\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 03, pp. 1--31.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2002 Southwest Texas State University.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/03\hfil Singular solutions to Protter's problem]
{Singular solutions to Protter's problem for the 3-D wave equation
involving lower order terms}

\author[M. K. Grammatikopoulos, T. D. Hristov, \& N. I. Popivanov
\hfil EJDE--2003/03\hfilneg]
{Myron. K. Grammatikopoulos, Tzvetan D. Hristov, \&  Nedyu I. Popivanov}

\address{Myron K. Grammatikopoulos \hfill\break
Department of Mathematics, University of Ioannina\\
451 10 Ioannina, Greece}
\email{mgrammat@cc.uoi.gr}

\address{Tzvetan D. Hristov \hfill\break
Department of Mathematics and Informatics, University of Sofia, 1164 Sofia,
Bulgaria}
\email{tsvetan@fmi.uni-sofia.bg}

\address{Nedyul I. Popivanov \hfill\break
Department of Mathematics and Informatics, University of Sofia,
1164 Sofia, Bulgaria}
\email{nedyu@fmi.uni-sofia.bg}

\date{}
\thanks{Submitted May 27, 2002. Published January 2, 2003.}

\subjclass[2000]{35L05,35L20, 35D05, 35A20}
\keywords{Wave equation, boundary value problems, generalized solutions,
\hfill\break\indent
singular solutions, propagation of singularities}

\begin{abstract}
  In 1952, at a conference in New York, Protter formulated
  some boundary value problems for the wave equation, which are
  three-dimensional analogues of the Darboux problems (or Cauchy-Goursat
  problems) on the plane.  Protter studied these problems
  in a 3-D domain $\Omega_0$, bounded by two characteristic cones
  $\Sigma_1$ and $\Sigma_{2,0}$, and by a plane region $\Sigma_0$.
  It is well known that, for an infinite number of smooth functions
  in the right-hand side, these problems do not have classical solutions.
  Popivanov and Schneider (1995) discovered the reason of this fact for the
  case of Dirichlet's and Neumann's conditions on $\Sigma_0$: the strong
  power-type singularity appears in the generalized solution on the
  characteristic cone $\Sigma_{2,0}$. In the present paper we consider the
  case of third boundary-value problem on $\Sigma_0$ and obtain the
  existence of many singular solutions for the wave equation involving
  lower order terms. Especifica ally, for Protter's problems in
  $\mathbb{R}^{3}$ it is shown here that for any $n\in N$ there exists a
  $C^{n}(\bar{\Omega}_0)$-function, for which the corresponding unique
  generalized solution belongs to $C^{n}(\bar{\Omega}_0\backslash O)$
  and has a strong power type singularity at the point $O$.
  This singularity is isolated  at the vertex $O$ of the characteristic
  cone $\Sigma_{2,0}$ and does not propagate along the cone. For the
  wave equation without lower order terms, we presented  the exact behavior of
  the singular solutions at the point $O$.
\end{abstract}

\maketitle

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

\section{Introduction}

Consider the hyperbolic partial differential equation, involving the wave
operator in its main part, with lower order terms of the form
\begin{equation}
Lu\equiv
u_{x_1x_1}+u_{x_2x_2}-u_{tt}+b_1u_{x_1}+b_2u_{x_2}+bu_{t}+cu=f
\label{eq:0p1}
\end{equation}
expressed in Cartesian coordinates $x_1,x_2,t$ in a simply
connected region $\Omega_0\subset \mathbb{R}^{3}$. The region
\begin{equation*}
\Omega_0:=\{(x_1,x_2,t):0<t<1/2,t<\sqrt{x_1^2+x_2^2}<1-t\}
\end{equation*}
is bounded by the disk
\begin{equation*}
\Sigma_0:=\{(x_1,x_2,t):t=0,x_1^2+x_2^2<1\}
\end{equation*}
with center at the origin $O(0,0,0)$ and the characteristic surfaces of
\eqref{eq:0p1}:
\begin{gather*}
\Sigma_1:=\{(x_1,x_2,t):0<t<1/2,\sqrt{x_1^2+x_2^2}=1-t\}, \\
\Sigma_{2,0}:=\{(x_1,x_2,t):0<t<1/2,\sqrt{x_1^2+x_2^2}=t\}.
\end{gather*}
In this work we are interested in finding sufficient conditions for the
existence and uniqueness of a generalized solution to the following problem.

\noindent \textbf{Problem $\mathbf{P_\alpha }.$}
 Find a solution to \eqref{eq:0p1} in $\Omega_0$ that satisfies the
boundary conditions
\begin{equation}
 u|_{\Sigma_1}=0,\quad \lbrack u_{t}+\alpha u]|_{\Sigma
_0\backslash O}=0,  \label{eq:0p2}
\end{equation}
where $\alpha \in C^{1}(\bar{\Sigma}_0\backslash O).$\bigskip

The adjoint problem to $\mathbf{P}_{\alpha }$ is as follows.

\noindent \textbf{Problem $\mathbf{P_\alpha ^{\ast }}$.}
 Find a solution of the adjoint equation
\begin{equation*}
L^{\ast }u\equiv
u_{x_1x_1}+u_{x_2x_2}-u_{tt}-(b_1u)_{x_1}-(b_2u)_{x_2}-(bu)_{t}+cu=g\quad
\text{ in }\Omega_0
\end{equation*}
with the boundary conditions:
\begin{equation}
 u|_{\Sigma_{2,0}}=0,\quad \lbrack
u_{t}+(\alpha +b)u]|_{\Sigma_0}=0.  \label{eq:0p3}
\end{equation}

The following problems were introduced by Protter (see \cite{Prot}).

\noindent\textbf{Protter's Problems.}
Find a solution of the wave equation
\begin{equation}
\square u\equiv \Delta_{x}u-u_{tt}\equiv
u_{x_1x_1}+u_{x_2x_2}-u_{tt}=f \quad \text{in }\Omega _0  \label{eq:0p30}
\end{equation}
with one of the following boundary conditions
\begin{equation}
\begin{alignedat}{2}
P1:&\quad u|_{\Sigma_0\cup \Sigma_1}=0, & \quad
P1^{\ast }:&\quad u|_{\Sigma_0\cup \Sigma_{2,0}}=0\,; \\
P2:&\quad u|_{\Sigma_1}=0,u_{t}|_{\Sigma_0}=0,&\quad
P2^{\ast}:&\quad u|_{\Sigma_{2,0}}=0,u_{t}|_{\Sigma_0}=0\;.
\end{alignedat}
\label{eq:0p4}
\end{equation}

Protter \cite{Prot} formulated and investigated both Problems $P1$
and $P1^{\ast }$ in $\Omega_0$ as multi-dimensional analogues of
the Darboux problem on the plane. It is well known that the
corresponding Darboux problems on $\mathbb{R}^2$ are well posed,
which is not true for the Protter's problems in $\mathbb{R}^{3}$.
The uniqueness of a classical solution of Problem $P1$ was proved
by Garabedian \cite{Gar}. For recent results concerning the
Protter's problems \eqref{eq:0p4} see the work of Popivanov,
Schneider \cite{PS95}, Grammatikopoulos, Hristov, Popivanov \cite
{GHP} and references therein. For more publications in this area
see, for example:
\cite{Ald93,Ald98,EP98,Kor,Kan95,Kan98,Kan981,NT}. Some different
statements of Darboux type problems can be found in
\cite{AS,BB,Bits,Kh95,Nak70} in bounded or unbounded domains,
different from $\Omega_0 $.

According to the ill-possedness of Protter's Problems $P1$ and $P2$, it is
interesting to find some their regularizations. A nonstandard, nonlocal
regularization of Problem P1, can be found in \cite{EP98}. In the present
paper we are looking for some other kind of regularization and formulate the
following problem.

\noindent\textbf{Open Question 1.} Is it possible to find
conditions for the coefficients $b_1,b_2,b_{3},c$ and $\alpha$,
under which for all smooth functions $f$  Problem $P_{\alpha }$
has only regular solutions?

\begin{remark} \label{rmk1.1} \rm
If the answer to the above question is positive, then, using an operator $%
L_k$ with lower order perturbations in the wave equation \eqref{eq:0p30},
we can find possible regularization for Problem $P2.\ $Solving the equation $%
L_ku_k=f$, with $L_k\to \square $ (i.e. $%
b_{1k},b_{2k},b_{3k},c_k\to 0)$ and $\alpha_k\to 0$, we
can find an approximated sequence $u_k$. Due to the fact that in this case
the cones $\Sigma_1$ and $\Sigma_{2,0}$ are again characteristics for $%
L_k$, this process,  with respect to our boundary value problem, looks to
be natural.
\end{remark}

For Problem {eq:0p1} with $P_{\alpha }$ and $\alpha (x)\neq 0$
there are only few publications, while for \eqref{eq:0p30}
with $P_{\alpha }$, we refer the reader to \cite{GHP}. Some results
of this type can be found also in Section 7 of this paper.

In the case of the equation \eqref{eq:0p1} , which involves either lower
order terms or some other type perturbations, Problem $P_{\alpha }$ in
$\Omega_0$ with $\alpha (x)\equiv 0$ has been studied by Aldashev in
\cite{Ald93,Ald98,Ald00}. For
comments, concerning Aldashev's results, we refer the reader to
Remark \ref{rmk6.1}.
Finally, we point out that in the case of \eqref{eq:0p1}, with nonzero lower
order terms, Karatoprakliev \cite{Kar82} obtained a priori estimates,
but only for solutions enough smooth  of Problem $P1$ in $\Omega_0$.

Next, we formulate the following well known result
\cite{Tong,PS88}, presented here in the terms of the polar
coordinates $x_1=\varrho \cos \varphi $, $x_2=\varrho \sin \varphi
$.

\begin{theorem} \label{thm1.1}
For all $n\in \mathbb{N}$, $n\geq 4$; $a_{n},b_{n}$ arbitrary constants, the
functions
\begin{equation}
v_{n}(\varrho ,\varphi ,t)=t\varrho ^{-n}\left( \varrho
^2-t^2\right) ^{n-\frac{3}{2}}(a_{n}\cos n\varphi +b_{n}\sin
n\varphi )  \label{eq:0p5}
\end{equation}
are classical solutions of the homogeneous problem $P1^{\ast }$ and the
functions
\begin{equation}
w_{n}(\varrho ,\varphi ,t)=\varrho ^{-n}\left( \varrho ^2-t^2\right) ^{n-%
\frac{1}{2}}(a_{n}\cos n\varphi +b_{n}\sin n\varphi )  \label{eq:0p6}
\end{equation}
are classical solutions of the homogeneous problem $P2^{\ast }$.
\end{theorem}

This theorem shows that for the classical solvability (see \cite{Bits}) of
the problem $P1$ (respectively, $P2$) the function $f$ at least must be
orthogonal to all smooth functions \eqref{eq:0p5} (respectively, %
\eqref{eq:0p6}). The reason of this fact has been found by Popivanov and
Schneider in \cite{PS88}, where they announced for Problems $P1$ and $P2$
that there exist singular solutions for the wave equation \eqref{eq:0p30}
with power type isolated singularities even for very smooth functions $f$.
Using Theorem \ref{thm1.1}, Popivanov and Schneider \cite{PS95} proved the existence
of \textsl{generalized solutions} of Problems $P1$ and $P2$, which have at
least power type singularities at the vertex $O$ of the cone $\Sigma_{2,0}$. Considering Problems $P1$ and $P2$, Popivanov and Schneider \cite{PS88}
announced the existence of singular solutions for both wave and degenerate
hyperbolic equation. First a priori estimates for singular solutions of
Protter's Problems $P1$ and $P2$, concerning the wave equation in $\mathbb{R}%
^{3}$, were obtained in \cite{PS95}. On the other hand, for the case of the
wave equation in $\mathbb{R}^{m+1}$, Aldashev \cite{Ald93} shows that there
exist solutions of Problem $P1$ (respectively, $P2$) in the domain $\Omega
_{\varepsilon }$, which grow up on the cone $\Sigma_{2,\varepsilon }$ like $%
\varepsilon ^{-(n+m-2)}$ (respectively, $\varepsilon ^{-(n+m-1)}$), when $%
\varepsilon \to 0$ and the cone $\Sigma_{2,\varepsilon }:=\{\varrho
=t+\varepsilon \}$ approximates $\Sigma_{2,0}$. It is obvious that for $m=2$
this results can be compared with the estimate \eqref{eq:0p101a} of Theorem
\ref{thm6.3} and with the analogous estimates of Theorems \ref{thm6.1} and
\ref{thm6.2}. For the
homogeneous Problem $P_{\alpha }^{\ast }$ (except the case $\alpha \equiv 0$, i.e. except Problem $P2^{\ast }$), even for the wave equation, we do not
know nontrivial solutions analogous to \eqref{eq:0p5} and \eqref{eq:0p6}.
Anyway, in the present paper under appropriate conditions for the
coefficients of the general equation \eqref{eq:0p1}, we derive results which
ensure the existence of many singular solutions of Problem $P_{\alpha }$.
Here we refer also to Khe Kan Cher \cite{Kan98}, who gives some nontrivial
solutions for the homogeneous Problems $P1^{\ast }$ and $P2^{\ast }$, but in
the case of Euler-Poisson-Darboux equation. These results are closely
connected to the such ones of Theorem \ref{thm1.1}.

In order to obtain our results, we give the following definition of a
generalized solution of Problem $P_{\alpha }$ with a possible
singularity at the point $O$.

\begin{definition} \label{def1.1} \rm
A function $u=u(x_{1,}x_2,t)$ is called a \textsl{generalized
solution} of $P_{\alpha }$: in $\Omega_0$, if
\begin{enumerate}
\item $u\in C^{1}(\bar{\Omega}_0\backslash O)$,
$[u_{t}+\alpha (x)u]\bigr\rvert_{\Sigma_0\backslash O}=0$
$u\bigr\rvert_{\Sigma_1}=0$,
\item The equality
\begin{equation} \label{eq:0p7}
\begin{aligned}
&\int_{\Omega_0}[u_{t}v_{t}-u_{x_1}v_{x_1}-u_{x_2}v_{x_2}
+(b_1u_{x_1}+b_2u_{x_2}+bu_{t}+cu-f)v]dx_1dx_2dt \\
&=\int_{\Sigma_0}\alpha (x)(uv)(x,0)dx_1dx_2
\end{aligned}
\end{equation}
holds for all $v$ in
\begin{equation*}
V_0:=\{ v\in C^1(\bar{\Omega}_0): [v_{t}+(\alpha +b)v]\bigr\rvert
_{\Sigma_0}=0,\; v=0\text{ in a neighborhood of  }\Sigma_{2,0}\}.
\end{equation*}
\end{enumerate}
\end{definition}

To deal with difficulties such as
singularities of \textsl{generalized solutions} on the cone $\Sigma_{2,0}$,
we introduce the region
\begin{equation*}
\Omega_{\varepsilon }=\Omega_0\cap \{\varrho -t>\varepsilon \},\quad
\varepsilon \in \lbrack 0,1),
\end{equation*}
which in polar coordinates becomes
\begin{equation}
\Omega_{\varepsilon }=\{(\varrho ,\varphi ,t):t>0,\,0\leq \varphi <2\pi
,\varepsilon +t<\varrho <1-t\}.  \label{eq:0p8}
\end{equation}
We define  \textsl{generalized solution} of Problem
$P_{\alpha }$ in $\Omega_{\varepsilon },\varepsilon \in (0,1)$, in
Definition \ref{def2.1} below. Note that, if a generalized solution $u$
belongs to {$C^{1}({\bar{\Omega}}_{\varepsilon })\cap
$}$C^2({\Omega }_{\varepsilon })$, it is called a
\textsl{classical solution} of Problem $P_{\alpha }$ in
$\Omega_{\varepsilon }$, $\varepsilon \in (0,1)$, and it satisfies
the equation \eqref{eq:0p1} in $\Omega_{\varepsilon }$. It should
be pointed
out that the case $\varepsilon =0$ is totally different from the case $%
\varepsilon \neq 0$.

This paper is a generalization, extension, and improvement of the
results obtained in \cite{GHP}. The paper, besides Introduction,
consists of six more sections. In Section 2, using some
appropriate techniques, we formulate the 2-D boundary value
problems $P_{\alpha ,1}$, $P_{\alpha ,2}$ and $P_{\alpha ,3}$,
corresponding to the 3-D Problem $P_{\alpha }$. The aim of Section
3 is to treat Problem $P_{\alpha ,3}$. For this reason, we
construct and study the system of integral equations, assigned to
the under consideration equation \eqref{eq:0p1}. Also, we present
results concerning the classical solutions of Problem $P_{\alpha
,3}$ in $\Omega_{\varepsilon },\varepsilon \in (0,1)$ and give
corresponding a priori estimates. We mention also here Lemma \ref{lm3.1},
which is actually a maximum principle for Problem $P_{\alpha ,3}$.
In Section 4 we prove Theorems \ref{thm4.1} and \ref{thm4.2}, which ensure the
existence and uniqueness of a generalized solution of Problem
$P_{\alpha ,1}$ in 2-D domain. Using the results of the previous
section, in Section 5 we study the existence and uniqueness of a
generalized solution of 3-D Problem $P_{\alpha }$. More precisely,
Theorem \ref{thm5.1} ensures the uniqueness of a generalized solution for
Problem $P_{\alpha }$ in $\Omega _{\varepsilon },\varepsilon \in
\lbrack 0,1)$, while Theorems \ref{thm5.2}, \ref{thm5.3} and \ref{thm5.4}
 ensure the
existence of a generalized solution for problem $P_{\alpha }$ in
$\Omega_0$, which is a classical one in each domain $\Omega
_{\varepsilon },\varepsilon \in \lbrack 0,1)\ $and satisfies some
a priori estimates in $C^2(\Omega_{\varepsilon })$. Comparing
these estimates with such ones of \cite{GHP}, we see that the new
estimates are better even in the case of the wave equation
\eqref{eq:0p30} without lower order terms. In Theorems \ref{thm6.1},
\ref{thm6.2} and \ref{thm6.3}  under different conditions, imposed on the coefficients
of the equation \eqref{eq:0p1}, we present some singular
\textsl{generalized solutions} which are smooth enough away from the point $%
O$, while at the point $O$ they have power type singularity of the type $%
(x_1^2+x_2^2+t^2)^{-n/2}$. More precisely, we formulate and prove
the following theorem.

\begin{theorem} \label{thm6.3}
Let the coefficients $b_1,b_2,b$ be constants,
$c(x_1,x_2,t)=c(|x|,t)\in C^{1}(\bar{\Omega}_0)$, $4c\leq
b_1^2+b_2^2-b^2$ in $\bar{\Omega}_0$
 and $\alpha =\alpha (|x|)\in C^{1}(0,1]\cap C[0,1]$.
Then for each $n\in \mathbb{N}$ there exists a function
$f_{n}(x_1,x_2,t)\in C^{n-2}(\bar{\Omega}_0)\cap C^{\infty
}(\Omega _0)$, for which the corresponding generalized solution
$u_{n}$
 of problem $P_{\alpha }$ belongs to $C^2(\bar{\Omega}_0\backslash O)$
and satisfies the estimate
\begin{equation}
|u_{n}(x_1,x_2,|x|)|\geq \frac{1}{2}|u_{n}(2x_1,2x_2,0)|+|x|^{-n}|%
\cos n(\arctan \frac{x_2}{x_1})|.  \label{eq:0p101a}
\end{equation}
\end{theorem}

On the other hand, in Theorems \ref{thm6.1} and \ref{thm6.2} the coefficients of the equation
\eqref{eq:0p1} are nonconstant.

