_{L_p(0,T)}=\sup\big\{\int_0^T\xi(t)\overline{\theta}(t)\,dt:
 \theta\in L_q(0,T), \|\theta\|_{L_q(0,T)}=1\big\}=0.
$$
This implies  $\xi=0$ for a.e. $t\in(0,T) $.
Hence,
 $$
\int_{\Omega} \sum_{|\alpha|,|\beta|=0}^m
a_{\alpha\beta}D_x^\beta\, u\overline{D_x^\alpha \psi}\,dx
=\int_{\Omega}(u_{tt} +f)\overline{\psi}\,dx
$$
for all $\psi\in {\mathaccent"7017 W}_q^{m}(\Omega)$,   for a.e.
$t\in (0,T)$. It follows that $ u(t)$ is a generalized solution in
${\mathaccent"7017 W}_p^m(\Omega)$ of the Dirichlet problem for
elliptic equation \eqref{eq4.1}, for a.e. $t\in (0,T)$.
\end{proof}

In this section, we present the main results  which is
based on our previous subsection and the results for elliptic
equations in cusp domains (cf. \cite{M1}).
For the start of this section, we denote by
$\mathcal{U}(\lambda,t) (\lambda\in \mathbb{C},t\in(0,T))$ the
operator corresponding to the parameter-depending boundary-value problem
\begin{equation}\label{eq4.4}
\begin{gathered}
\widehat{L}(y',t;\,D_{y'}, \lambda)v=0 \quad \text{in } \omega; \\
\partial^j_\nu v=0\quad \text{on } \partial \omega,\; j=1,\dots, m-1.
\end{gathered}
\end{equation}
Where $\widehat{L}(y',t;\,D_{y'},\,\lambda )$ is the Fourier
transformation $y_n\to \lambda$ of
$\widehat{L}(y',t;\,D_{y'},\,D_{y_n})$.

For each $t\in(0,T)$, the operator pencil $\mathcal{U}(\lambda,t) $
is Fredholm, and its spectrum consists of a countable numbers
of isolated eigenvalues.
The similarly, to Theorem 9.1 in \cite{M1}, we have the following lemma.

\begin{lemma}\label{lem.1}
Assume that $f_1\in W^k_{p,\beta,\gamma}(\Omega)$, where
$\beta, \gamma $ are real numbers. Suppose further that no
eigenvalues of  $ \;\mathcal{U}(\lambda,t), t\in(0,T))$ line in strip
 $$
\mathop{\rm Im}\lambda_{-}\leq \mathop{\rm Im}\lambda
\leq\mathop{\rm Im}\lambda_{+};\quad
  \mathop{\rm Im}\lambda_{-}<\beta<\text{ Im}\lambda_{+}
$$
 where $\lambda_{+},\,\lambda_{-}$ are eigenvalues of
$\mathcal{U}(\lambda,t)$, and
$\mathop{\rm Im}\lambda_{-}<0<\mathop{\rm Im}\lambda_{+}$.
Then the generalized solution $u$ of the Dirichlet problem for the
elliptic equation \eqref{eq4.1}, $u\equiv 0$ if $x_n>1$, belongs
 to the space $W^{2m+k}_{p,\beta,\gamma}(\Omega)$ and  satisfies
the inequality
\begin{equation}\label{eq4.5}
\|u\|^2_{W^{2m+k}_{p,\beta,\gamma}(\Omega)}
\leq C\|f_1\|^2_{W^k_{p,\beta,\gamma}(\Omega)}
\end{equation}
where the constant $C$ is independent of $f_1$.
\end{lemma}

\begin{proof}
Setting
$\omega_\tau=\varphi(\tau)\omega$
by the Friederichs inequality, we have
$$
\int_{\omega_\tau}|u|^pdx'\leq C \varphi(\tau)^{pk}
\sum_{|\gamma|=k}\int_{\omega_\tau}|D^\gamma_{x'}u|^pdx'\,;
$$
 therefore,
$$
\varphi(x_n)^{p(|\gamma|-m)}\int_{\omega_{x_n}}|D^\gamma_{x'}u |^pdx'
\leq C \sum_{|\alpha|=m}\int_{\omega_{x_n}}|D^\alpha_{x'}u|^pdx'
$$
for all $|\gamma|\leq m$. Hence,
\begin{equation}\label{eq4.6}
 \sum_{|\gamma|\leq m}\int_{\Omega}\varphi(x_n)^{p(|\gamma|-m)}
|D_x^\gamma u |^pdx\leq C \sum_{|\alpha|\leq m}
\int_{\Omega}|D_x^\alpha u|^pdx
\end{equation}