Finally, in Theorem \ref{thm7.1}  for the wave equation \eqref{eq:0p30} we
present two-sided estimates for the singularities of the
\textsl{generalized solution }of Problem $P_{\alpha }$. In
particular, the exact behavior for the singular solution
$u_{n}(x_1,x_2,t)$ around $O$ is $(x_1^2+x_2^2+t^2)^{-n/2}\cos
n(\arctan \frac{x_2}{x_1})$.

\begin{remark} \label{rmk1.2} \rm
Actually, all these results state some conditions on the coefficients of
the equation \eqref{eq:0p1}, under which we do not have a positive answer to
Open Question 1. For example, Theorem \ref{thm7.1} ensures that for any parameter
$\alpha (x)$, involved to the boundary condition \eqref{eq:0p2} on
$\Sigma_0$, there are infinitely many singular solutions of the wave equation
\eqref{eq:0p30}. That means that, it is impossible to give a positive answer
to Open Question 1, by using the wave operator only. Possibly, it is
necessary to ask some of  the nonzero lower order perturbations of the wave
equation to be involved to the more general equation \eqref{eq:0p1}. This is
one of the reasons of  the existence of the present paper, where we use and
developed further the ideas of \cite{GHP}. Note also that, each one of \ the
singular solutions has a strong singularity at the vertex $O$ of the cone $%
\Sigma_{2,0}$. The singularities of the generalized solutions do not
propagate in the direction of the bicharacteristics on the characteristic
cone. It is traditionally assumed that the wave equation, with right-hand
side sufficiently smooth in $\bar{\Omega}_0$, cannot have a solution with
an isolated singular point. For results, concerning the propagation of
singularities for second order operators, see H\"{o}rmander
\cite[Chapter 24.5]{H}.
\end{remark}

We conclude this section with the following four more questions.

\noindent\textbf{Open Questions:}
\begin{enumerate}
\item Find the exact behavior of all singular solutions at the point $O$,
different from those ones which appear in Theorems \ref{thm6.1}, \ref{thm6.2}
 and \ref{thm7.1}.

\item Find appropriate conditions for the function $f$ under which the Problem $%
P_{\alpha }$, even for the wave equation, has only classical solutions. We
do not know any kind of such results even for Problem $P2$.
\item  In all results, concerning the existence of singular solutions (except
Theorem \ref{thm6.3}), we assume that $a_2\equiv 0$. Is it possible to find
any singular solution, when $a_2\neq 0?$ Even in the case $a_2\neq
0$, Theorem \ref{thm5.3}  ensures the existence of a
generalized solution for
any function $f$, but we do not know the behavior of a such solution at $%
(0,0,0)$.
\item  From the a priori estimates, obtained in Theorems
\ref{thm5.2}--\ref{thm5.4} for all
solutions of Problem $P_{\alpha }$, including singular ones, it follows
that, as $\rho \to 0$, none of these solutions can grow up faster
than exponential one. The arising question is: are there singular solutions
of Problem $P_{\alpha }$ with exponential growth as $\rho \to 0$ or
any such solution is of polynomial growth satisfying \eqref{eq:0p101a}?
\end{enumerate}
In the case of Problem $P1$, for the wave equation \eqref{eq:0p1} the answer
to Open Questions 1, 2 and 4 above can be found in \cite{NT}.

\section{Preliminaries}

In this section we consider \eqref{eq:0p1} in polar coordinates
$x_1=\varrho \cos \varphi $, $x_2=\varrho \sin \varphi $ and $t$
\begin{equation}
Lu=\frac{1}{\varrho }(\varrho u_{\varrho })_{\varrho }+\frac{1}{\varrho ^2}%
u_{\varphi \varphi }-u_{tt}+a_1u_{\varrho }+a_2u_{\varphi
}+bu_{t}+cu=f \label{eq:1p1}
\end{equation}
in a simply connected region
\begin{equation}
\Omega_{\varepsilon }:=\big\{(\varrho ,\varphi ,t):0<t<(1-\varepsilon
)/2,\,0\leq \varphi <2\pi ,\,\varepsilon +t<\varrho <1-t\big\},  \label{eq:1p2}
\end{equation}
$0<\varepsilon <1$, bounded by the disc $\Sigma_0:=\{(\varrho ,\varphi
,t):t=0,\varrho <1\}$ and the characteristic surfaces of \eqref{eq:1p1}
\begin{gather*}
\Sigma_1 :=\{(\varrho ,\varphi ,t):0\leq \varphi <2\pi ,\varrho =1-t\},
\\
\Sigma_{2,\varepsilon } :=\{(\varrho ,\varphi ,t):0\leq \varphi <2\pi
,\varrho =\varepsilon +t\}.
\end{gather*}
The coefficients $a_1,a_2$ depend on $b_1,b_2$ in an obvious way.
We seek sufficient conditions for the existence and uniqueness of
a generalized solution of the equation \eqref{eq:1p1} with $f\in
C({\bar{\Omega}}_{\varepsilon })$, which satisfies the following
boundary conditions
\begin{gather}
 P_{\alpha }: \quad u\bigr\rvert_{\Sigma_1\cap \partial \Omega
_{\varepsilon }} =0,  \quad \lbrack u_{t}+\alpha u]\bigr\rvert_{\Sigma
_0\cap \partial \Omega_{\varepsilon }} =0; \\
P_{\alpha }^{\ast }:  \quad u\bigr\rvert_{\Sigma_{2,\varepsilon }} =0,
\quad \lbrack u_{t}+(\alpha +b)u]\bigr\rvert_{\Sigma_0\cap \partial
\Omega_{\varepsilon }} =0.
\end{gather}
Here, for the sake of simplicity, we assume that all coefficients of %
\eqref{eq:1p1}\ depend only on $\varrho $ and $t$, and we set $\alpha
(x)\equiv \alpha (|x|)=\alpha (\varrho )\in C^{1}(0,1]$. The problem $%
P_{\alpha }^{\ast }$ is the adjoint one to Problem $P_{\alpha }$ in $\Omega
_{\varepsilon }$.

\begin{remark} \label{rmk2.1} \rm
In what follows, we consider the domain $\Omega_{\varepsilon }$ and its
boundary in Cartesian coordinates. Nevertheless, for convenience we use the
polar coordinates in the sense that the intersections $\varphi =0$ and $%
\varphi =2\pi $ do not belong to the boundary of $\Omega_{\varepsilon }$
and all the functions, which we use here, are considered as periodical ones.
\end{remark}

Now, in order to obtain our results, we define the notion of a generalized
solution as follows.

\begin{definition} \label{def2.1}\rm
A function $u=u(\varrho ,\varphi ,t)$ is called a \textsl{generalized
solution} of Problem $P_{\alpha }$ in $\Omega_{\varepsilon }$, $\varepsilon
>0$, if: \begin{enumerate}

\item  $u\in C^{1}({\bar{\Omega}}_{\varepsilon })$, $u\bigr\rvert_{\Sigma
_1\cap \partial \Omega_{\varepsilon }}=0$; $[u_{t}+\alpha (\varrho )u]
\bigr\rvert_{\Sigma_0\cap \partial \Omega_{\varepsilon }}=0$;

\item The equality
\begin{equation} \label{eq:1p6}
\begin{aligned}
&\int_{\Omega_{\varepsilon }}[u_{t}v_{t}-u_{\varrho }v_{\varrho }
-\frac{1}{\varrho ^2}u_{\varphi }v_{\varphi }+(a_1u_{\varrho }
+a_2u_{\varphi}+bu_{t}+cu-f)v]\varrho \,d\varrho \,d\varphi \,dt  \\
&=\int_{\Sigma_0\cap \partial \Omega_{\varepsilon }}\alpha (\varrho
)uv\varrho \,d\varrho d\varphi
\end{aligned}
\end{equation}
holds for all
\begin{equation*}
v\in V_{\varepsilon }:=\{v\in C^{1}({\bar{\Omega}}_{\varepsilon
}):[v_{t}+(\alpha +b)v]\bigr\rvert_{\Sigma_0\cap \partial \Omega
_{\varepsilon }}=0,v\bigr\rvert_{\Sigma_{2,\varepsilon }}=0\}.
\end{equation*}
\end{enumerate}
\end{definition}

The following lemma describes the properties of generalized solutions
of Problem $P_{\alpha }$ in $\Omega_{\varepsilon }$.

\begin{lemma} \label{lm2.1}
 Each \textsl{generalized solution} of Problem $P_{\alpha }$
in $\Omega_0$ is also a \textsl{generalized solution} of the same problem
in $\Omega_{\varepsilon }$ for $\varepsilon >0$.
\end{lemma}

The proof of this lemma follows from the proof in \cite[Lemma 2.1]{GHP}.
In the special, but main case, when
\begin{equation}
f(\varrho ,\varphi ,t)=f_{n}^{(1)}(\varrho ,t)\cos n\varphi
+f_{n}^{(2)}(\varrho ,t)\sin n\varphi ,  \label{eq:1p7}
\end{equation}
we ask the generalized solution to be of the form
\begin{equation*}
u(\varrho ,\varphi ,t)=u_{n}^{(1)}(\varrho ,t)\cos n\varphi
+u_{n}^{(2)}(\varrho ,t)\sin n\varphi .
\end{equation*}
Then, in view of \eqref{eq:1p1}, we obtain the 2-D system
\begin{equation} \label{eq:1p9}
\begin{gathered}
\frac{1}{\varrho }(\varrho u_{n,\varrho }^{(1)})_{\varrho
}-u_{n,tt}^{(1)}+a_1u_{n,\varrho
}^{(1)}+bu_{n,t}^{(1)}+(c-\frac{n^2}
{\varrho ^2})u_{n}^{(1)}+na_2u_{n}^{(2)}=f_{n}^{(1)}, \\
\frac{1}{\varrho }(\varrho u_{n,\varrho }^{(2)})_{\varrho
}-u_{n,tt}^{(2)}+a_1u_{n,\varrho
}^{(2)}+bu_{n,t}^{(2)}+(c-\frac{n^2} {\varrho
^2})u_{n}^{(2)}-na_2u_{n}^{(1)}=f_{n}^{(2)}.
\end{gathered}
\end{equation}
We consider this system in the domain
\begin{equation*}
G_{\varepsilon }=\{(\varrho ,t):t>0,\varepsilon +t<\varrho <1-t\}
\end{equation*}
which is bounded by the sets:
\begin{equation}\label{eq:1p10}
\begin{gathered}
S_0 =\{(\varrho ,t):t=0,0<\varrho <1\}, \\
S_1=\{(\varrho ,t):\varrho =1-t\},\\
S_{2,\varepsilon }=\{(\varrho ,t):\varrho =t+\varepsilon \}.
\end{gathered}
\end{equation}
In this case, for $u=(u^{(1)},u^{(2)})(\varrho ,t)$, the 2-D problem
corresponding to $P_{\alpha }$ is $P_{\alpha ,1}$:
\begin{equation}
\begin{gathered}
\frac{1}{\varrho }(\varrho u_{\varrho }^{(1)})_{\varrho
}-u_{tt}^{(1)}+a_1u_{\varrho
}^{(1)}+bu_{t}^{(1)}+(c-\frac{n^2}{\varrho
^2})u^{(1)}+na_2u^{(2)}=f^{(1)}\text{ \quad in }G_{\varepsilon }, \\
\frac{1}{\varrho }(\varrho u_{\varrho }^{(2)})_{\varrho
}-u_{tt}^{(2)}+a_1u_{\varrho
}^{(2)}+bu_{t}^{(2)}+(c-\frac{n^2}{\varrho
^2})u^{(2)}-na_2u^{(1)}=f^{(2)}\text{ \quad in }G_{\varepsilon }, \\
u^{(i)}\bigr\rvert_{S_1\cap \partial G_{\varepsilon }}=0,\quad \lbrack
u_{t}^{(i)}+\alpha (\varrho )u^{(i)}]\bigr\rvert_{S_0\cap \partial
G_{\varepsilon }}=0,\quad i=1,2.
\end{gathered} \label{eq:1p12}
\end{equation}
The generalized solution of the Problem $P_{\alpha ,1}$ is as follows.

\begin{definition} \label{def2.2} \rm
A function $u=(u^{(1)},u^{(2)})(\varrho ,t)$ is called a \textsl{generalized
solution} of Problem $P_{\alpha ,1}$ in $G_{\varepsilon }$, $\varepsilon >0$,
if:
\begin{enumerate}
\item $u\in C^{1}({\bar{G}}_{\varepsilon })$, $[u_{t}^{(i)}+\alpha (\varrho
)u^{(i)}]\bigr\rvert_{S_0\cap \partial G_{\varepsilon }}=0$, $u^{(i)}%
\bigr\rvert_{S_1\cap \partial G_{\varepsilon }}=0,i=1,2;$

\item The equalities
\begin{equation}
\begin{gathered}
\int_{G_{\varepsilon }}\big[u_{t}^{(1)}v_{1,t}-u_{\varrho
}^{(1)}v_{1,\varrho }+\big( a_1u_{\varrho }^{(1)}+bu_{t}^{(1)}+(c
-\frac{n^2}{\varrho ^2})u^{(1)}+na_2u^{(2)}
-f^{(1)}\big) v_1\big] \varrho d\varrho \,dt \\
=\int_{S_0\cap \partial G_{\varepsilon }}\alpha
(\varrho )u^{(1)}v_1\varrho \,d\varrho ,\\
\int_{G_{\varepsilon }}[u_{t}^{(2)}v_{2,t}-u_{\varrho
}^{(2)}v_{2,\varrho }+\big( a_1u_{\rho }^{(2)}+bu_{t}^{(2)}
+(c-\frac{n^2}{\varrho ^2})u^{(2)}-na_2u^{(1)}
 -f^{(2)}\big) v_2\big] \varrho d\varrho \,dt\\
=\int_{S_0\cap \partial G_{\varepsilon }}\alpha (\varrho
)u^{(2)}v_2\varrho \,d\varrho
\end{gathered} \label{eq:1p14}
\end{equation}
hold for all
\begin{equation*}
v_1,v_2\in V_{\varepsilon }^{(1)}=\{{v\in
C^{1}({\bar{G}}_{\varepsilon
}):[v_{t}+(\alpha +b)v]\bigr\rvert_{S_0\cap \partial G_{\varepsilon }}=0,v%
\bigr\rvert_{S_{2,\varepsilon }\cap \partial G_{\varepsilon }}=0\}.}
\end{equation*}
\end{enumerate}
\end{definition}

Introducing a new function
\begin{equation}
z^{(i)}(\varrho ,t)=\varrho ^{\frac{1}{2}}u^{(i)}(\varrho ,t),\ i=1,2,
\label{eq:1p15}
\end{equation}
we transform the system \eqref{eq:1p12} to the system
\begin{equation}
\begin{gathered}
 z_{\varrho \varrho }^{(1)}-z_{tt}^{(1)}+a_1z_{\varrho
}^{(1)}+bz_{t}^{(1)}+\big( c-\frac{1}{2\varrho }a_1-\frac{4n^2-1}{%
4\varrho ^2}\big) z^{(1)}+na_2z^{(2)}=\varrho
^{\frac{1}{2}}f^{(1)},
\\
z_{\varrho \varrho }^{(2)}-z_{tt}^{(2)}+a_1z_{\varrho}^{(2)}+bz_{t}^{(2)}+\big( c-\frac{1}{2\varrho }a_1-\frac{4n^2-1}{%
4\varrho ^2}\big) z^{(2)}-na_2z^{(1)}=\varrho
^{\frac{1}{2}}f^{(2)}.
\end{gathered}   \label{eq:1p16}
\end{equation}
with the string operator in the main part. Substituting the new coordinates
\begin{equation}
\xi =1-\varrho -t,\quad \eta =1-\varrho +t,  \label{eq:1p17}
\end{equation}
from \eqref{eq:1p16} we derive
\begin{equation}
\begin{gathered}
U_{\xi \eta }^{(1)}-A_1U_{\xi }^{(1)}-B_1U_{\eta
}^{(1)}-C_1U^{(1)}-D_1U^{(2)}=F^{1}(\xi ,\eta )\text{ \quad in }%
D_{\varepsilon }, \\
U_{\xi \eta }^{(2)}-A_2U_{\xi }^{(2)}-B_2U_{\eta
}^{(2)}-C_2U^{(2)}-D_2U^{(1)}=F^2(\xi ,\eta )\text{ \quad in }%
D_{\varepsilon },
\end{gathered} \label{eq:1p18}
\end{equation}
where $D_{\varepsilon }=\{(\xi ,\eta ):0<\xi <\eta <1-\varepsilon \}$ and
for $i=1,2$:
\begin{equation}
\begin{gathered}
U^{(i)}(\xi ,\eta )=z^{(i)}(\varrho (\xi ,\eta ),t(\xi ,\eta )), \\
 F^{(i)}(\xi ,\eta )=\frac{1}{4\sqrt{2}}(2-\eta -\xi )^{\frac{1%
}{2}}f^{(i)}(\varrho (\xi ,\eta ),t(\xi ,\eta )).
\end{gathered} \label{eq:1p19}
\end{equation}
Note that, in the case of the equation \eqref{eq:1p1}, the corresponding
coefficients of the system \eqref{eq:1p18} are
\begin{equation}
\begin{gathered}
A_1=A_2=\frac{1}{4}(a_1+b),B_1=B_2=\frac{1}{4}
(a_1-b),D_2=-D_1=\frac{1}{4}na_2, \\
C_1=C_2=\frac{1}{4}\big( \frac{4n^2-1}{(2-\xi -\eta )^2%
}+\frac{a_1}{2-\xi -\eta }-c\big) .
\end{gathered} \label{eq:1p101}
\end{equation}
As we see, Problem $P_{\alpha ,1}$ is reduced to the Darboux-Goursat
problem for the system \eqref{eq:1p18} with the same boundary conditions.
That is, we consider the following question.

\noindent\textbf{Problem $P_{\alpha ,2}$.} Find a solution
$(U^{1},U^2)$ of the system \eqref{eq:1p18} in $D_{\varepsilon }$
with the boundary conditions
\begin{equation}
\ U^{(i)}(0,\eta )=0,\quad (U_{\eta }^{(i)}-U_{\xi }^{(i)})(\xi ,\xi
)+\alpha (1-\xi )U^{(i)}(\xi ,\xi )=0,\ i=1,2.  \label{eq:1p20}
\end{equation}

To investigate the smoothness or the singularity of a solution of
the original 3-D problem $P_{\alpha }$ on $\Sigma_{2,0}$, we are seeking
a classical solution of the corresponding 2-D problem $P_{\alpha ,2}$ not
only in the domain $D_{\varepsilon }$, but also in the domain
\begin{equation}
D_{\varepsilon }^{(1)}:=\{(\xi ,\eta ):0<\xi <\eta <1,0<\xi <1-\varepsilon
\},\quad \varepsilon >0.
\end{equation}
Clearly, $D_{\varepsilon }\subset D_{\varepsilon }^{(1)}$.

Consider now an appropriate boundary value problem for the system of
equations whose coefficients are continuous in
${\bar{D}}_{\varepsilon}^{(1)}$, $\varepsilon >0$:

\noindent\textbf{Problem $P_{\alpha ,3}$.} Find a solution
$(U^{1},U^2)$ of the system
\begin{equation}
\begin{gathered}
U_{\xi \eta }^{(1)}-A_1U_{\xi }^{(1)}-B_1U_{\eta
}^{(1)}-C_1U^{(1)}-D_1U^{(2)}=F^{1}(\xi ,\eta )\text{ \quad in }{\bar{D}}%
_{\varepsilon }^{(1)}, \\
U_{\xi \eta }^{(2)}-A_2U_{\xi }^{(2)}-B_2U_{\eta
}^{(2)}-C_2U^{(2)}-D_2U^{(1)}=F^2(\xi ,\eta )\text{ \quad in }{\bar{D}}%
_{\varepsilon }^{(1)},
\end{gathered} \label{eq:1p21}
\end{equation}
with the boundary conditions
\begin{equation}
U^{(i)}(0,\eta )=0,(U_{\eta }^{(i)}-U_{\xi }^{(i)})(\xi ,\xi )+\alpha (1-\xi
)U^{(i)}(\xi ,\xi )=0, \label{bound}
\end{equation}
for $i=1,2$ and  $\xi \in (0,1-\varepsilon )$.

\section{The system of integral equations corresponding to Problem
$P_{\protect\alpha ,3}$}

First of all, we construct an equivalent system of integral equations in
such a way that any solution to Problem $P_{\alpha ,3}$ is a solution of the
constructed system and vice-verca. For this reason, following the ideas of
[9], concerning the representation of the solutions of the Protter's problem
for the wave equation, we will try to find a corresponding representation of
the solutions in the case, where the equation \eqref{eq:0p1} involves lower
order terms. In this case, because of the appearance in the equation %
\eqref{eq:1p1} of the term $a_2u_{\varphi }$, we have to deal not
with a single scalar equation, but with a system of equations. In
order to realize these ideas, for any $(\xi_0,\eta_0)\in
D_{\varepsilon }^{(1)}$, we consider the sets
\begin{equation*}
\Pi :=\{(\xi ,\eta ):0<\xi <\xi_0,\xi_0<\eta <\eta_0\},\quad
T:=\{(\xi ,\eta ):0<\xi <\xi_0,\xi <\eta <\xi_0\}
\end{equation*}
and the following integrals:
\begin{gather*}
I_0^{(i)}:=\iint_{\Pi }U_{\xi \eta }^{(i)}(\xi ,\eta )\,d\xi d\eta
\,=\int_0^{\xi_0}\int_{\xi_0}^{\eta_0}U_{\xi \eta }^{(i)}(\xi
,\eta )\,d\eta \,d\xi\,, \\
I_1^{(i)}:=\iint_{T}U_{\xi \eta }^{(i)}(\xi ,\eta )\,d\xi d\eta
\,=\int_0^{\xi_0}\int_{\xi }^{\xi_0}U_{\xi \eta }^{(i)}(\xi ,\eta
)\,d\eta \,d\xi \,.
\end{gather*}
As it has been shown in \cite{GHP},
\begin{equation*}
I_0^{(i)}+2I_1^{(i)}=U^{(i)}(\xi_0,\eta_0)-\int_0^{\xi
_0}\alpha (1-\xi )U^{(i)}(\xi ,\xi )\,d\xi .
\end{equation*}
Set $p^{(i)}:=U_{\xi }^{(i)}$, $q^{(i)}:=U_{\eta }^{(i)}$. Then, in view of
the last relation and \eqref{eq:1p21}, we obtain
\begin{equation} \label{eq:2p80}
\begin{aligned}
U^{(1)}(\xi_0,\eta_0)=&\int_0^{\xi_0}\int_{\xi_0}^{\eta
_0}[F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi ,\eta
)\,d\eta \,d\xi\\
&+2\int_0^{\xi_0}\int_0^{\eta
}[F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi ,\eta
)\,d\xi \,d\eta \\
&+\int_0^{\xi_0}\alpha (1-\xi )U^{(1)}(\xi ,\xi )\,d\xi ,\quad
\text{for }(\xi_0,\eta_0)\in {\bar{D}}_{\varepsilon }^{(1)},
\end{aligned}
\end{equation}
\begin{equation}\label{eq:2p801}
\begin{aligned}
U^{(2)}(\xi_0,\eta_0)=&\int_0^{\xi_0}\int_{\xi_0}^{\eta
_0}[F^2+A_2p^{(2)}+B_2q^{(2)}+C_2U^{(2)}+D_2U^{(1)}](\xi ,\eta
)\,d\eta \,d\xi \\
&+2\int_0^{\xi_0}\int_0^{\eta
}[F^2+A_2p^{(2)}+B_2q^{(2)}+C_2U^{(2)}+D_2U^{(1)}](\xi ,\eta
)\,d\xi \,d\eta \\
&+\int_0^{\xi_0}\alpha (1-\xi )U^{(2)}(\xi ,\xi )\,d\xi ,\text{ \ \ \
for }(\xi_0,\eta_0)\in {\bar{D}}_{\varepsilon }^{(1)},
\end{aligned}
\end{equation}
which is the first couple of desired integral equations.
From $\eqref{eq:2p80}$ and \eqref{eq:2p801} we derive for the first
derivatives $p^{(i)}$ and $q^{(i)}$ ($i=1,2$) the next four integral
equations:
\begin{equation}
\begin{aligned}
p^{(1)}(\xi_0,\eta_0)=&\int_0^{\xi
_0}[F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi ,\xi
_0)\,d\xi \\
&+\int_{\xi_0}^{\eta
_0}[F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi
_0,\eta )\,d\eta \\
+\alpha (1-\xi_0)U^{(1)}(\xi_0,\xi_0),
\end{aligned}
\label{eq:2p91}
\end{equation}
\begin{equation}
q^{(1)}(\xi_0,\eta_0)=\int_0^{\xi
_0}[F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi ,\eta
_0)\,d\xi ,  \label{eq:2p92}
\end{equation}
\begin{equation}
\begin{aligned}
p^{(2)}(\xi_0,\eta_0)=&\int_0^{\xi
_0}[F^2+A_2p^{(2)}+B_2q^{(2)}+C_2U^{(2)}+D_2U^{(1)}](\xi ,\xi
_0)\,d\xi \\
&+\int_{\xi_0}^{\eta
_0}[F^2+A_2p^{(2)}+B_2q^{(2)}+C_2U^{(2)}+D_2U^{(1)}](\xi
_0,\eta )\,d\eta \\
&+\alpha (1-\xi_0)U^{(2)}(\xi_0,\xi_0),
\end{aligned} \label{eq:2p911}
\end{equation}
\begin{equation}
q^{(2)}(\xi_0,\eta_0)=\int_0^{\xi
_0}[F^2+A_2p^{(2)}+B_2q^{(2)}+C_2U^{(2)}+D_2U^{(1)}](\xi ,\eta
_0)\,d\xi .  \label{eq:2p921}
\end{equation}
Now, we set
\begin{equation}
\begin{gathered}
M:=\max \left( \sup_{D_{\varepsilon }^{(1)}}\lvert F^{1}\rvert
,\sup_{D_{\varepsilon }^{(1)}}\lvert F^2\rvert \right) ,\quad
M_{\alpha}:=\sup_{[0,1-\varepsilon ]}\lvert \alpha (\xi )\rvert \\
c(\varepsilon ):=\max_{i=1,2}\left\{ \sup_{D_{\varepsilon
}^{(1)}}\lvert A_{i}\rvert ,\ \sup_{D_{\varepsilon }^{(1)}}\lvert
B_{i}\rvert ,\ \sup_{D_{\varepsilon }^{(1)}}\lvert C_{i}\rvert ,\
\sup_{D_{\varepsilon }^{(1)}}\lvert D_{i}\rvert \right\} ,
\end{gathered} \label{eq:2p5}
\end{equation}
and formulate the following results.

\begin{theorem} \label{thm3.1}
Let $F^{i},A_{i},B_{i},C_{i},D_{i}\in C({\bar{D}}_{\varepsilon }^{(1)})$,
$i=1,2$, $\varepsilon >0$. Then there exists a classical solution
$(U^{(1)},U^{(2)})\in C^{1}({\bar{D}}_{\varepsilon }^{(1)})$ of the Problem
$P_{\alpha ,3}$ for which $U_{\xi \eta }^{(i)}\in C({\bar{D}}_{\varepsilon
}^{(1)}),i=1,2$ and
\begin{equation}
\begin{gathered}
\lvert U^{(i)}(\xi_0,\eta_0)\rvert \leq M[4c(\varepsilon )+M_{\alpha
}]^{-2}\exp \left\{ 8c(\varepsilon )+2M_{\alpha }\right\} \quad {\text{in }}
D_{\varepsilon }^{(1)},\ i=1,2, \\
\sup_{D_{\varepsilon }^{(1)}}\{\lvert U_{\xi }^{(i)}\rvert
,\lvert U_{\eta }^{(i)}\rvert \}\leq M[4c(\varepsilon )+M_{\alpha
}]^{-1}\exp \left\{ 8c(\varepsilon )+2M_{\alpha }\right\} ,\ i=1,2.
\end{gathered}
\label{eq:2p7}
\end{equation}
\end{theorem}

\noindent\textbf{Proof.}
To get our results, we will solve the system of integral
equations \eqref{eq:2p80}--\eqref{eq:2p921}. For this reason we use
sequence of successive approximations
$(U_{m}^{(i)},p_{m}^{(i)},q_{m}^{(i)}),m=1,2,\dots$, defined by the
formula{e}
\begin{equation}
\begin{aligned}
U_{m+1}^{(i)}(\xi_0,\eta_0)
=&\int_0^{\xi_0}\int_{\xi _0}^{\eta_0}E_{m}^{(i)}(\xi ,\eta )
\,d\eta \,d\xi +2\int_0^{\xi_0}\int_0^{\eta }E_{m}^{(i)}(\xi ,\eta )
\,d\xi \,d\eta \\
&+\int_0^{\xi_0}\alpha (1-\xi )U_{m}^{(i)}(\xi ,\xi )\,d\xi
,\quad \ i=1,2;\quad m=0,1,2\dots \\
p_{m+1}^{(i)}(\xi_0,\eta_0)
=&\int_0^{\xi_0}E_{m}^{(i)}(\xi ,\xi_0)\,d\xi +\int_{\xi_0}^{\eta
_0}E_{m}^{(i)}(\xi_0,\eta )\,d\eta \\
&+\alpha (1-\xi_0)U_{m}^{(i)}(\xi_0,\xi_0),\ \quad i=1,2;\quad
m=0,1,2\dots  \\
q_{m+1}^{(i)}(\xi_0,\eta_0)=&\int_0^{\xi
_0}E_{m}^{(i)}(\xi ,\eta_0)\,d\xi ,\quad  i=1,2;\quad m=0,1,2\dots \\
U_0^{(i)}(\xi_0,\eta_0)=&0,\quad p_0^{(i)}(\xi_0,\eta_0)=0,\\
q_0^{(i)}(\xi_0,\eta_0)=&0,\quad i=1,2,\quad
\text{in $D_{\varepsilon}^{1},$}
\end{aligned}
\label{eq:2p820}
\end{equation}
where
\begin{align*}
E_{m}^{(1)}(\xi ,\eta )
&:=[F^{1}+A_1p_{m}^{(1)}+B_1q_{m}^{(1)}+C_1U_{m}^{(1)}+D_1U_{m}^{(2)}](\xi ,\eta ),
\\
E_{m}^{(2)}(\xi ,\eta )
&:=[F^2+A_2p_{m}^{(2)}+B_2q_{m}^{(2)}+C_2U_{m}^{(2)}+D_2U_{m}^{(1)}](\xi
,\eta ).
\end{align*}
We will show that each of the functions $U_{m}^{(i)},p_{m}^{(i)}$ and
$q_{m}^{(i)},i=1,2$, is continuous in ${\bar{D}}_{\varepsilon }^{(1)}$ and
for any $(\xi_0,\eta_0)\in {\bar{D}}_{\varepsilon }^{(1)}$ and
$m\in \mathbb{N}$
\begin{equation}
{\lvert (U_{m}^{(i)}-U_{m-1}^{(i)})(\xi_0,\eta_0)\rvert \leq M\frac{%
[4c(\varepsilon )+M_{\alpha }]^{m-1}}{(m+1)!}}\text{ }{(\xi_0+\eta_0)}%
^{m+1}{,}  \label{eq:2p8}
\end{equation}
\begin{equation}
\begin{aligned}
&\max \left\{ {\lvert (p_{m}^{(i)}-p_{m-1}^{(i)})|(\xi_0,\eta_0),\lvert
(q_{m}^{(i)}-q_{m-1}^{(i)})|(\xi_0,\eta_0)}\right\}\\
&\leq {M\frac{[4c(\varepsilon )+M_{\alpha }]^{m-1}}{m!}}
(\xi_0+\eta_0)^{m}
\end{aligned}\label{eq:2p9}
\end{equation}
Indeed, by induction: 1) For ${m=1}$
\begin{equation*}
{U_1^{(i)}(\xi_0,\eta_0)=\int_0^{\xi_0}\int_{\xi }^{\eta_0}}%
\text{ }{F}^{(i)}{(\xi ,\eta )\,d\eta \,d\xi +\int_0^{\xi_0}\int_{\xi
}^{\xi_0}}\text{ }{F}^{(i)}{(\xi ,\eta )\,d\eta \ d\xi \,,}
\end{equation*}
and hence{\
\begin{equation*}
\lvert U^{(1)}(\xi_0,\eta_0)\rvert \leq M\xi_0\eta_0\leq M(\xi
_0+\eta_0)^2/2.
\end{equation*}
Similarly one can estimate }$p_1^{(i)}$ and $q_1^{(i)}$.
{2) Let now, by the induction hypothesis, \eqref{eq:2p8} and \eqref{eq:2p9}
be satisfied for} some{\ }$m\in \mathbb{N}$. Then for $i=1,2${\
\begin{equation*}
\lvert (U_{m}^{(i)}-U_{m-1}^{(i)})(\xi_0,\eta_0)\rvert \leq M\frac{%
[4c(\varepsilon )+M_{\alpha }]^{m-1}}{m!}(\xi_0+\eta_0)^{m}:=Q_{m}(\xi
_0+\eta_0)^{m}.
\end{equation*}
It follows that
\begin{align*}
&\lvert (U_{m+1}^{(i)}-U_{m}^{(i)})(\xi_0,\eta_0)\rvert\\
& \leq Q_{m}\Big[4c(\varepsilon )\Big( \int_0^{\xi_0}\int_{\xi_0}^{\eta_0}(\xi
+\eta )^{m}\,d\eta \,d\xi
+2\int_0^{\xi_0}\int_0^{\eta }(\xi +\eta )^{m}\,d\xi\,d\eta \Big) \\
&\quad +\frac{M_{\alpha }}{m+1}\int_0^{\xi_0}(2\xi
)^{m+1}\,d\xi \Big]\\
&\leq \frac{Q_{m}}{(m+1)(m+2)}\big[4c(\varepsilon )\big((\xi_0+\eta
_0)^{m+2}-\eta_0^{m+2}-\xi_0^{m+2}\big)+M_{\alpha }(2\xi
_0)^{m+2}\big] \\
&\leq \frac{Q_{m+1}}{(m+2)}(\xi_0+\eta_0)^{m+2}.
\end{align*}
By \eqref{eq:2p91} and \eqref{eq:2p911}, we have}
\begin{align*}
&\lvert (p_{m+1}^{(i)}-p_{m}^{(i)})|(\xi_0,\eta_0) \\
&\leq \frac{Q_{m}}{m+1}\big[4c(\varepsilon )\big((\xi
_0+\eta_0)^{m+1}-\xi_0{}^{m+1}\,\big)+M_{\alpha }(2\xi_0)^{m+1}%
\big]\leq Q_{m+1}(\xi_0+\eta_0)^{m+1}.
\end{align*}
A similar estimate holds for (${q_{m+1}^{(i)}-q_{m}^{(i)})}$. So that
\eqref{eq:2p8} and \eqref{eq:2p9} hold and hence the uniform
convergence of the sequences $\{U_{m}^{(i)}(\xi ,\eta )\}_{m\in \mathbb{N}}$,
$\{p_{m}^{(i)}(\xi ,\eta )\}_{m\in \mathbb{N}}$ and $\{q_{m}^{(i)}(\xi ,\eta
)\}_{m\in \mathbb{N}}$ in ${\bar{D}}_{\varepsilon }^{(1)}$ follows. For
the limit functions $U^{(i)},p^{(i)},q^{(i)}\in C({\bar{D}}_{\varepsilon
}^{(1)})$ we obtain the integral equalities 
\eqref{eq:2p80}--\eqref{eq:2p921} with the obvious condition 
$U^{(i)}(0,\eta_0)=0$. From the integral equalities 
\eqref{eq:2p80}--\eqref{eq:2p921} it follows
that $p^{(i)}=U_{\xi }^{(i)}$ and $q^{(i)}=U_{\eta }^{(i)}$ in 
${\bar{D}}_{\varepsilon }^{(1)}$. 
Therefore, $U^{(i)}\in C^{1}({\bar{D}}_{\varepsilon}^{(1)}),i=1,2$.