  Let $v=v(y)$ be the function that arises from
$\varphi(x_n)^{m-\frac{n}{p}}u(x)$ via the coordinate change $x\to y$.
We set $\overline{\varphi}(y_n)=\varphi(x_n)$, from the properties
of the mapping (\ref{map}) and from  inequality (\ref{eq4.6}), it
follows that $(\overline{\varphi})^{-m+\frac{n}{p}}v\in
{\mathaccent"7017 W}_p^m(\mathcal{C_+})$. Since
$(\overline{\varphi})^{-m+\frac{n}{p}}v$ is the solution of an
elliptic equation in $\mathcal{C_+}$ with coefficients which
stabilize for $y_n\to+\infty$, i.e.
 $$
\widehat{L}(\overline{\varphi})^{-m+\frac{n}{p}}v=\widehat{f_1}
$$
where $\widehat{f_1}=(\overline{\varphi})^{2m }f_1$,
we obtain
$(\overline{\varphi})^{-m+\frac{n}{p}}v\in{\mathaccent"7017
W}_p^{2m+k}(\mathcal{C_+})$ (cf. \cite[Theorem 8.1, 8.2]{M1}).
This implies $u\in W^{2m+k}_{p,0,m+k}(\Omega)$. Using the fact that
$$
\varphi(x_n)^{\gamma-m+k}e^{-\epsilon y_n(x_n)}\to 0
$$
as $x_n\to 0, $ if $0<\epsilon<\beta$, we conclude that
$u\in W^{2m+k}_{p,-\epsilon,\gamma}(\Omega)$. In a similar manner,
Theorem 8.2 in  \cite{M1} it follows that
$u\in W^{2m+k}_{p,\beta,\gamma}(\Omega)$. Furthermore, \eqref{eq4.5}
 is valid.
\end{proof}

\begin{lemma}\label{lem.2}
Suppose that $f,f_t\in L_p(Q_T), f(x,0)=0$, and the strip
$\mathop{\rm Im}\lambda_{-}\leq \mathop{\rm Im}\lambda
\leq\mathop{\rm Im}\lambda_{+}$
does not contain  eigenvalues of
$\mathcal{U}(\lambda,t), t\in(0,T))$.
  Then  the generalized solution $u$ of problem \eqref{eq.2}-\eqref{eq.4},
$u\equiv 0$ if $x_n>1$, belongs to the $V_p^{2m,2}(Q_T)$ and
satisfies the inequality
\begin{equation}\label{eq.9}
\|u\|_{V_p^{2m,2}(Q_T)}\leq C[\|f\|_{L_p(Q_T)   }+\|f_t\|_{L_p(Q_T)}],
\end{equation}
where the constant $C$ is independent of $f$.
\end{lemma}

\begin{proof}
Using the smoothness of the  generalized solution of
\eqref{eq.2}-\eqref{eq.4} with respect to $t$ in Theorem
\ref{Theo4} and  Proposition \ref{pro.1}, we can see that for a.e.
$t\in (0,T), u\in{\mathaccent"7017 W}_p^m(\Omega)$ is the
generalized solution of Dirichlet problem for equation
\eqref{eq4.1} with  compact support, where $f_1=u_{tt}+f\in
L_p(\Omega)=W^0_{p,0,0}(\Omega)=V_p^0(\Omega)$. From Lemma
\ref{lem.1}, it implies that $u\in V_p^{2m}(\Omega)$ for a.e.
$t\in(0,T)$ and satisfies the inequality
$$
\|u\|_{V_p^{2m}(\Omega)}\leq C_1\|f_1\|_{L_p(\Omega)}
\leq C\left(\|f\|_{L_p(\Omega)}+\|u_{tt}\|_{L_p(\Omega)}\right) .
$$
By integrating the inequality above with respect to $t$ from $0$ to $T$,
 and using the estimates for derivatives of $u$ with respect to $t$
again, we obtain $u\in V_p^{2m,2}(Q_T)$, which satisfies
(\ref{eq.9}).
 \end{proof}