Also, in view of \eqref{eq:2p8}, we see that
\begin{align*}
\lvert (U^{(i)}(\xi_0,\eta_0)\rvert
& =\big| \sum_{m=0}^{\infty}(U_{m+1}^{(i)}-U_{m}^{(i)})(\xi_0,\eta_0)\big|
\leq M\sum_{m=0}^{\infty }\frac{[4c(\varepsilon )+M_{\alpha }]^{m}}{(m+2)!}
(\xi_0+\eta_0)^{m+2} \\
& \leq M[4c(\varepsilon )+M_{\alpha }]^{-2}\exp \left\{
8c(\varepsilon )+2M_{\alpha }\right\}, \quad i=1,2.
\end{align*}
So, using {\eqref{eq:2p9}, }for the derivatives $U_{\xi_0}^{(i)}(\xi
_0,\eta_0)$ and $U_{\eta_0}^{(i)}(\xi_0,\eta_0)$ we get the
estimates
\begin{align*}
\lvert U_{\xi_0}^{(i)}(\xi_0,\eta_0)\rvert
&=\big|\sum_{m=0}^{\infty }(p_{m+1}^{(i)}-p_{m}^{(i)})(\xi_0,\eta_0)\big|
\leq M\sum_{m=0}^{\infty }\frac{[4c(\varepsilon )+M_{\alpha
}]^{m}}{(m+1)!}(\xi_0+\eta_0)^{m+1} \\
&\leq M[4c(\varepsilon )+M_{\alpha }]^{-1}\exp \left\{ 8c(\varepsilon
)+2M_{\alpha }\right\} ,\quad i=1,2
\end{align*}
and
\begin{equation*}
\lvert U_{\eta_0}^{(i)}(\xi_0,\eta_0)\rvert \leq M[4c(\varepsilon
)+M_{\alpha }]^{-1}\exp \left\{ 8c(\varepsilon )+2M_{\alpha }\right\} ,\quad
i=1,2,
\end{equation*}
which shows \eqref{eq:2p7}. Also, by \eqref{eq:2p91}--\eqref{eq:2p921},
it follows that
\begin{align*}
U_{\xi_0\eta_0}^{(1)} &\equiv U_{\eta_0\xi
_0}^{(1)}=F^{1}+A_1U_{\xi_0}^{(1)}+B_1U_{\eta
_0}^{(1)}+C_1U^{(1)}+D_1U^{(2)}, \\
U_{\xi_0\eta_0}^{(2)} &\equiv U_{\eta_0\xi_0}^{(2)}=F^2\text{
}+A_2U_{\xi_0}^{(2)}+B_2U_{\eta_0}^{(2)}+C_2U^{(2)}+D_2U^{(1)}.
\end{align*}
Thus, the vector-valued function {$U(\xi_0,\eta_0)$ }is a solution to
the system \eqref{eq:1p21} and{\ $U_{\xi \eta }\in C({\bar{D}}_{\varepsilon
}^{(1)})$. Finally, using representations \eqref{eq:2p91}--\eqref{eq:2p921}
for the first derivatives of }$U^{(i)}${, we conclude that each function
$U^{(i)}(\xi_0,\eta_0)$ satisfies the boundary condition \eqref{bound}
of the Problem }${P}_{\alpha ,3}$ for $\eta =\xi $. \hfill$\square$


The next lemma is very important for the investigation of the singularity of
a generalized solution of Problem $P_{\alpha }$.

\begin{lemma} \label{lm3.1}
Let $F^{i},A_{i},B_{i},C_{i},D_{i}\in C({\bar{D}}_{\varepsilon }^{(1)})$, $%
i=1,2,$
\begin{equation}
A_{i}\geq 0,B_{i}\geq 0,C_{i}\geq 0,D_{i}\geq 0,\text{ }\alpha (1-\xi )\geq 0%
\text{ in ${\bar{D}}_{\varepsilon }^{(1)}$, $\ i=1,2$}  \label{eq:2p101}
\end{equation}
and
\begin{equation}
\mathbf{(a)}\text{\ }p_1^{(i)}\geq 0\text{ \ and \ \ }q_1^{(i)}\geq 0,
\quad\text{or}\quad(\mathbf{b})\text{\ }F^{(i)}\geq 0\text{\ in }{\bar{D}%
}_{\varepsilon }^{(1)},\quad \text{$i=1,2.$}  \label{eq:2p101a}
\end{equation}
Then for the solution $(U^{(1)},U^{(2)})$ of Problem $P_{\alpha ,3}$
(already found in Theorem \ref{thm3.1}) we have
\begin{equation}
U^{(i)}(\xi ,\eta )\geq 0,\quad U_{\eta }^{(i)}(\xi ,\eta )\geq 0,\quad
U_{\xi }^{(i)}(\xi ,\eta )\geq 0\quad \text{for }(\xi ,\eta )\in \text{${%
\bar{D}}_{\varepsilon }^{(1)}$},\ i=1,2\text{.}
\end{equation}
\end{lemma}

\noindent\textbf{Proof.}
First, suppose that the condition $\mathbf{(b)}$ is satisfied.
Then, in view of \eqref{eq:2p820}, for
$(\xi_0,\eta_0)\in {\bar{D}}_{\varepsilon }^{(1)}$ we have
\begin{gather}
U_1^{(i)}(\xi_0,\eta_0)=\int_0^{\xi_0}\int_{\xi_0}^{\eta
_0}F^{(i)}(\xi ,\eta )\,d\eta \,d\xi +2\int_0^{\xi_0}\int_0^{\eta
}F^{(i)}(\xi ,\eta )\,d\xi \,d\eta \geq 0,  \label{eq:2p201}
\\
p_1^{(i)}(\xi_0,\eta_0)=\int_0^{\xi_0}F^{(i)}(\xi ,\xi
_0)\,\,d\xi +\int_{\xi_0}^{\eta_0}F^{(i)}(\xi_0,\eta )\,\,d\eta
\geq 0,  \label{eq:2p202}
\\
q_1^{(i)}(\xi_0,\eta_0)=\int_0^{\xi_0}F^{(i)}(\xi ,\eta
_0)\,d\xi \geq 0,\quad i=1,2.  \label{eq:2p203}
\end{gather}
Thus, the condition $\mathbf{(b)}$\ is {stronger, than }$\mathbf{(a).}$ {%
Assume now that} $\mathbf{(a)}$ $p_1^{(i)}\geq 0$ and $q_1^{(i)}\geq 0$
in ${\bar{D}}_{\varepsilon }^{(1)}$. Then, using {\eqref{eq:2p203}}, we find
that $U_1^{(i)}\geq 0$. Thus, in both cases $\mathbf{(a)}$\textbf{\ }or $%
\mathbf{(b)}$, the inequalities {\eqref{eq:2p201} - \eqref{eq:2p203}} hold.
Suppose next that {for some }$m\in \mathbb{N}$%
\begin{equation*}
{(U_{m}^{(i)}-U_{m-1}^{(i)})\geq 0,\ (p_{m}^{(i)}-p_{m-1}^{(i)})\geq 0,\
(q_{m}^{(i)}-q_{m-1}^{(i)})\geq 0}\text{ \ \ in}\ {\bar{D}}_{\varepsilon
}^{(1)},\ i=1,2.
\end{equation*}
Then
\begin{align*}
E_{m}^{(i)}-E_{m-1}^{(i)}
=&A_{i}{(p_{m}^{(i)}-p_{m-1}^{(i)})}+B_{i}{(q_{m}^{(i)}-q_{m-1}^{(i)})}
+C_{i}{(U_{m}^{(i)}-U_{m-1}^{(i)})} \\
&{+D_{i}{(U_{m}^{(i+1)}-U_{m-1}^{(i+1)})}\geq 0}
\quad\text{in } {\bar{D}}_{\varepsilon }^{(1)},\quad i=1,2,
\end{align*}
where we denote ${{U_{m}^{(3)}:=U_{m}^{(1)}}}$. Therefore, we see that
\begin{align*}
(U_{m+1}^{(i)}-U_{m}^{(i)})(\xi_0,\eta_0)
=&\int_0^{\xi_0}\int_{\xi_0}^{\eta_0}(E_{m}^{(i)}-E_{m-1}^{(i)})(\xi ,\eta
)\,d\eta \,d\xi \\
&+2\int_0^{\xi_0}\int_0^{\eta
}(E_{m}^{(i)}-E_{m-1}^{(i)})(\xi ,\eta )\,d\xi \,d\eta \\
&+\int_0^{\xi_0}\alpha (1-\xi )({U_{m}^{(i)}-U_{m-1}^{(i)}})(\xi ,\xi )\,d\xi
\geq 0.
\end{align*}
In the same manner,
\begin{align*}
(p_{m+1}^{(i)}-p_{m}^{(i)})(\xi_0,\eta_0)
=&\int_0^{\xi_0}(E_{m}^{(i)}-E_{m-1}^{(i)})(\xi ,\xi_0)\,d\xi
+\int_{\xi_0}^{\eta_0}(E_{m}^{(i)}-E_{m-1}^{(i)})(\xi_0,\eta )\,\,d\eta \\
&+\alpha (1-\xi_0)({U_{m}^{(i)}-U_{m-1}^{(i)}})(\xi
_0,\xi_0)\geq 0,
\end{align*}
\[
(q_{m+1}^{(i)}-q_{m}^{(i)})(\xi_0,\eta_0)=\int_0^{\xi
_0}(E_{m}^{(i)}-E_{m-1}^{(i)})(\xi ,\eta_0)\,d\xi \geq 0.
\]
Finally, by induction, we conclude that
\begin{gather*}
U^{(i)}(\xi_0,\eta_0)=\sum_{m=0}^{\infty
}(U_{m+1}^{(i)}-U_{m}^{(i)})(\xi_0,\eta_0)\geq 0, \\
 p^{(i)}(\xi_0,\eta_0)\geq 0,\quad
  q^{(i)}(\xi_0,\eta_0)\geq 0,\quad
(\xi_0,\eta_0)\in {\bar{D}}_{\varepsilon }^{(1)}.
\end{gather*}
\hfill$\square$

\begin{remark} \label{rmk3.1} \rm
Note here that Lemma \ref{lm3.1}, which is actually a maximum principle for Problem $%
P_{\alpha ,3}$, describes the behavior of the system {\eqref{eq:1p21} around
the point }$(1,1)$. Thus, this lemma becomes particularly useful in Sections
6 and 7 in finding singular solutions of the equation {\eqref{eq:1p1}. None
the less, when the equation \eqref{eq:1p1}} transforms to the system
\eqref{eq:1p18}, by \eqref{eq:1p101}, we see that $D_2=-D_1=na_2/4$.
Since, in view of Lemma \ref{lm3.1}, $D_1\geq 0$ and $D_2\geq 0$, it should be $%
a_2\equiv 0$. Because of this fact, we are able to find singular
solutions only when $a_2\equiv 0$ (see also Introduction, Open
Questions, 3).
\end{remark}

As a consequence of Theorem \ref{thm3.1} and representations
\eqref{eq:2p91}--\eqref{eq:2p921}, we have the following smoothness result:

\begin{theorem} \label{thm3.2}
Let $F^{i},A_{i},B_{i},C_{i},D_{i}\in C^{1}({\bar{D}}_{\varepsilon
}^{(1)})$, $i=1,2$, $\varepsilon >0$. Then there exists a
classical solution $U\in C^2({\bar{D}}_{\varepsilon }^{(1)})$ of
Problem $P_{\alpha ,3}$.
\end{theorem}

\noindent\textbf{Proof.} Since we have already shown that
\begin{equation}
p_{\eta }^{(1)}(\xi_0,\eta_0)\equiv q_{\xi }^{(1)}(\xi_0,\eta_0)
=[F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi_0,\eta_0)
\label{eq:2p93}
\end{equation}
and that similar representations for $p_{\eta }^{(2)}$ and $q_{\xi }^{(2)}$
hold, we have to prove only the fact that $p_{\xi }^{(i)}$ and $q_{\eta
}^{(i)}$ exist and belong to $C({\bar{D}}_{\varepsilon }^{(1)})$. Indeed, to
do this, we observe the following:

1. For fixed $\eta_0$ the equality {\eqref{eq:2p93}} is a linear ODE for
the function $q^{(1)}(\xi_0,\eta_0)$. So, using the well known formula
for the solution with the initial Cauchy data $q^{(1)}(0,\eta_0)=0$ from {%
\eqref{eq:2p92}, we find that}
\begin{equation}
q^{(1)}(\xi_0,\eta_0)=
\int_0^{\xi_0}[F^{1}+A_1p^{(1)}+C_1U^{(1)}+D_1U^{(2)}](\xi ,\eta_0)\exp
\Big( \int_{\xi }^{\xi_0}B_1(\tau ,\eta_0)d\tau \Big) d\xi .
\label{eq:2p94}
\end{equation}
Since $F^{1},A_1,B_1,C_1,D_1,U^{(1)},U^{(2)}\in C^{1}({\bar{D}}%
_{\varepsilon }^{(1)})$ and $p^{(1)},p_{\eta }^{(1)}\in C({\bar{D}}%
_{\varepsilon }^{(1)})$, by {\eqref{eq:2p94}, }we conclude that $q^{(1)}\in
C^{1}({\bar{D}}_{\varepsilon }^{(1)})${.}

2. For fixed $\xi_0$ the equality {\eqref{eq:2p93}} is a linear ODE for
the function $p^{(1)}(\xi_0,\eta_0)$. So, arguments similar to those
above lead to
\begin{equation} \label{eq:2p95}
\begin{aligned}
p^{(1)}(\xi_0,\eta_0)=& G_1(\xi_0)\exp
\Big(\int_{\xi_0}^{\eta_0}A_1(\xi_0,\eta )d\eta \Big)\\
&+\int_{\xi_0}^{\eta_0}\left[
F^{1}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}\right] (\xi_0,\eta )\\
&\times\exp\Big( \int_{\eta }^{\eta_0}A_1(\xi_0,\tau )d\tau \Big) d\eta .
\end{aligned}
\end{equation}
The function $G_1(\xi_0)$, which is defined implicitly by
\eqref{eq:2p91}, is of the form
\begin{align*}
&G_1(\xi_0)\\
&=\int_0^{\xi_0}\big[
F^{1}+A_1p^{(1)}+B_1q^{(1)}+C_1U^{(1)}+D_1U^{(2)}\big] (\xi ,\xi
_0)d\xi +\alpha (1-\xi_0)U^{(1)}(\xi_0,\xi_0).
\end{align*}
Obviously $G_1\in C^{1}({\bar{D}}_{\varepsilon }^{(1)})$,
because $F^{1}$, $A_1$, $B_1$, $C_1$, $D_1$, $\alpha $, $U^{(1)}$,
$U^{(2)}$, $q^{(1)}\in C^{1}({\bar{D}}_{\varepsilon }^{(1)})$ and
$p^{(1)},p_{\eta}^{(1)}\in C({\bar{D}}_{\varepsilon }^{(1)})$.
Finally, by \eqref{eq:2p95}, we see that
$p^{(1)}\in C^{1}({\bar{D}}_{\varepsilon }^{(1)})$.
\hfill$\square$

\begin{remark} \label{rmk3.2} \rm
By studying a solution of the Problem $P_{\alpha ,3}$ in the domain
${\bar{D}}_{\varepsilon }^{(1)}$, we are actually investigating the
behavior of the solution of Problem $P_{\alpha ,2}$ in the domain
${\bar{D}}_{\delta }$,
when $\delta \to 0$, around the line $\eta =1$. It is easy to show
that, for any $\varepsilon \in (0,1)$ and $\delta \in (0,1)$, the solutions
of these two problems coincide in their common domain
${\bar{D}}_{\varepsilon }^{(1)}\cap {\bar{D}}_{\delta }$.
\end{remark}

\section{Existence and uniqueness theorems for the 2-D Problem
$\mathbf{P_{\alpha ,1}}$} %4

Consider the 2-D problem $P_{\alpha ,1}:$
\begin{equation}
\begin{gathered}
\frac{1}{\varrho }(\varrho u_{\varrho }^{(1)})_{\varrho
}-u_{tt}^{(1)}+a_1u_{_{\varrho }}^{(1)}+bu_{t}^{(1)}+(c-\frac{n^2}{%
\varrho ^2})u^{(1)}+na_2u^{(2)}=f^{(1)}\text{ in }G_{\varepsilon }, \\
\frac{1}{\varrho }(\varrho u_{\varrho }^{(2)})_{\varrho
}-u_{tt}^{(2)}+a_1u_{\varrho
}^{(2)}+bu_{t}^{(2)}+(c-\frac{n^2}{\varrho
^2})u^{(2)}-na_2u^{(1)}=f^{(2)}\text{ in }G_{\varepsilon }, \\
u^{(i)}\bigr\rvert_{S_1\cap \partial G_{\varepsilon }}=0,\quad \lbrack
u_{t}^{(i)}+\alpha (\varrho )u^{(i)}]\bigr\rvert_{S_0\cap \partial
G_{\varepsilon }}=0,\quad i=1,2.
\end{gathered} \label{eq:3p1}
\end{equation}

Note that, the generalized solution of the problem $P_{\alpha ,1}$
in the domain $G_{\varepsilon }$ , $\varepsilon \in (0,1)$, was
defined by Definition \ref{def2.2}.

\begin{theorem} \label{thm4.1}
Let $a_1,a_2,b,c,f^{(1)},f^{(2)}\in C^{1}(\bar{G}_0\setminus (0,0))$.
Then there exists a generalized solution
$u=(u^{(1)},u^{(2)})\in C^2(\bar{G}_0\setminus (0,0))$ of problem
$P_{\alpha ,1}$ in $G_0$, which is a classical solution of the problem
$P_{\alpha ,1}$ in any domain $G_{\varepsilon }$, $\varepsilon \in (0,1)$.
\end{theorem}

\noindent\textbf{Proof.}
 In view of \eqref{eq:1p15} and \eqref{eq:1p17}, i.e. $z(\varrho
,t)=\varrho ^{1/2}u(\varrho ,t)$ and $\xi =1-\varrho -t$, $\eta =1-\varrho
+t$, we introduce the function
\begin{equation*}
U^{(i)}(\xi,\eta)=z^{(i)}(\varrho(\xi,\eta),t(\xi ,\eta )).
\end{equation*}
Then Problem $P_{\alpha ,1}$, in the new terms, becomes $P_{\alpha ,2}$, i.e.
\begin{equation}
\begin{gathered}
U_{\xi \eta }^{(1)}-A_1U_{\xi }^{(1)}-B_1U_{\eta
}^{(1)}-C_1U^{(1)}-D_1U^{(2)}=F^{1}(\xi ,\eta )\text{ in }D_{\varepsilon
}, \\
U_{\xi \eta }^{(2)}-A_2U_{\xi }^{(2)}-B_2U_{\eta
}^{(2)}-C_2U^{(2)}-D_2U^{(1)}=F^2(\xi ,\eta )\text{ in
}D_{\varepsilon },
\end{gathered} \label{eq:3p5}
\end{equation}
\begin{equation}
U^{(i)}(0,\eta )=0,(U_{\eta }^{(i)}-U_{\xi }^{(i)})(\xi ,\xi )+\alpha (1-\xi
)U^{(i)}(\xi ,\xi )=0,\quad i=1,2,  \label{eq:3p3}
\end{equation}
where the connection between the coefficients is given by \eqref{eq:1p101}.
For each fixed $\varepsilon \in (0,1)$ Theorem \ref{thm3.2} ensures the
existence of a classical solution $(U^{(1)},U^{(2)})\in
C^2(\bar{D}_{\varepsilon
}^{(1)})$ of the problem $P_{\alpha ,3}$. More precisely, for any fixed $%
\varepsilon_1,\varepsilon_2$ with $0<\varepsilon_1<\varepsilon
_2<1$ the corresponding vector-valued solution $U_{\varepsilon_2}$
is a
restriction of $U_{\varepsilon_1}$ in the region $D_{\varepsilon_2}$. So, essentially we have a function of class $C^2\left( \overset{\_}{D}%
_0\backslash (0,0)\right) $, which in any region $D_{\varepsilon }$
coincides with the corresponding solution $U_{\varepsilon }$ and is a
classical solution of Problem $P_{\alpha ,3}$. We remark that the inverse
transformations {\eqref{eq:1p15} and \eqref{eq:1p17} }lead to a
vector-valued function $(u^{(1)},u^{(2)})\in C^2\left( \bar{G}%
_0\backslash (0,0)\right) $, which is a classical solution of Problem $%
P_{\alpha ,1}$\ in each $G_{\varepsilon }$. This solution is also a
generalized solution of the same problem in $G_0$, because for each
concrete test function $v\in $\ $V_0$ there is an $\varepsilon_{v}>0$ for
which $v\equiv 0$ in $G_0\backslash G_{\varepsilon_{v}}$ and {%
\eqref{eq:0p7}} coincides with \eqref{eq:1p6}.
The proof of the theorem is complete. \hfill$\square$

\begin{theorem} \label{thm4.2}
Let $a_1,a_2,b,c\in C^{1}(\bar{G}_0\setminus
(0,0))$. Then for each fixed $\varepsilon \in (0,1)$ there exists
at most one generalized solution of the problem $P_{\alpha ,1}$ in
$G_{\varepsilon }$.
\end{theorem}

\noindent\textbf{Proof.}
 Let $(u_1^{(1)},u_1^{(2)})$ and $(u_2^{(1)},u_2^{(2)})$ be
two generalized solutions of $P_{\alpha ,1}$ in $G_{\varepsilon}$.
Then for $u^{(i)}:=u_1^{(i)}-u_2^{(i)}$, $i=1,2$, we see that
\begin{enumerate}
\item
 $u^{(i)}\in C^{1}({\bar{G}}_{\varepsilon })$, $[u_{t}^{(i)}+\alpha
(\varrho )u^{(i)}]\bigr\rvert_{S_0\cap \partial G_{\varepsilon }}=0$,
$u^{(i)}\bigr\rvert_{S_1\cap \partial G_{\varepsilon }}=0$, $i=1,2$;

\item The equalities
\begin{equation}
\begin{aligned}
&\int_{G_{\varepsilon }}\Big[ u_{t}^{(1)}v_{t}^{(1)}-u_{%
\varrho }^{(1)}v_{\varrho }^{(1)}+\big( a_1u_{\varrho
}^{(1)}+bu_{t}^{(1)}+(c-\frac{n^2}{\varrho ^2})u^{(1)}+na_2u^{(2)}%
\big) v^{(1)}\Big] \varrho d\varrho \,dt \\
&=\int_{S_0\cap \partial G_{\varepsilon }}\alpha
(\varrho )u^{(1)}v^{(1)}\varrho \,d\varrho , \\
&\int_{G_{\varepsilon }}\Big[ u_{t}^{(2)}v_{t}^{(2)}
-u_{\varrho }^{(2)}v_{\varrho }^{(2)}+\big( a_1u_{\varrho
}^{(2)}+bu_{t}^{(2)}+(c-\frac{n^2}{\varrho ^2})u^{(2)}-na_2u^{(1)}
\big) v^{(2)}\Big] \varrho d\varrho \,dt \\
&=\int_{S_0\cap \partial G_{\varepsilon }}\alpha
(\varrho )u^{(2)}v^{(2)}\varrho \,d\varrho
\end{aligned}
\label{eq:3p101}
\end{equation}
hold for all {functions }$v^{(1)},v^{(2)}\in V_{\varepsilon }^{(1)}$.
\end{enumerate}
 If the functions $v^{(i)}\in C^2({\bar{G}}_{\varepsilon })$,
 then from \eqref{eq:3p101} we conclude that
\begin{equation}
\begin{aligned}
\int_{G_{\varepsilon }}\Big[ \big( \frac{1}{\varrho }
(\rho v_{\varrho }^{(1)})_{\varrho }-v_{tt}^{(1)}-\frac{1}{\varrho }(\varrho
a_1v^{(1)})_{\varrho }-(bv^{(1)})_{t}&\\
+(c-\frac{n^2}{\varrho ^2})v^{(1)}\big) u^{(1)}
+na_2v^{(1)}u^{(2)}\Big] \varrho d\varrho \,dt&=0,
\\
\int_{G_{\varepsilon }}\Big[ \big( \frac{1}{\rho }
(\rho v_{\varrho }^{(2)})_{\varrho }-v_{tt}^{(2)}-\frac{1}{\varrho }(\varrho
a_1v^{(2)})_{\varrho }-(bv^{(2)})_{t} &\\
+(c-\frac{n^2}{\varrho ^2})v^{(2)}\big) u^{(2)}
-na_2v^{(2)}u^{(1)}\Big] \varrho d\varrho \,dt&=0.
\end{aligned} \label{eq:3p120}
\end{equation}
For $h^{(1)},h^{(2)}\in C^{1}\left( \bar{G}_0\backslash (0,0)\right) $
 we state the following problem.

\textbf{Problem }$P_{\alpha ,1}^{\ast }$. Find a solution
$v^{(1)},v^{(2)}\in V_{\varepsilon }^{(1)}\cap
C^2({\bar{G}}_{\varepsilon }) $ of the system
\begin{gather*}
\frac{1}{\varrho }(\rho v_{\varrho }^{(1)})_{\varrho
}-v_{tt}^{(1)}-\frac{1}{\varrho }(\varrho a_1v^{(1)})_{\varrho
}-(bv^{(1)})_{t}+(c-\frac{n^2}{\varrho
^2})v^{(1)}-na_2v^{(2)}=h^{(1)},
\\
\frac{1}{\varrho }(\varrho v_{\varrho }^{(2)})_{\varrho
}-v_{tt}^{(2)}-\frac{1}{\varrho }(\varrho a_1v^{(2)})_{\varrho
}-(bv^{(2)})_{t}+(c-\frac{n^2}{\varrho
^2})v^{(2)}+na_2v^{(1)}=h^{(2)}.
\end{gather*}
For $z^{(i)}=\varrho ^{1/2}v^{(i)}$, $\xi_1=1-\varepsilon -\eta $, $\eta
_1=1-\varepsilon -\xi $, and
\begin{equation}
V^{(i)}(\xi_1,\eta_1)=z^{(i)}(1-\varepsilon -\eta_1,1-\varepsilon
-\xi_1),
\end{equation}
the domain $G_{\varepsilon }$ maps into $D_{\varepsilon }$, and
for appropriate coefficients $A_{i},B_{i},C_{i},D_{i}$ and $\beta
=\alpha +b$ the above Problem $P_{\alpha ,1}^{\ast }$ transforms
to the Darboux--Goursat Problem $P_{\beta ,3}$. But for this
problem Theorem \ref{thm3.2} ensures the solvability in
$C^2({\bar{D}}_{\varepsilon })$. Consequently, there exists a
classical solution $(V^{(1)},V^{(2)})\in
C^2({\bar{D}}_{\varepsilon })$
and so the inverse transformations\emph{\ }{\eqref{eq:1p15} and %
\eqref{eq:1p17} }lead to a classical solution $(v^{(1)},v^{(2)})\in C^2(%
\bar{G}_{\varepsilon })$ of Problem $P_{\alpha ,1}^{\ast }$. Moreover, with
these functions $v^{(1)},v^{(2)}$ the system \eqref{eq:3p120} becomes
\begin{equation}
\begin{gathered}
\int_{G_{\varepsilon }}\left[ \left( h^{(1)}+na_2v^{(2)}\right)
u^{(1)}+na_2v^{(1)}u^{(2)}\right] \varrho
d\varrho \,dt=0, \\
\int_{G_{\varepsilon }}\left[ \left( h^{(2)}-na_2v^{(1)}\right)
u^{(2)}-na_2v^{(2)}u^{(1)}\right] \varrho d\varrho \,dt=0.
\end{gathered} \label{eq:3p102}
\end{equation}
Since the functions $h^{(1)}(\varrho ,t),h^{(2)}(\varrho ,t)\in C^{1}
\left( {\bar{G}}_0\backslash (0,0)\right) $ are arbitrary,
\eqref{eq:3p102} gives $u^{(1)}(\varrho ,t)=u^{(2)}(\varrho ,t)=0$
in $G_{\varepsilon }$, i.e.
$(u_1^{(1)},u_1^{(2)})\equiv (u_2^{(1)},u_2^{(2)})$. The proof is
complete. \hfill$\square$

\section{Existence and uniqueness theorems for the 3-D Problem
$P_{\protect\alpha }$} %5

In this section we consider the following 3-D boundary value problem.

\noindent\textbf{Problem $P_{\alpha }$.}
 Find a solution to the equation
\begin{equation}
Lu=\frac{1}{\varrho }(\varrho u_{\varrho })_{\varrho }+\frac{1}{\varrho ^2}%
u_{\varphi \varphi }-u_{tt}+a_1u_{\varrho }+a_2u_{\varphi
}+bu_{t}+cu=f(\varrho ,\varphi ,t)\text{ in }\Omega_{\varepsilon
}, \label{eq:4p1}
\end{equation}
which satisfies the boundary conditions
\begin{equation}
\quad u\bigr\rvert_{\Sigma_1\cap \partial \Omega_{\varepsilon }}=0,\quad
\lbrack u_{t}+\alpha (\varrho )u]\bigr\rvert_{\Sigma_0\cap \partial
\Omega_{\varepsilon }}=0.  \label{eq:4p2}
\end{equation}
For this problem we formulate the following theorems.

\begin{theorem} \label{thm5.1}
Let $a_1,a_2,b,c\in C^{1}(\bar{\Omega}_0\backslash
O)$. Then for $0\leq \varepsilon <1$ there exists at most one
generalized solution of Problem $P_{\alpha }$ in\
$\Omega_{\varepsilon }$. \end{theorem}

\noindent\textbf{Proof.} \textit{Case} $0<\varepsilon <1$.
If $u_1,u_2$ are two
generalized solutions of $P_{\alpha }${\ in\ $\Omega_{\varepsilon
}$, then
$u:=u_1-u_2\in C^{1}(\bar{\Omega}_{\varepsilon })$} satisfies
\eqref{eq:4p2} and
\begin{equation}
\begin{aligned}
&\int_{\Omega_{\varepsilon }}[u_{t}v_{t}-u_{\varrho }v_{\varrho
}-\frac{1}{\varrho ^2}u_{\varphi }v_{\varphi }+(a_1u_{\varrho
}+a_2u_{\varphi }+bu_{t}+cu)v]\varrho \,d\varrho \,d\varphi \,dt\\
&=\int_{\Sigma_0\cap \partial \Omega_{\varepsilon }}\alpha
(\varrho )uv\varrho \,d\varrho d\varphi
\end{aligned}\label{eq:4p3}
\end{equation}
holds for all $v\in V_{\varepsilon }$. We will show that in the Fourier
expansion\
\begin{equation}
u(\varrho ,\varphi ,t)=\sum_{n=0}^{\infty }\left\{ u_{n}^{(1)}(\varrho
,t)\cos n\varphi +u_{n}^{(2)}(\varrho ,t)\sin n\varphi \right\}
\label{eq:4p4a}
\end{equation}
the coefficients satisfy $u_{n}^{(i)}(\varrho ,t)$ $\equiv 0$ in
$\Omega_{\varepsilon }$, $i=1,2$, i.e. $u\equiv 0$ in
$\Omega_{\varepsilon}$.
Since $u\in C^{1}(\bar{\Omega}_{\varepsilon })$, using the substitution
\begin{equation*}
v_1(\varrho ,\varphi ,t)=w_1(\varrho ,t)\cos n\varphi \in
V_{\varepsilon }\text{ \ \ \ or \ \ \ }v_2(\varrho ,\varphi
,t)=w_2(\varrho ,t)\sin n\varphi \in V_{\varepsilon }
\end{equation*}
in \eqref{eq:4p3}, we derive the system
\begin{equation}
\begin{aligned}
&\int_{G_{\varepsilon }}\Big[ u_{n,t}^{(1)}w_{1,t}-u_{n,
\varrho }^{(1)}w_{1,\varrho }+\big( a_1u_{n,\varrho
}^{(1)}+bu_{n,t}^{(1)}+(c-\frac{n^2}{\varrho ^2}
)u_{n}^{(1)}+na_2u_{n}^{(2)}\big) w_1\Big] \varrho d\varrho \,dt \\
&=\int_{S_0\cap \partial G_{\varepsilon }}\alpha
(\varrho )u_{n}^{(1)}w_1\varrho \,d\varrho ,\\
&\int_{G_{\varepsilon }}\left[ u_{n,t}^{(2)}w_{2,t}-u_{n,
\varrho }^{(2)}w_{2,\varrho }+\left( a_1u_{n,\varrho
}^{(2)}+bu_{n,t}^{(2)}+(c-\frac{n^2}{\varrho ^2}
)u_{n}^{(2)}-na_2u_{n}^{(1)}\right) w_2\right] \varrho \,d\varrho dt \\
&=\int_{S_0\cap \partial G_{\varepsilon }}\alpha (\varrho
)u_{n}^{(2)}w_2\varrho \,d\varrho
\end{aligned}
\label{eq:4p5}
\end{equation}
for all $w_1,w_2\in V_{\varepsilon }^{(1)}$ and $n\in
\mathbb{N}\cup \{0\}$. By Definition \ref{def2.2}, the function
$(u_{n}^{(1)},u_{n}^{(2)})(\varrho ,t)$ is a generalized solution
of the homogeneous problem $P_{\alpha ,1}$. Clearly, Theorem \ref{thm4.2}
implies $u_{n}^{(1)}(\varrho ,t)\equiv u_{n}^{(2)}(\varrho
,t)\equiv 0$ in $\Omega_{\varepsilon }$ for $n\in \mathbb{N}\cup
\{0\}$ and so $u^{(1)}=u_1-u_2\equiv 0$ in $\Omega _{\varepsilon
}$. {\textit{Case }}$\varepsilon =0$. Let $\varepsilon_0$ be an
arbitrary fixed number of $(0,1)$. Then, by Lemma \ref{lm2.1}, it follows
that the generalized
solution $u\in C^{1}(\bar{\Omega}_0\setminus (0,0,0))$ of Problem $%
P_{\alpha }$ in $\Omega_0$ is also a generalized solution of the
homogeneous problem $P_{\alpha }$ in $\Omega_{\varepsilon_0}$. Since, by
the previous case, $u\equiv 0$ in $\Omega_{\varepsilon_0}$ and $%
\varepsilon_0>0$ is arbitrary, we see that $u=u_1-u_2\equiv 0$ in $%
\Omega_0$.
This completes the proof of the theorem.\hfill $\square$