\begin{theorem} \label{Theo5}
Let the assumptions of Lemma \ref{lem.2} be satisfied, and
$f_{t^k}\in L_p(Q_T)$, $k\leq 2m$, $f_{t^k}(x,0)=0$, for
$k=0,1,\dots,2m-1$.
  Then  the generalized solution $u$ of problem
\eqref{eq.2}-\eqref{eq.4}, $u\equiv 0$ if $x_n>1$,  belongs to the
$V_p^{2m}(Q_T)$ and  satisfies the inequality
\begin{equation}\label{eq.9a}
\|u\|_{V_p^{2m}(Q_T)}\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}
\end{equation}
where the constant $C$ is independent of $f$.
\end{theorem}

\begin{proof}
 Let us first prove that $u_{t^s}$ belongs to $V_p^{2m,0}(Q_T)$
for $s=0,\dots,2m-1$ and satisfy
\begin{equation}\label{eq.10}
 \|u_{t^s}\|_{V_p^{2m,0}(Q_T)}\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}.
\end{equation}
The proof is by done induction on $s$. According to
Lemma \ref{lem.2}, it is valid for $s=0$. Now let this assertion
be true for $s-1$, we will prove that this also holds for $s$.
Due to Lemma \ref{lem.2} then $u$ satisfies  \eqref{eq.2}, by
differentiating both sides of \eqref{eq.2} with respect to $t, s$
times, we obtain
\begin{equation}\label{eq.11}
Lu_{t^s}=f_{t^s}+u_{t^{s+2}}-\sum_{k=1}^{s}\binom{s}{k}L_{t^k}u_{t^{s-k}}
\end{equation}
where
$$
L_{t^k}=L_{t^k}(x,t;D_x)=\sum_{\alpha,\beta=0}^{m}
D_x^\alpha\Big(\frac{\partial^ka_{\alpha\beta}(x,t) }
{\partial t^k}\,D_x^\beta\Big).
$$
By the supposition of the theorem and the inductive assumption,
the right-hand side of (\ref{eq.11}) belongs to $L_p(Q_T)$.
By the arguments analogous to the proof of Lemma \ref{lem.2},
we get $u_{t^s}\in V_p^{2m,0}(Q_T)$ and
\begin{equation}\label{eq.12}
\|u_{t^s}\|_{V_p^{2m,0}(Q_T)}\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}
\end{equation}
where $C$ is a constant independent of $u,\,f$, and $s\leq m-1$.

Using (\ref{eq.12})  and estimates for derivatives of $u$ with
respect to $t$ in Theorem \ref{Theo3}, we have
\[
\|u\|_{V_p^{2m}(Q_T)} \leq \sum_{k=0}^{2m-1}
 \|u_{t^k}\|_{V_p^{2m,0}(Q_T)}+\|u_{t^{2m}}\|_{L_p(Q_T)}
\leq  C\sum_{k=0}^{2m}\|f_{t^k}\|_{L_p(Q_T)}.
\]
\end{proof}