\begin{theorem} \label{thm5.2}
Let $a_1,a_2,b,c\in C^{1}\left( \bar{\Omega}_0\backslash
O\right) $ and the function $f\in C(\overset{-}{\Omega }_0)\cap
C^{1}( \overset{-}{\Omega }_0\backslash O) $ be of the
form
\begin{equation}
f(\varrho ,\varphi ,t)=\sum_{n=0}^{k}\big\{ f_{n}^{(1)}(\varrho ,t)\cos
n\varphi +f_{n}^{(2)}(\varrho ,t)\sin n\varphi \big\} ,\;
k\in \mathbb{N}\cup \{0\}.  \label{eq:4p6}
\end{equation}
Then there exists one and only one generalized solution
\begin{equation}
u(\varrho ,\varphi ,t)=\sum_{n=0}^{k}\left\{ u_{n}^{(1)}(\varrho ,t)\cos
n\varphi +u_{n}^{(2)}(\varrho ,t)\sin n\varphi \right\}  \label{eq:4p7}
\end{equation}
of the problem $P_{\alpha }$ in $\Omega_0$. This solution $u\in
C^2\left( \bar{\Omega}_0\backslash O\right) $ is a classical
solution of the problem $P_{\alpha }$ in each domain
$\Omega_{\varepsilon },\varepsilon \in (0,1)$. Moreover, if
\begin{equation*}
|a_1|\leq d\varrho ^{-1},\quad |a_2|\leq d\varrho ^{-2},
\quad |b|\leq d\varrho^{-2},\quad |c|\leq d\varrho ^{-2},
\quad |\alpha |\leq d\varrho ^{-2}\quad\text{in }\bar{\Omega}_0\backslash O,
\end{equation*}
then, in view of \eqref{eq:4p7}, for a fixed $n$, the corresponding
trigonometric polynomial $u_{n}$ of degree $n$, satisfies the following a
priori estimates:
For $n=0$,
\begin{equation}
\| u_0(x_1,x_2,t)\|_{C^{1}(\Omega_{\varepsilon }^{(1)})}
=\sum_{|\alpha |\leq 1}{\sup_{\Omega_{\varepsilon }^{(1)}}}
|D^{\alpha }u_0|
\leq  6\varepsilon ^{3/2}
\exp\big( \frac{32d+2}{\varepsilon ^2}\big)
\| f_0^{(1)} \|_{C^0(\bar{G}_0)};
\label{eq:4p8}
\end{equation}
while for $n\in \mathbb{N}$,
\begin{equation}
\| u_{n}(x_1,x_2,t)\|_{C^{1}(\Omega_{\varepsilon }^{(1)})} \\
\leq \frac{6\varepsilon ^{3/2}}{n(n+2d)}\exp \big(
\frac{8n(n+3d)}{\varepsilon ^2}\big)
\big( \| f_{n}^{(1)}\|_{C^0(\bar{G}_0)}+\| f_{n}^{(2)}
\|_{C^0(\bar{G}_0)}\big) ,
\label{eq:4p9}
\end{equation}
where $\Omega_{\varepsilon }^{(1)}\ =\Omega_0\cap \{(\varrho ,t):\varrho
+t>\varepsilon \}$.
\end{theorem}

\noindent\textbf{Proof.}
It suffices to consider the case of a fixed number $n$. As
in Section 2, we make the substitutions
\begin{equation}
\xi =1-\varrho -t,\quad \eta =1-\varrho +t,  \label{eq:4p11a}
\end{equation}
and introduce the new function
\begin{equation}
U^{(i)}(\xi ,\eta )=\varrho ^{1/2}u^{(i)}(\varrho (\xi ,\eta ),t(\xi ,\eta
)).  \label{eq:4p11}
\end{equation}
Denote
\begin{equation*}
F^{(i)}(\xi ,\eta ):=\frac{1}{4\sqrt{2}}(2-\eta -\xi
)^{1/2}f_{n}^{(i)}(\xi ,\eta )\in C^{1}\left( \bar{D}_0\backslash
(1,1)\right) ,
\end{equation*}
and use the notation of \eqref{eq:1p101}. Then the problem reduces to
Problem $P_{\alpha ,3}$. Thus, we can use Theorems \ref{thm3.1} and
\ref{thm3.2} to ensure the
existence of a classical solution $(U^{(1)},U^{(2)})(\xi ,\eta )$ of this
problem with the estimates \eqref{eq:2p7}.

\noindent\textit{Case} $n\in \mathbb{N}$. In view of \eqref{eq:2p5},
\eqref{eq:1p101}, it is easy to see that we can chose
\begin{gather*}
c(\varepsilon ):=  \frac{n(n+2d)}{\varepsilon ^2},\quad
M_{\alpha }:=\frac{4d}{\varepsilon ^2} \\
M  \leq \frac{1}{4}\max \left\{ \|f_{n}^{(1)}\|_{C^0(\bar{G}_0)},
\|f_{n}^{(2)}\|_{C^0(\bar{G}_0)}\right\} :=M_{n},
\end{gather*}
Hence, Theorems \ref{thm3.1} and \ref{thm3.2} ensure the smoothness of the solution $U$ of
Problem $P_{\alpha ,3}$ in $D_{\varepsilon }^{(1)}=\{(\xi ,\eta ):0<\xi
<\eta <1,\;0<\xi <1-\varepsilon \},\;\varepsilon >0$, i.e.
\begin{equation}
(U_{n}^{(1)},U_{n}^{(2)})(\xi ,\eta ):=U(\xi ,\eta )\in C^2(\bar{D}%
_{\varepsilon }^{(1)})\,.  \label{eq:4p14}
\end{equation}
On the other hand, these theorems ensure the a priori estimates
\begin{align*}
\sup_{D_{\varepsilon }^{(1)}}|U_{n}^{(i)}(\xi ,\eta )|
&\leq  M_{n}\left[ 4c(\varepsilon )+M_{\alpha }\right] ^{-2}\exp
\left\{ 8c(\varepsilon )+2M_{\alpha }\right\} \\
&\leq M_{n}\varepsilon ^{4}[4n(n+2d)]^{-2}\exp \left\{ (8n(n+3d)\varepsilon
^{-2}\right\} ,\\
\sup_{D_{\varepsilon }^{(1)}}\{|U_{n,\xi }^{(i)}|,\;|U_{n,\eta}^{(i)}|\}
&\leq M_{n}\varepsilon ^2[4n(n+2d)]^{-1}\exp \left\{
(8n(n+3d)\varepsilon ^{-2}\right\} .
\end{align*}
Also, by \eqref{eq:4p11a} and \eqref{eq:4p11}, we have
$u_{n}^{(i)}(\varrho ,t)=\varrho ^{-\frac{1}{2}}U_{n}^{(i)}(\xi ,\eta )$.
Since $\varrho \geq \varepsilon /2$ for $(\xi ,\eta )\in D_{\varepsilon
}^{(1)}$, by the inverse transformation we see that
\begin{equation}
\begin{gathered}
|u_{n}^{(i)}(\varrho ,t)|  \leq   M_{n}\frac{\varepsilon
^{7/2}}{8n^2(n+2d)^2}\exp \big( \frac{8n(n+3d)}{\varepsilon ^2}
\big) , \\
|u_{n,t}^{(i)}(\varrho ,t)|  \leq   M_{n}\frac{\varepsilon
^{3/2}}{n(n+2d)}\exp \big( \frac{8n(n+3d)}{\varepsilon ^2}\big) , \\
|u_{n,\varrho }^{(i)}(\varrho ,t)|  \leq   M_{n}\frac{
\varepsilon ^{3/2}}{n(n+2d)}\exp \big( \frac{8n(n+3d)}{\varepsilon ^2}
\big) .
\end{gathered} \label{eq:4p16}
\end{equation}
Therefore, in view of \eqref{eq:4p7} and \eqref{eq:4p16}, for the
trigonometrical polynomial
\begin{equation}
u_{n}(\varrho ,\varphi ,t)=u_{n}^{(1)}(\varrho ,t)\cos n\varphi
+u_{n}^{(2)}(\varrho ,t)\sin n\varphi  \label{eq:4p201}
\end{equation}
we derive
\begin{equation}
\| \frac{1}{\varrho }u_{n,\varphi }(\varrho ,\varphi
,t)\|_{C(\Omega_{\varepsilon }^{(1)})}\leq  M_{n}\frac{%
\varepsilon ^{5/2}}{4n(n+2d)^2}\exp \left( \frac{8n(n+3d)}{\varepsilon ^2%
}\right) .  \label{eq:5p20}
\end{equation}
Since $ u_{n}(\varrho \cos \varphi ,\varrho \sin \varphi
,t)=u_{n}^{(1)}(\varrho ,\varphi ,t)$, obviously one has
\begin{equation*}
|u_{n,x_{i}}(x_1,x_2,t)|\leq 2M_{n}\frac{\varepsilon ^{3/2}}{n(n+2d)}%
\exp \left( \frac{8n(n+3d)}{\varepsilon ^2}\right) ,\quad i=1,2.
\end{equation*}
So, the estimate \eqref{eq:4p9} holds in $\Omega_{\varepsilon }^{(1)}$.

\noindent\textit{Case $n=0$.}
By \eqref{eq:4p6} and
\eqref{eq:4p7}, we have $f_0(\varrho ,\varphi
,t)=f_0^{(1)}(\varrho ,t)$ and $u_0(x_1,x_2,t)=u_0(\varrho
,\varphi ,t)=u_0^{(1)}(\varrho ,t)$. Take
\begin{equation*}
c(\varepsilon ):=\frac{8d+1}{4\varepsilon ^2},\quad
M_{\alpha }:=\frac{4d}{\varepsilon ^2},\quad
M:=\frac{1}{4}\| f_0^{(1)}\|_{C^0(\bar{G}_0)}\,.
\end{equation*}
Then, as in the previous case, we obtain \eqref{eq:4p8}.
\hfill$\square$

\begin{theorem} \label{thm5.3}
Let the conditions of Theorem \ref{thm5.2} be fulfilled. Also, for the sake of
simplicity, suppose that $a_1,a_2,b,c\in C^{1}(\bar{\Omega}_0)$ and $%
|\alpha ^{\prime }(\varrho )|\leq d_1/\varrho ^{-3}$. Then for a fixed $%
n\in \mathbb{N}$ \ the corresponding trigonometric polynomial $u_{n}$ of
degree $n$ from \eqref{eq:4p201} satisfies the following a priori estimate
\begin{equation}
\| u_{n}(x_1,x_2,t)\|_{C^2(\bar{\Omega}_{\varepsilon}^{(1)})}
\leq C_1\varepsilon ^{-1/2}\exp \big( \frac{8n(n+3d)}{\varepsilon ^2}\big)
\big( \| f_{n}^{(1)}\|_{C^0(\bar{G}_0)}+\|
f_{n}^{(2)}\|_{C^0(\bar{G}_0)}\big) ,
\end{equation}
where the constant $C_1$ does not depend on $n$ and $\varepsilon $.
\end{theorem}

\noindent\textbf{Proof.}
 We will use the estimates of Theorem \ref{thm5.2} and the representations of
the second derivatives of Theorem \ref{thm3.2}. Following the same arguments, as in
Theorem \ref{thm5.2}, we obtain the estimates
\begin{equation*}
\sup_{D_{\varepsilon }^{(1)}}\{|U_{n,\xi \eta
}^{(i)}|,|U_{n,\eta \eta }^{(i)}|,\;|U_{n,\xi \xi }^{(i)}|\}\leq
C_1M_{n}\exp \left\{ 8n(n+3d)\varepsilon ^{-2}\right\} ,\text{ }i=1,2
\end{equation*}
and conclude that
\begin{equation*}
\sup_{D_{\varepsilon
}^{(1)}}\{|u_{n,x_{i}x_{j}}|,|u_{n,tx_{i}}|\}\leq
C_1M_{n}\varepsilon ^{-1/2}\exp \big( \frac{8n(n+3d)}{\varepsilon
^2}\big).
\end{equation*}
\hfill$\square$

The next theorem is an immediate consequence of Theorems \ref{thm5.1},
\ref{thm5.2} and \ref{thm5.3}.

\begin{theorem} \label{thm5.4}
Let the conditions of Theorem \ref{thm5.3} be fulfilled and let
$f\in C^{1}(\bar{\Omega}_0)$ be of the form
\begin{equation}
f(\varrho ,\varphi ,t)=\sum_{n=0}^{\infty }\{f_{n}^{(1)}(\varrho ,t)\cos
n\varphi +f_{n}^{(2)}(\varrho ,t)\sin n\varphi \}.  \label{eq:4p17}
\end{equation}
Suppose that the Fourier coefficients $f_{n}^{(1)}(\varrho ,t)$ and $%
f_{n}^{(2)}(\varrho ,t)$ satisfy
\begin{equation}
\begin{aligned}
\| f\|_{\exp \,(\varepsilon )}
:=&\exp \big( \frac{32d+2}{\varepsilon ^2}\big)
\| f_0^{(1)}\|_{C^0(\bar{G}_0)}
+\sum_{n=1}^{\infty }\frac{1}{n(n+2d)}\exp
\big(\frac{8n(n+3d)}{\varepsilon ^2}\big)\\
&\times\big( \| f_{n}^{(1)}\|_{C^0(\bar{G}_0)}+\| f_{n}^{(2)}
\|_{C^0(\bar{G}_0)}\big) <\infty \,.
\end{aligned} \label{eq:4p18}
\end{equation}
Then there exists one and only one generalized solution $u\in C^{1}\left(
\bar{\Omega}_{\varepsilon }^{(1)}\right) $ of the problem $P_{\alpha }$ in $%
\Omega_{\varepsilon }$ and satisfies the a priori estimate
\begin{equation}
\| u\|_{C^{1}(\Omega_{\varepsilon }^{(1)})}\leq 6\varepsilon
^{3/2}\| f\|_{\exp \,(\varepsilon )}.  \label{eq:4p19}
\end{equation}
 Moreover, if
\begin{equation}
\begin{aligned}
\| f\|_{\exp_1\,(\varepsilon )}:=&\exp
\big( \frac{32d+2}{\varepsilon ^2}\big) \| f_0^{(1)}\|_{C^0(\bar{G}_0)}
+\sum_{n=1}^{\infty }\exp \big( \frac{8n(n+3d)}{\varepsilon ^2}\big) \\
&\times \big( \| f_{n}^{(1)}\|_{C^0(\bar{G}_0)}
+\| f_{n}^{(2)}\|_{C^0(\bar{G}_0)}\big) <\infty ,
\end{aligned} \label{eq:4p20}
\end{equation}
then $u\in C^2\left( \bar{\Omega}_{\varepsilon }^{(1)}\right) $,
$u(x,t)$ is a classical solution of the problem $P_{\alpha }$ in
$\Omega _{\varepsilon }$ and satisfies the a priori estimate
\begin{equation}
\| u\|_{C^2(\Omega_{\varepsilon }^{(1)})}\leq C_2\varepsilon
^{-1/2}\| f\|_{\exp_1\,(\varepsilon )}. \label{eq:4p21}
\end{equation}
\end{theorem}

\begin{remark} \label{rmk5.1} \rm
It is obvious that the estimates \eqref{eq:4p19} or \eqref{eq:4p21} hold, if
the series \eqref{eq:4p18} and \eqref{eq:4p20} are finite. In this case we
have a solution, which is of class $C^{1}(\bar{\Omega}_0\backslash O)$ or
of class $C^2(\bar{\Omega}_0\backslash O)$. For example, the condition %
\eqref{eq:4p20} is valid for each $\varepsilon \in (0,1)$, if there exists a
sequence $a_{n}$ $\to +\infty $ as $n\to +\infty $ such that
\begin{equation}
\sum_{n=1}^{\infty }\exp \left( n^2a_{n}\right) \left( \|
f_{n}^{(1)}\|_{C^0(\bar{G}_0)}+\| f_{n}^{(2)}\|_{C^0(\bar{G}%
_0)}\right) <\infty .   \label{eq:4p22}
\end{equation}
\end{remark}

To show this, it is enough to see that the inequality $n(\varepsilon
^2a_{n}-8)\geq 24d$ holds, for all large enough $n\in \mathbb{N}$.


\section{On the singularity of solutions of Problem $P_{\alpha }$} %6

For the the equation
\begin{equation}
Lu=\frac{1}{\varrho }(\varrho u_{\varrho })_{\varrho }+\frac{1}{\varrho ^2}%
u_{\varphi \varphi }-u_{tt}+a_1u_{\varrho }+a_2u_{\varphi
}+bu_{t}+cu=f(\varrho ,\varphi ,t)\text{ in }\Omega_0,
\label{eq:5p1}
\end{equation}
we consider the boundary conditions of Problem $P_{\alpha }$, i.e.
\begin{equation}
P_{\alpha }:\quad u\bigr\rvert_{\Sigma_1}=0,\quad \lbrack u_{t}+\alpha
(\varrho )u]\bigr\rvert_{\Sigma_0\backslash O}=0  \label{eq:5p2}
\end{equation}
and prove the following result.

\begin{theorem} \label{thm6.1}
Let $\alpha (\varrho )\geq 0;$ $a_1$, $b$,
$c\in C^{1}({\bar{\Omega}}_0\backslash O)$, $a_2\equiv 0$ and
\begin{equation}
a_1(\varrho ,t)\geq |b|(\varrho ,t),\quad
 a_1(\varrho ,t)\geq 2\varrho c(\varrho ,t),\quad
(\varrho ,t)\in {\Omega }_0.   \label{eq:5p90}
\end{equation}
Then for each function
\begin{equation*}
f_{n}(\varrho ,\varphi ,t)=\varrho ^{-n}(\varrho
^2-t^2)^{n-1/2}\cos
n\varphi \in C^{n-2}({\bar{\Omega}}_0)\cap C^{\infty }({\Omega }%
_0),\quad n\in \mathbb{N},
\end{equation*}
the corresponding generalized solution $u_{n}$ of the problem
$P_{\alpha }$ belongs to $C^2({\bar{\Omega}}_0\backslash O)$ and
satisfies the estimate
\begin{equation}
|u_{n}(\varrho ,\varphi ,\varrho )|\geq \frac{1}{2}|u_{n}(2\varrho ,\varphi
,0)|+\varrho ^{-n}|\cos n\varphi |\geq \varrho ^{-n}|\cos n\varphi |,\quad
0<\varrho <1\,.  \label{eq:5p3}
\end{equation}
\end{theorem}

\noindent\textbf{Proof.}
 Note that, by Theorem \ref{thm1.1}, the functions
 \begin{equation*}
w_{n}(\varrho ,\varphi ,t)=\varrho ^{-n}(\varrho
^2-t^2)^{n-1/2}(a_{n}\cos n\varphi +b_{n}\sin n\varphi ),\quad
n\geq 4,
\end{equation*}
are classical solutions of the homogeneous Problem $P_{\alpha }^{\ast }$
for the wave equation, when $\alpha \equiv 0$.

Now consider the special case of Problem $P_{\alpha }$:
\begin{equation}
Lu=f_{n}\equiv \varrho ^{-n}(\varrho ^2-t^2)^{n-1/2}\cos n\varphi
\quad\text{in $\Omega_0.$}  \label{eq:5p5a}
\end{equation}
Observe also that
\begin{equation*}
f_{n}(x_1,x_2,t)=(x_1^2+x_2^2)^{-n}(x_1^2+x_2^2-t^2)^{n-1/2}%
\mathop{\rm Re}(x_1+ix_2)^{n}
\end{equation*}
and obviously $f_{n}\in C^{n-2}({\bar{\Omega}}_0)\cap C^{\infty }({\Omega }
_0)$, $n\in \mathbb{N}$. Theorem \ref{thm5.1} states that the equation
\eqref{eq:5p5a} with boundary conditions \eqref{eq:5p2} has at most one
generalized solution. On the other hand, from Theorem \ref{thm5.2} it is known that,
for the above right-hand side, there exists a generalized solution in
$\Omega_0$ of the form
\begin{equation*}
u_{n}(\varrho ,\varphi ,t)=u_{n}^{(1)}(\varrho ,t)\cos n\varphi \in C^2({%
\bar{\Omega}}_0\backslash O),
\end{equation*}
which is classical solution in $\Omega_{\varepsilon }$, $\varepsilon \in
(0,1)$. By setting $u_{n}^{(2)}(\varrho ,t)=\varrho ^{\frac{1}{2}%
}u_{n}^{(1)}(\varrho ,t)$ and substituting
\begin{equation}
\xi =1-\varrho -t,\quad \eta =1-\varrho +t,  \label{eq:5p6}
\end{equation}
the equation \eqref{eq:5p5a}, with boundary conditions \eqref{eq:5p2}, in
view of
\begin{equation}
U(\xi ,\eta )=u_{n}^{(2)}(\varrho (\xi ,\eta ),t(\xi ,\eta )),
\label{eq:5p7}
\end{equation}
becomes a Darboux-Goursat problem $P_{\alpha ,3}$:
\begin{gather}
U_{\xi \eta }-AU_{\xi }-BU_{\eta }-CU=F(\xi ,\eta ),  \label{eq:5p8}
\\
U(0,\eta )=0,\quad (U_{\eta }-U_{\xi })(\xi ,\xi )+\alpha (1-\xi )U(\xi
,\xi )=0.  \label{eq:5p8a}
\end{gather}
Note that, because of the condition $a_2\equiv 0$ and the
special right-hand side of \eqref{eq:5p5a}, we do not obtain
a system as in the general case of Section 3, but a single equation
\eqref{eq:5p8}. According to \eqref{eq:1p101}, the coefficients of
\eqref{eq:5p8} are defined as follows:
\begin{equation}
\begin{gathered}
A=\frac{1}{4}(a_1+b)\geq 0,\quad B=\frac{1}{4}(a_1-b)\geq 0, \\
C(\xi ,\eta )=\frac{1}{4}\Big( \frac{4n^2-1}{(2-\eta -\xi
)^2}+\frac{a_1(\xi ,\eta )}{2-\eta -\xi }-c(\xi ,\eta )\Big)
\geq 0,\ n\in \mathbb{N},
\end{gathered}
\label{eq:5p89}
\end{equation}
\begin{equation}
F(\xi ,\eta )=2^{n-\frac{5}{2}}\Big[ \frac{(1-\xi )(1-\eta )}{2-\eta -\xi }%
\Big] ^{n-\frac{1}{2}}\in C^{n-1}({\bar{D}}_{\varepsilon }^{(1)}),\quad
F(\xi ,\eta )\geq 0,  \label{eq:5p10}
\end{equation}
where we preserve the same notations for $a_1,b$ and $c$ in the new
coordinates $(\xi ,\eta )$. Next, in view of Theorem \ref{thm3.2} and
Lemma \ref{lm3.1}, we formulate the following result.

\begin{proposition} \label{prop6.1}
There exists a classical solution
$U(\xi,\eta )\in C^2(\bar{D}_0\setminus (1,1))$ for the problem
\eqref{eq:5p8}, \eqref{eq:5p8a} for which
\begin{equation*}
U(\xi ,\eta )\geq 0,\quad U_{\xi }(\xi ,\eta )\geq 0,\quad
U_{\eta } (\xi ,\eta )\geq 0 \quad \text{in } \bar{D}_{\varepsilon }^{(1)}.
\end{equation*}
\end{proposition}

Set
\begin{equation}
K=\int_{D_{\frac{1}{2}}^{(1)}}F^2(\xi ,\eta )\,\,d\xi d\eta >0.
\label{eq:5p10a}
\end{equation}
Then from \eqref{eq:5p8} for $0<\varepsilon <1/2$ it follows that
\begin{equation}
\begin{aligned}
0<K\leq &\int_{D_{\varepsilon }^{(1)}}F^2(\xi ,\eta )\,\,d\xi d\eta\\
=&\int_{D_{\varepsilon }^{(1)}}(U_{\xi \eta }F)(\xi ,\eta
)d\xi \,d\eta
-\int_{D_{\varepsilon }^{(1)}}[(AU_{\xi }+BU_{\eta }+CU)F](\xi
,\eta )\,d\xi d\eta \\
=:&I_1-I_2.
\end{aligned}\label{eq:5p90a}
\end{equation}
Using the properties of $F(\xi ,\eta )$ from \eqref{eq:5p10} and following
\cite{GHP}, we find that
\begin{equation}
\begin{aligned}
I_1:=&\int_0^{1-\varepsilon }\int_{\xi }^{1}(U_{\xi \eta
}F)(\xi ,\eta )\,d\eta \,d\xi \\
=&\int_{D_{\varepsilon }^{(1)}}(UF_{\xi \eta
})(\xi ,\eta )\,d\xi \,d\eta
-\int_0^{1-\varepsilon }[U_{\xi }(\xi ,\xi )F(\xi ,\xi)
+U(\xi ,\xi )F_{\eta }(\xi ,\xi )]\,d\xi\\
&-\int_{1-\varepsilon}^{1}U(1-\varepsilon ,\eta )F_{\eta }(1-\varepsilon ,
\eta )\,d\eta .
\end{aligned}
\label{eq:5p90b}
\end{equation}
An elementary calculation shows that
\begin{equation}
F_{\xi }(\xi ,\eta )\leq 0,F_{\eta }(\xi ,\eta )\leq 0,
\end{equation}
which actually follows from Lemma \ref{lm3.1}, and
\begin{equation}
F_{\xi }(\xi ,\xi )=F_{\eta }(\xi ,\xi )=\frac{1}{16}(1-2n)(1-\xi )^{n
-\frac{3}{2}}.  \label{eq:5p16}
\end{equation}
From \eqref{eq:5p90a} and \eqref{eq:5p90b} it follows that
\begin{equation}
\begin{aligned}
0<K\leq I_1-I_2=&-\int_0^{1-\varepsilon }[U_{\xi }(\xi
,\xi )F(\xi ,\xi )+U(\xi ,\xi )F_{\xi }(\xi ,\xi )]\,d\xi \\
&-\int_{1-\varepsilon }^{1}U(1-\varepsilon ,\eta )F_{\eta
}(1-\varepsilon ,\eta )\,d\eta \\
&+\int_{{D}_{\varepsilon }^{(1)}}\left\{
(F_{\xi \eta }-CF)U-F(AU_{\xi }+BU_{\eta })\right\} \,d\xi \,d\eta .
\end{aligned}\label{eq:5p17}
\end{equation}
Also, it is easy to check that
\begin{equation}
F_{\xi \eta }(\xi ,\eta )-\frac{4n^2-1}{4(2-\eta -\xi )^2}F(\xi
,\eta )=0 \label{eq:5p92}
\end{equation}
and so, because of \eqref{eq:5p90}, \eqref{eq:5p89} and Proposition
\ref{prop6.1},
\begin{equation*}
(F_{\xi \eta }-CF)U-F(AU_{\xi }+BU_{\eta })=-\big( \frac{a_1(\xi ,\eta )}{%
2-\eta -\xi }-c(\xi ,\eta )\big) \frac{FU}{4}-F(AU_{\xi }+BU_{\eta })\leq
0.
\end{equation*}
Thus, we find
\begin{equation}
\begin{split}
0<K\leq I_1-I_2 \leq &-\int_0^{1-\varepsilon }[U_{\xi }(\xi ,\xi
)F(\xi ,\xi )+U(\xi ,\xi )F_{\xi }(\xi ,\xi )]\,d\xi \\
& -\int_{1-\varepsilon }^{1}U(1-\varepsilon ,\eta )g_{\eta
}(1-\varepsilon ,\eta )\,d\eta ,
\end{split} \label{eq:5p18}
\end{equation}
where, as it is easy to check,
\begin{equation}
F_{\xi }(\xi ,\xi )=\frac{1}{2}[F(\xi ,\xi )]_{\xi }.  \label{eq:5p19}
\end{equation}
The function $U(\xi ,\eta )$ is a classical solution of \eqref{eq:5p8}, %
\eqref{eq:5p8a} in ${\bar{D}}_{\varepsilon }$, $\varepsilon \in (0,1)$ with
\begin{equation}
U_{\xi }(\xi ,\xi )=\frac{1}{2}[U(\xi ,\xi )]_{\xi }+\frac{1}{2}\alpha
(1-\xi )U(\xi ,\xi ).  \label{eq:5p23}
\end{equation}
If we substitute \eqref{eq:5p19} and \eqref{eq:5p23} into \eqref{eq:5p18},
we get
\begin{equation}
\begin{aligned}
K\leq& I_1-I_2=-\frac{1}{2}\int_0^{1-\varepsilon }[F(\xi
,\xi )U(\xi ,\xi )]_{\xi }\,d\xi \\
&-\frac{1}{2}\int_0^{1-\varepsilon }\alpha (1-\xi )U(\xi ,\xi
)F(\xi ,\xi )\,d\xi -\int_{1-\varepsilon }^{1}U(1-\varepsilon ,\eta )F_{\eta
}(1-\varepsilon ,\eta )\,d\eta \\
=&-\frac{1}{2}(FU)(1-\varepsilon ,1-\varepsilon )-\frac{1}{2}%
\int_0^{1-\varepsilon }\alpha (1-\xi )U(\xi ,\xi )F(\xi ,\xi )\,d\xi \\
&-\int_{1-\varepsilon }^{1}U(1-\varepsilon ,\eta )F_{\eta
}(1-\varepsilon ,\eta )\,d\eta .
\end{aligned} \label{eq:5p24}
\end{equation}
Next, in view of Proposition \ref{prop6.1} and the properties of the function
$F(\xi ,\eta )$, we find
\begin{equation*}
U(\xi ,\eta )\geq 0,\quad U_{\eta }(\xi ,\eta )\geq 0,\quad
 \alpha (\xi )\geq 0,\quad F(\xi ,\eta )\geq 0,\quad
F_{\eta }(\xi ,\eta )\leq 0\text{ in }{\bar{D}}_{\varepsilon }^{(1)},
\end{equation*}
which together with \eqref{eq:5p24} and because of $F(1-\varepsilon ,1)=0$
implies
\begin{align*}
K&\leq I_1+I_2\leq -\int_{1-\varepsilon }^{1}U(1-\varepsilon ,\eta
)F_{\eta }(1-\varepsilon ,\eta )\,d\eta
-\frac{1}{2}(FU)(1-\varepsilon,1-\varepsilon ) \\
&=\int_{1-\varepsilon }^{1}U(1-\varepsilon ,\eta )\lvert
F_{\eta }(1-\varepsilon ,\eta )\rvert \,d\eta -\frac{1}{2}(FU)(1-\varepsilon
,1-\varepsilon ) \\
&\leq \int_{1-\varepsilon }^{1}U(1-\varepsilon ,1)\lvert
F_{\eta }(1-\varepsilon ,\eta )\rvert \,d\eta -\frac{1}{2}(FU)(1-\varepsilon
,1-\varepsilon ) \\
&=\big[ U(1-\varepsilon ,1)-\frac{1}{2}U(1-\varepsilon
,1-\varepsilon )\big] F(1-\varepsilon ,1-\varepsilon ).
\end{align*}
Since $F(1-\varepsilon ,1-\varepsilon )
=\frac{1}{4}\varepsilon ^{n-\frac{1}{2}}$, we have
\begin{equation*}
0<K\leq \big[ U(1-\varepsilon ,1)-\frac{1}{2}U(1-\varepsilon ,1-\varepsilon
)\big] \frac{1}{4}\varepsilon ^{n-\frac{1}{2}}.
\end{equation*}
For $\xi =1-\varepsilon $, $\eta =1$ we have $\varrho =t=\varepsilon /2$ and
so
\begin{equation}
0<4K\varepsilon ^{\frac{1}{2}-n}\leq u_{n}^{(2)}\left( \frac{\varepsilon }{2}%
,\frac{\varepsilon }{2}\right) -\frac{1}{2}u_{n}^{(2)}(\varepsilon ,0).
\label{eq:5p26}
\end{equation}
Finally, the inverse transformation gives
\begin{equation*}
u_{n}^{(1)}\left( \frac{\varepsilon }{2},\frac{\varepsilon }{2}\right) \geq
\frac{1}{2}u_{n}^{(1)}(\varepsilon ,0)+{\tilde{C}}_1\varepsilon ^{-n}\geq {%
\tilde{C}}_1\varepsilon ^{-n},\quad 0<\varepsilon <\frac{1}{2},
\end{equation*}
where ${\tilde{C}}_1=2^{\frac{5}{2}}K$. Multiplying the function $u_{n}$
by ${\tilde{C}}_1^{-1}$, we see that \eqref{eq:5p3} holds.
The proof is complete. \hfill$\square$\smallskip

Note that the conditions of Theorem \ref{thm6.1} are only sufficient, and not
invariant with respect to change of variables. Now, we use this fact in
order to find some new singular solutions. For this reason, consider the
special form of the equation \eqref{eq:5p1}
\begin{equation}
Lu=\frac{1}{\varrho }(\varrho u_{\varrho })_{\varrho }+\frac{1}{\varrho ^2}%
u_{\varphi \varphi }-u_{tt}+a_1(u_{\varrho }-u_{t})+cu=f(\varrho ,\varphi
,t)\text{ in }\Omega_{\varepsilon },  \label{eq:5p101}
\end{equation}
with the boundary conditions \eqref{eq:5p2}.

\begin{theorem} \label{thm6.2}
Let $a_1$, $c\in C^{1}({\bar{\Omega}}_0\backslash O);a_1\in C({\bar{%
\Omega}}_0)$ and $\alpha (\varrho )\in C^{1}(0,1]\cap C[0,1]$, without any
conditions imposed on the sign. Suppose that $a_1(\varrho ,t)\geq 2\varrho
c(\varrho ,t)$ for $(\varrho ,t)\in {\Omega }_0$. Then there exists
appropriate constant $\Lambda_0$, such that for any other constant $%
\Lambda \geq \Lambda_0$ and function
\begin{equation*}
f_{n}(\varrho ,\varphi ,t)=\exp \left\{ \Lambda (\varrho
+t)\right\} \varrho ^{-n}(\varrho ^2-t^2)^{n-1/2}\cos n\varphi
,n\in \mathbb{N},
\end{equation*}
the corresponding singular \textsl{generalized solution} $u_{n}$
of the problem $P_{\alpha }$ belongs to
$C^2({\bar{\Omega}}_0\backslash O)$ and the estimate
\eqref{eq:5p3}\ holds.
\end{theorem}