\noindent\textbf{Remark.}
Let $\beta$ be a sufficiently small positive number. Suppose that
$$
e^{\beta y_n(x_n)}f\in L_p(Q_T),  \quad
\mathop{\rm Im}\lambda_{-}< \beta<\mathop{\rm Im}\lambda_{+}
$$
and the strip
$$ \mathop{\rm Im}\lambda_{-}\leq \mathop{\rm Im}\lambda
\leq\mathop{\rm Im}\lambda_{+}
$$
 contains no eigenvalues of $\mathcal{U}(\lambda,t), t\in(0,T))$;
then  the generalized solution $u$ of  \eqref{eq.2} -\eqref{eq.4},
$u\equiv 0$ if $x_n>1$, belongs to the $W_{p,\beta,0}^{2m}(Q_T)$.
In fact that, setting $u=e^{-\beta y_n(x_n)}U$,
we obtain the first initial boundary value problem which
differs little from \eqref{eq.2}-\eqref{eq.4}. Therefore,
$U\in V_p^{2m}(Q_T)$, and then
$u\in W_{p,\beta,0}^{2m}(Q_T)$. Using the remark above and
Lemma \ref{lem.1}, we obtain the following theorem.

\begin{theorem}\label{Theo6}
Let the assumptions of Lemma \ref{lem.1} be satisfied.
Furthermore, we assume that
$f_{t^k}\in W^0_{p,\beta,\gamma}(Q_T), k\leq 2m$ and
$f_{t^k}(x,0)=0$, for $k=0,1,\dots,2m-1$.
  Then  the generalized solution $u$ of  \eqref{eq.2}-\eqref{eq.4},
such that $u\equiv 0$ if $x_n>1$,  belongs to the
$W_{p,\beta,\gamma}^{2m}(Q_T)$ and  satisfies the inequality
\begin{equation}\label{eq.12a}
\|u\|_{W_{p,\beta,\gamma}^{2m}(Q_T)}
\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W^0_{p,\beta,\gamma}(Q_T)}
\end{equation}
where the constant $C$ is independent of $f$.
\end{theorem}

This theorem is proved by arguments analogous to those proofs
of Lemma \ref{lem.2} and Theorem \ref{Theo4}.
Next, we will prove the  regularity of the generalized
solution of problem \eqref{eq.2}-\eqref{eq.4}.

\begin{theorem}\label{Theo6b}
Let the assumptions of Lemma \ref{lem.1} be satisfied.
Furthermore, we assume that
$f_{t^k}\in W_{p,\beta,\gamma}^{h}(Q_T), k\leq 2m+h$ and
$f_{t^k}(x,0)=0$, for $k=0,1,\dots,2m+h-1,\; h\in \mathbb{N}$.
Then  the generalized solution $u$ of  \eqref{eq.2}-\eqref{eq.4},
such that $u\equiv 0$ if $x_n>1$, belongs to
$W_{p,\beta,\gamma}^{2m+h}(Q_T)$ and  satisfies the inequality
\begin{equation}\label{eq.13}
\|u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)}
\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}
\end{equation}
where the constant $C$ is independent of $u$ and $f$.
\end{theorem}

\begin{proof}
The theorem is proved by induction on $h$. Thanks to
Theorem \ref{Theo5}, this theorem is obviously valid for $h=0$.
Assume that the theorem is true for $h-1$, we will prove that
it also holds for $h$. It is only needed to show that
\begin{equation}\label{eq.14}
\begin{gathered}
u_{t^s}\in W_{p,\beta,\gamma}^{{2m+h-s},0}(Q_T) \quad\text{for }
 s=h,h-1\dots, 0;\\
\|u_{t^s}\|_{W_{p,\beta,\gamma}^{2m+h-s}(Q_T)}
 \leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}.
\end{gathered}
\end{equation}
Differentiating both sides of \eqref{eq.2} again with respect to $t, h$
times, we obtain
\begin{equation}\label{eq.14a}
Lu_{t^h}=f_{t^h}+u_{t^{h+2}}-\sum_{k=1}^{h}\binom{h}{k}L_{t^k}u_{t^{h-k}}
\end{equation}
By the supposition of the theorem and the inductive assumption,
the right-hand side of (\ref{eq.14a}) belongs to
$W^0_{p,\beta,\gamma}(\Omega)$ for a.e. $t\in (0,T)$.
Using Lemma \ref{lem.1},  we conclude that
$u_{t^h}\in W_{p,\beta,\gamma}^{{2m},0}(Q_T)$.
It implies that (\ref{eq.14}) holds for $s=h$.
Suppose that (\ref{eq.14}) is true for $s=h, h-1,\dots, j+1$
and set $v=u_{t^j}$, we obtain
\begin{equation}\label{eq.15}
Lv=F_j,
\end{equation}
 where $F_j=f_{t^j}+v_{tt}-\sum_{k=1}^{j}\binom{j}{k}L_{t^k}u_{t^{j-k}}$.
By the inductive assumption with respect to $s$, $v_{tt}$ belongs to
$ W_{p,\beta,\gamma}^{{h-j}}(\Omega)$ for a.e. $t\in(0,T)$.
Thus, the right-hand side of (\ref{eq.15}) belongs to
$W_{p,\beta,\gamma}^{h-j}(\Omega)$. Applying Lemma \ref{lem.1}
again for $k=h-j$, we get that
$v=u_{t^j}\in W_{p,\beta,\gamma}^{2m+h-j}(\Omega)$ for a.e.
$t\in(0,T)$. It means that $v=u_{t^j}$ belongs to
$ W_{p,\beta,\gamma}^{{2m+h-j},0}(Q_T)$.
Furthermore, we have
\begin{equation}
\|v\|_{W_{p,\beta,\gamma}^{{2m+h-j},0}(Q_T)}
\leq C\|F_j\|_{W^{h-j,0}_{p,\beta,\gamma}(Q_T)} 
\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}.
\end{equation}
Therefore,
\begin{align*}
\|u_{t^j}\|_{W_{p,\beta,\gamma}^{2m+h-j}(Q_T)}
&\leq\|u_{t^{j+1}}\|_{W_{p,\beta,\gamma}^{2m+h-j-1}(Q_T)}
+\|u_{t^j}\|_{W_{p,\beta,\gamma}^{{2m+h-j},0}(Q_T)}\\
&\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}.
\end{align*}
It implies that (\ref{eq.14}) holds for $s=j$. The proof is complete.
\end{proof}

Now we  prove the global regularity of the solution.

\begin{theorem}\label{Theo7}
Let the hypotheses of Lemma \ref{lem.1} be satisfied. Furthermore,
assume that $f_{t^k}\in W_{p,\beta,\gamma}^{h}(Q_T)$, $k\leq 2m+h$ and
$f_{t^k}(x,0)=0$, for $k=0,1,\dots,2m+h-1$, $h\in \mathbb{N}$.
  Then  the generalized solution $u$ of  \eqref{eq.2}-\eqref{eq.4}
belongs to  $W_{p,\beta,\gamma}^{2m+h}(Q_T)$ and  satisfies the inequality
\begin{equation}\label{eq.15a}
\|u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)}
\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}
\end{equation}
where the constant $C$ is independent of $u$ and $f$.
\end{theorem}

\begin{proof}
We denote by $B$ the unit ball and suppose that
$\zeta\in C^\infty_0(B)$, and $\zeta\equiv 1$ in the neighborhood
of the origin $\mathcal{O}$.
We have
$$
L(\zeta u)-(\zeta u)_{tt}=\zeta f+L_1u
$$
where $L_1$ is a differential operator, whose coefficients have
 compact support in a neighborhood of the origin.
By Theorem \ref{Theo6}, we obtain
$$
\|\zeta u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)}
\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}.
$$
Setting $\zeta_1u=(1-\zeta)u$, then  $\zeta_1u\equiv 0$  in a
neighborhood of the origin and $u=\zeta u+(1-\zeta)u$, and
using the smoothness of the solution of this problem in domain
with smooth boundary, we get
$$
\|\zeta_1u\|_{W_p^{2m+h}(Q_T)}\sim
\|\zeta_1u\|_{W_{p,\beta,\gamma}^{2m+h}(Q_T)}
\leq C\sum_{k=0}^{2m}\|f_{t^k}\|_{W_{p,\beta,\gamma}^h(Q_T)}.
$$
The proof is complete .
\end{proof}

\section{An example}\label{sec5}

In this section, we apply the results of the previous section to
 the Cauchy-Dirichlet problem for the beam equation.
Suppose that $\Omega$ is a bounded domain in
$\mathbb{R}^n,\partial \Omega\backslash\{\mathcal{O}\}$ is smooth,
and
$$
\{x\in\Omega: 0<x_n<1\}\equiv\{x\in\mathbb{R}^n:\;0<x_n<1,
|x'|<\varphi(x_n)\},
$$
where $x'=(x_1,\dots,x_{n-1}), \varphi\in C^\infty[0,1),
\varphi'(x_n)\to 0, \varphi(x_n)\varphi''(x_n)\to 0$ as
 $x_n\to 0$ and $\varphi(0)=0$.
Set $Q_T=\Omega\times(0,T), S_T=\partial\Omega
\setminus\{\mathcal{O}\}\times(0,T)$.

We consider the Cauchy-Dirichlet problem for the beam equation in $Q_T$:
\begin{gather}
 \Delta^2 u-\Delta u-u_{tt}=-f \quad\text{in } Q_T,\label{eq.2a} \\
u =0, u_{t} =0\quad \text{on }\Omega,\label{eq.3a}\\
u =0, \partial_\nu u=0\quad \text{on }S_T \label{eq.4a}
 \end{gather}
where $f: Q_T\to \mathbb{C}$ is given and $Lu= \Delta_x^2 u-\Delta_x u$.
By using Green's formula, we get
$$
B[u,u;\,t]=\int_{\Omega}\left( |D^2_xu|^2+|D_x u|^2\right)dx
$$
for all $u\in {\mathaccent"7017 W}^2_2(\Omega)$. On other hand, by
the Friedrich inequality
$$
\int_{\Omega}|u|^2\,dx\leq C\int_{\Omega}|D_xu|^2\,dx,
$$
it implies that there exists a constant $\gamma_0>0$ such that
$$
B[u,u;\,t]=\int_{\Omega}\left( |D^2_xu|^2+|D_x u|^2\right)dx
\geq \gamma_0|u|^2_{W{}^2_2(\Omega)}.
$$
Hence,  \eqref{GD} is satisfied for all $u\in
{\mathaccent"7017 W}^2_2(\Omega)$, for all $t\in(0,T)$.

For simplicity, we consider  \eqref{eq.2a}-\eqref{eq.4a} in the
two-dimensional case $(n=2)$, and let $\varphi(\tau)=\tau^2$;
  Then $\Omega$ is a bounded domain in
$\mathbb{R}^2$, $\partial \Omega\backslash\{\mathcal{O}\}$ is
smooth, and
$$
\{(x,y)\in\Omega: 0<x<1\}\equiv\{(x,y)\in\mathbb{R}^2:\;0<x<1, |y|<x^2\},
$$
on the change of variables
\begin{equation}
\xi=\int_x^1\cfrac{d\tau}{\tau^2}= x^{-1}-1,\quad \eta= yx^{-2},
\end{equation}
which transforms
 $\{(x,y)\in\Omega: 0<x<1\}$  onto
\[
\mathcal{C_+}:=\{(\xi,\eta): \xi>0, \,\eta\in(-1,1)\}.
\]
 With the notation  $v(\xi,\eta,t)=u(x,y,t)$, we have
$$
u(x,y,t)=v( x^{-1}-1,yx^{-2},t)
$$
and
\begin{align*}
\partial_y u&=x^{-2}  \partial_\eta v, \\
  \partial_xu &=-x^{-2}\partial_\xi v -2yx^{-3}\partial_\eta v,\\
  \partial^2_{yy}u&=x^{-4}\partial_{\eta\eta}^2v\\
\partial_{xx}^2u &=x^{-3}\partial_\xi v +6yx^{-4}\partial_\eta v
+x^{-4}\partial_{\xi\xi}^2v +4yx^{-5}\partial_{\xi\eta}^2v+4y^2x^{-6}
\partial^2_{\eta\eta} v\\
\\
&=x^{-4}\left[x\partial_\xi v +6y\partial_\eta v +\partial_{\xi\xi}^2v
+4yx^{-1}\partial_{\xi\eta}^2v+4y^2x^{-3}\partial^2_{\eta\eta} v\right]\\
\\
&=x^{-4}\big[\partial_{\xi\xi}^2v +4\eta(\xi+1)^{-1}
\partial_{\xi\eta}^2v+4\eta^2(\xi+1)^{-2}\partial^2_{\eta\eta} v\\
\\
&\quad+(\xi+1)^{-1}\partial_\xi v +6\eta(\xi+1)^{-2}\partial_\eta v \big].
\end{align*}
Hence, the differential operator $\widehat{\Delta}$, which arises from the 
differential operator
$x^{8}\Delta u, (\varphi(x)=x^2, 2m=4) $ via the coordinate change 
$(x,y)\to (\xi,\eta)$, turns out to be
\begin{align*}
 \widehat{\Delta}v&=(\xi+1)^{-4}\left(\partial_{\xi\xi}^2v+\partial_{\eta\eta}^2v\right)
+4\eta(\xi+1)^{-5}\partial_{\xi\eta}^2v+4\eta^2(\xi+1)^{-6}\partial^2_{\eta\eta} v\\
\\
&\quad+(\xi+1)^{-5}\partial_\xi v +6\eta(\xi+1)^{-6}\partial_\eta v;
\end{align*}
the similar calculation for $\widehat{\Delta^2} $.
 Clearly, coefficients of differential operator
$ \widehat{L}=\widehat{\Delta^2} -\widehat{\Delta}$ stabilize for
$\xi\to+\infty$ and
the limit differential operator of $ \widehat{L} $ (denote by $\widehat{L}$ 
for convenience) is
$$
\widehat{L}=\widehat{\Delta^2}v
=\partial_{\xi^4}^4v+2\partial_{\eta^2\xi^2}^4v+\partial_{\eta^4}^4v.
$$
We denote also by $\mathcal{U}(\lambda) (\lambda\in \mathbb{C})$
the operator corresponding to the parameter-depending boundary
value problem
\begin{gather*}
\frac{d^4v}{d\eta^4}-2\lambda^2\frac{d^2v}{d\eta^2}+\lambda^4v=0,\\
v(-1)=v(1)=0,\\
v'(-1)=v'(1)=0.
\end{gather*}
It is easy to see that $\mathcal{U}(\lambda)$ is invertible for all
$\lambda\in \mathbb{C}$.
 From arguments above in combination with  Theorem \ref{Theo6} and
Theorem \ref{Theo7}, we obtain the following results.

\begin{theorem} \label{thm4.1}
Suppose that
$e^{ \beta(\frac{1}{x}-1)}x^{2\gamma} f_{t^k}\in L_p(Q_T)$,
$k\leq 2, \beta, \gamma$ are real numbers
and $f_{t^k}(x,0)=0$, for $k=0,1$.
 Then  \eqref{eq.2a}-\eqref{eq.4a} has a unique solution $u$
in $W^2_{p,\beta,\gamma}(Q_T)$ and
$$
\|u\|_{W^2_{p,\beta,\gamma}(Q_T)}\leq
C\sum_{k=0}^2\|e^{ \beta(\frac{1}{x}-1)}x^{2\gamma}
f_{t^k}\|_{L_p(Q_T)}.
$$
Moreover,
if $f_{t^k}\in W^h_{p,\beta,\gamma}(Q_T)$, $k\leq 2+h$, and
$f_{t^k}(x,0)=0$ for $k=0,1,\dots, 1+h$,
then $u\in W^{2+h}_{p,\beta,\gamma}(Q_T)$ and satisfies
$$
\|u\|_{W^{2+h}_{p,\beta,\gamma}(Q_T)}
\leq C\sum_{k=0}^2 \|f_{t^k}\|_{W^h_{p,\beta,\gamma}(Q_T)}.
$$
\end{theorem}

In case boundary when $\Omega$ has  cuspidal points,
then by arguments analogous to Section \ref{sec3}, we obtain
 the similar results.

\subsection*{Acknowledgments}
This work was supported by National Foundation for Science and
Technology Development (NAFOSTED), Vietnam.


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\end{document}