\noindent\textbf{Proof.} We are looking for appropriate right-hand side
functions $f_{n}$ of the equation \eqref{eq:5p101}, for which singular
solutions exist. Set
\begin{equation*}
u(\varrho ,\varphi ,t)=\exp \left\{ \lambda (\varrho +t)\right\} w(\varrho
,\varphi ,t),
\end{equation*}
where the function $\lambda (s)$ will be chosen later. Then the equation %
\eqref{eq:5p101} becomes
\begin{equation} \label{eq:5p202}
\begin{aligned}
L_1w &=\frac{1}{\varrho }(\varrho w_{\varrho })_{\varrho }
+\frac{1}{\varrho ^2}w_{\varphi \varphi }-w_{tt}+(a_1+2\lambda ')
(w_{\varrho }-w_{t})+(c+\lambda ^{\prime }\varrho ^{-1})w\\
&=\exp \left\{ -\lambda (\varrho +t)\right\} f(\varrho ,\varphi ,t)
\quad\text{in }\Omega_{\varepsilon }
\end{aligned}
\end{equation}
and so we lead to the following boundary value problem:
\begin{equation}
P_{\beta }:\left\{
\begin{array}{l}
L_1w=g:=\exp \left\{ -\lambda (\varrho +t)\right\} f\text{ \quad in\ $%
\Omega_0$}, \\
w\bigr\rvert_{\Sigma_1}=0,[w_{t}+\beta (\varrho )w]\bigr\rvert_{\Sigma
_0\backslash O}=0
\end{array}
\right.
\end{equation}
with $\beta (\varrho )=\alpha (\varrho )+\lambda ^{\prime }(\varrho )$. In
order to apply Theorem \ref{thm6.1} to Problem $P_{\beta }$, we need the following
conditions
\begin{equation*}
\alpha (\varrho )+\lambda ^{\prime }(\varrho )\geq 0,\quad a_1(\varrho
,t)+2\lambda ^{\prime }(\varrho )\geq 0,\quad a_1(\varrho ,t)-2\varrho
c(\varrho ,t)\geq 0,
\end{equation*}
which are satisfied, for example, for $\lambda (\varrho )=\Lambda \varrho $
and $\Lambda >0$ large enough. Following the proof of Theorem \ref{thm6.1} and using
the transformations \eqref{eq:5p6} and \eqref{eq:5p7}, we lead to the
function $W(\xi ,\eta )$ of \eqref{eq:5p7}, for which the equation %
\eqref{eq:5p8} reduces to
\begin{equation}
W_{\xi \eta }-\frac{1}{2}(a_1+2\lambda ^{\prime })W_{\eta }-CW=F(\xi ,\eta
).  \label{eq:5p103}
\end{equation}
Here $C(\xi ,\eta )$ and $F(\xi ,\eta )$ are functions from
\eqref{eq:5p10} and \eqref{eq:5p89}.

We formulate now a result to be used in the proof of
Theorem \ref{thm7.1}.

\begin{proposition} \label{prop6.2}
There exists a classical solution
$W(\xi,\eta )\in C^2(\bar{D}_0\backslash (1,1))$ of the problem
\eqref{eq:5p103}, \eqref{eq:5p8a}  for which
\begin{equation}
W(\xi ,\eta )\geq 0,\quad W_{\xi }(\xi ,\eta )\geq 0,\quad
W_{\eta }(\xi ,\eta )\geq 0 \quad \mbox{in }\bar{D}_{\varepsilon}^{(1)}.
 \label{eq:5p104}
\end{equation}
\end{proposition}

For the function
\begin{equation}
g_{n}(\varrho ,\varphi ,t)=\varrho ^{-n}(\varrho
^2-t^2)^{n-1/2}\cos n\varphi ,  \label{eq:6p301}
\end{equation}
as the right-hand side of the equation $L_1w=g$, Theorem \ref{thm6.1} gives a
singular solution $w_{n}$ satisfying \eqref{eq:5p3}; that is,
\begin{equation}
|w_{n}(\varrho ,\varphi ,\varrho )|\geq \frac{1}{2}|w_{n}(2\varrho ,\varphi
,0)|+\varrho ^{-n}|\cos n\varphi |\geq \varrho ^{-n}|\cos n\varphi |,\
0<\varrho <1.
\end{equation}
Now, the inverse transform $u_{n}=\exp \left\{ \Lambda (\varrho +t)\right\}
w_{n}$ gives
\begin{equation}
|u_{n}(\varrho ,\varphi ,\varrho )|\geq \frac{1}{2}|u_{n}(2\varrho ,\varphi
,0)|+\varrho ^{-n}|\cos n\varphi |\geq \varrho ^{-n}|\cos n\varphi |,\
0<\varrho <1,
\end{equation}
where the function $u_{n}(\varrho ,\varphi ,t)$ is a solution of the problem %
\eqref{eq:5p101}, \eqref{eq:5p2} with
\begin{equation*}
f(\varrho ,\varphi ,t)=\exp \left\{ \Lambda (\varrho +t)\right\} \text{ }%
\varrho ^{-n}(\varrho ^2-t^2)^{n-1/2}\cos n\varphi .\text{ }
\end{equation*}
The proof is complete. \hfill$\square$\smallskip

Next, we find singular solutions for the original Problem $P_{\alpha}$,
formulated in Section 1.

\noindent\textbf{Proof of Theorem \ref{thm6.3}}
Recall that we are looking for a suitable right-hand side function $f_{n}$
of  \eqref{eq:0p1} for which singular solutions exist. Set
\begin{equation}
u(x_1,x_2,t)=\exp \left\{ (bt-b_1x_1-b_2x_2)/2\right\}
v(x_1,x_2,t)\,.  \label{eq:6p305}
\end{equation}
Then equation \eqref{eq:1p1} becomes
\begin{equation}
v_{x_1x_1}+v_{x_2x_2}-v_{tt}+c_1v=h\equiv \exp \left\{
(b_1x_1+b_2x_2-bt)/2\right\} f\text{\quad in }\Omega_0,\text{\ }
\label{eq:6p302}
\end{equation}
where $c_1:=c+(b^2-b_1^2-b_2^2)/4$. In order to apply Theorem \ref{thm6.2},
we rewrite \eqref{eq:6p302} in polar coordinates and obtain
the problem $P_{\gamma }$:
\begin{equation*}
\begin{gathered}
\frac{1}{\varrho }(\varrho v_{\varrho })_{\varrho }+\frac{1}{\varrho ^2}
v_{\varphi \varphi }-v_{tt}+c_1v=h(\varrho ,\varphi ,t)\quad\text{in }
\Omega_0, \\
v\bigr\rvert_{\Sigma_1}=0,\quad \lbrack v_{t}+\gamma (\varrho )v]%
\bigr\rvert_{\Sigma_0\backslash O}=0
\end{gathered}
\end{equation*}
with $\gamma (\varrho ):=\alpha (\varrho )+b/2$. In order to apply
Theorem \ref{thm6.2} to the Problem $P_{\gamma }$, the only condition we
need is $c_1\leq 0, $ i.e. $c+(b^2-b_1^2-b_2^2)/4\leq 0$, which is
satisfied. If we now choose
\begin{equation*}
h_{n}(\varrho ,\varphi ,t)=\exp \left\{ \Lambda (\varrho
+t)\right\} \varrho ^{-n}(\varrho ^2-t^2)^{n-1/2}\cos n\varphi
\end{equation*}
according to Theorem \ref{thm6.2} with the constant $\Lambda $ large enough, then the
Problem $P_{\gamma }$ has a corresponding singular solution $v_{n}\in C^2(%
\bar{\Omega}_0\backslash O)$ with the estimate
\begin{equation*}
|v_{n}(\varrho ,\varphi ,\varrho )|\geq \frac{1}{2}|v_{n}(2\varrho ,\varphi
,0)|+\varrho ^{-n}|\cos n\varphi |,\quad 0<\varrho <1.
\end{equation*}
Now the inverse transform of \eqref{eq:6p305} gives a singular \textsl{%
generalized solution} $u_{n}\in C^2(\bar{\Omega}_0\backslash O)$
of the Problem $P_{\alpha }$ with the right-hand side
\begin{align*}
f_{n}=&\exp \left\{ (bt-b_1x_1-b_2x_2)/2+\Lambda (\varrho
+t)\right\}\\
&\times |x|^{-n}(x_1^2+x_2^2-t^2)^{n-1/2}\cos n(\arctan \frac{x_2}{x_1}).
\end{align*}
The proof is complete. \hfill$\square$

\begin{remark} \label{rmk6.1}
Aldashev in \cite{Ald98} considered \eqref{eq:1p1} and studied
the \textit{homogeneous} Problems $P_{\alpha }$ and $P_{\alpha }^{\ast }$.
Unfortunately, as it is easy to check, the procedure which he follows leads
to a correct conclusion only in the case of the wave equation, i.e. only in
the case where all the lower order terms in \eqref{eq:1p1} are identically
zero. Otherwise, this procedure leads to systems of differential equations
which are not equivalent to those which should be
solved (see \eqref{eq:1p9}). This is due to the fact that, in the systems obtained in \cite{Ald98}
by integration with respect to $\varphi $, the Fourier coefficients $u_k$
of degree $k$ depend on the coefficients $u_{k-1}$ of degree $k-1$.
\end{remark}

\section{Applications to the wave equation, singular solutions}

In this section we consider the wave equation
\begin{equation}
\square u=u_{x_1x_1}+u_{x_2x_2}-u_{tt}=f(x_1,x_2,t) \label{eq:6p1}
\end{equation}
subject to the boundary-value problem $P_{\alpha }$, i.e.
\begin{equation}
\square u=f\text{ in\ $\Omega_0$,}\quad u\bigr\rvert%
_{\Sigma_1}=0,\quad \lbrack u_{t}+\alpha (|x|)u]\bigr\rvert_{\Sigma
_0\backslash O}=0.  \label{eq:6p2}
\end{equation}
As an application to the wave equation of the results of the previous
section, we have the following statement.

\begin{theorem} \label{thm7.1}
 Let $\alpha \in C^{\infty }(0,1]$ $\cap C[0,1]$ be an arbitrary
function. Then: \begin{itemize}
\item[(i)] For each $n\in \mathbb{N}$, $n\geq 4$, there exists
a function\ $f_{n}\in C^{n-2}({\bar{\Omega}}_0)\cap C^{\infty }(\Omega
_0)$, \ for which the corresponding generalized solution $u_{n}$ of the
problem $P_{\alpha }$ belongs to $C^{n}({\bar{\Omega}}_0\backslash O)$ and
satisfies the estimate
\begin{equation}
|u_{n}(x_1,x_2,|x|)|\geq \frac{1}{2}|u_{n}(2x_1,2x_2,0)|+|x|^{-n}|%
\cos n(\arctan \frac{x_2}{x_1})|.  \label{eq:6p3}
\end{equation}
\item[(ii)] In the case $\alpha (\varrho )\leq 0$ an upper estimate of
the singular solution $u_{n}$ is
\begin{equation}
|u_{n}(x_1,x_2,t)|\leq C_{\mu }|x|^{-1/2}\Big( \frac{|x|}{%
x_1^2+x_2^2-t^2}\Big) ^{n-\frac{1}{2}}|\cos n(\arctan \frac{x_2%
}{x_1})|,\quad (|x|,t)\in D_1^{\mu },  \label{eq:6p4a}
\end{equation}
where $C_{\mu }$ is a constant. and
\begin{equation*}
D_1^{\mu }:=\left\{ (\varrho ,t):0<\varrho -t\leq \varrho +t\leq \mu
(\varrho -t)\right\} ,\mu <2^{\frac{2n+1}{2n-1}}-1.
\end{equation*}
\end{itemize}
Thus, for\ $\alpha (\varrho )\leq 0$ we have two-sided estimates, which in
the limit cases $t=|x|$ and $t=0$ are:
\begin{equation}
\begin{gathered}
|x|^{-n}|\cos n(\arctan \frac{x_2}{x_1})|\leq |u_{n}(x_1,x_2,|x|)|, \\
|u_{n}(x_1,x_2,0)|\leq C|x|^{-n}|\cos n(\arctan \frac{x_2}{x_1})|,
\end{gathered} \label{eq:6p4}
\end{equation}
with $C$ a constant. That is, in the case of $\alpha (\varrho )\leq 0$
the exact behavior of $u_{n}(x_1,x_2,t)$ around $O$ is
$(x_1^2+x_2^2+t^2)^{-n/2}\cos n(\arctan \frac{x_2}{x_1})$.
\end{theorem}

\noindent\textbf{Proof.}
 \textbf{(i)} Note that, the wave equation \eqref{eq:6p1} is of the
form \eqref{eq:5p101} and so the first part of Theorem \ref{thm7.1} follows from
Theorem \ref{thm6.2} and \cite{GHP}. Actually, according to this theorem we choose
the function $f_{n}$ to be of the special form
\begin{equation}
\square u=f_{n}=\exp \left\{ \Lambda (\varrho +t)\right\} \text{
}\varrho ^{-n}(\varrho ^2-t^2)^{n-1/2}\cos n\varphi \quad\text{in }
\Omega _0, \label{eq:6p85a}
\end{equation}
where $\Lambda >0$ is large enough and such that $\Lambda +\alpha (\varrho
)\geq 0,\varrho \in \lbrack 0,1]$. Then by Theorems \ref{thm5.1} and
\ref{thm5.2} there exists
a unique generalized solution $u_{n}(\varrho ,\varphi ,t)$ of the equation
\eqref{eq:6p85a}, satisfying the boundary conditions \eqref{eq:6p2} and the
estimates \eqref{eq:6p3} (see Theorem \ref{thm6.2}). On the other hand,
by \cite[Theorem 5.2]{GHP}, for the equation \eqref{eq:6p85a} there exists a generalized
solution in $\Omega_0$ of the form
\begin{equation*}
u_{n}(\varrho ,\varphi ,t)=u_{n}^{(1)}(\varrho ,t)\cos n\varphi \in C^{n}({%
\bar{\Omega}}_0\backslash O),
\end{equation*}
which is a classical solution of Problem $P_{\alpha }$ in $\Omega
_{\varepsilon }$, $\varepsilon \in (0,1)$.
\\
\textbf{(ii)} By setting $u_{n}^{(2)}(\varrho ,t)=\varrho ^{\frac{1}{2}%
}u_{n}^{(1)}(\varrho ,t)$ and substituting
\begin{equation}
\xi =1-\varrho -t,\quad \eta =1-\varrho +t,  \label{eq:6p86}
\end{equation}
the problem \eqref{eq:6p85a}, \eqref{eq:6p2}, in view of
\begin{equation}
U_{n}(\xi ,\eta )=u_{n}^{(2)}(\varrho (\xi ,\eta ),t(\xi ,\eta )),
\label{eq:6p87}
\end{equation}
becomes a Darboux-Goursat problem $P_{\alpha ,3}:$%
\begin{gather}
U_{n,\xi \eta }-C(\xi ,\eta )U_{n}=G(\xi ,\eta )\equiv \exp \left\{ \Lambda
(1-\xi )\right\} F(\xi ,\eta ),  \label{eq:6p88}
\\
U_{n}(0,\eta )=0,\quad (U_{n,\eta }-U_{n,\xi })(\xi ,\xi )+\alpha (1-\xi
)U_{n}(\xi ,\xi )=0.  \label{eq:7p201}
\end{gather}
Here, the coefficients
\begin{equation}
C(\xi ,\eta )=\frac{4n^2-1}{4(2-\eta -\xi )^2}\in C^{\infty }({\bar{D}}
_{\varepsilon }^{(1)}),\quad n\geq 4,  \label{eq:6p89}
\end{equation}
and
\begin{equation}
F(\xi ,\eta )=2^{n-\frac{5}{2}}\Big[ \frac{(1-\xi )(1-\eta )}{2-\eta -\xi }
\Big] ^{n-\frac{1}{2}}\in C^{n-1}({\bar{D}}_{\varepsilon }^{(1)})
\label{eq:6p90}
\end{equation}
are defined by \eqref{eq:1p101} and \eqref{eq:1p19}. Now, we need some
information about the behavior of the function $U_{n}(\xi ,\eta )$. Since,
by Theorem \ref{thm6.2},
\begin{equation*}
U_{n}(\xi ,\eta )=\exp \left\{ \Lambda (\varrho +t)\right\} \ W(\xi ,\eta
)=\exp \left\{ \Lambda (1-\xi )\right\} \ W(\xi ,\eta ),
\end{equation*}
$W(\xi ,\eta )\geq 0$ and $W_{\eta }(\xi ,\eta )\geq 0$\ in $\bar{D}%
_{\varepsilon }^{(1)}$, in view of Proposition \ref{prop6.2}, we formulate the
following result.

\begin{proposition} \label{prop7.1}
There exists a classical solution
$U_{n}(\xi ,\eta )\in C^{n}(\bar{D}_0\setminus (1,1))$
for \eqref{eq:6p88}, \eqref{eq:7p201} for which
\begin{equation}
U_{n}(\xi ,\eta )\geq 0,\quad U_{n,\eta }(\xi,\eta )\geq 0,
\quad (\xi ,\eta )\in \bar{D}_{\varepsilon }^{(1)}.
\label{eq:6p93}
\end{equation}
\end{proposition}

Put
\begin{equation}
K_1=\int_{D_0}G^2(\xi ,\eta )\,d\xi \,d\eta >0.  \label{eq:6p90a}
\end{equation}
Then, by \eqref{eq:6p88}, for $0<\varepsilon <1$ it follows that
\begin{equation}
\begin{aligned}
K_1&\geq \int_{D_{\varepsilon }^{(1)}}G^2(\xi ,\eta )\,d\xi \,d\eta
\geq \int_{D_{\varepsilon }^{(1)}}G(\xi ,\eta )\,F(\xi ,\eta
)\,d\xi\,d\eta \\
&=\int_{D_{\varepsilon }^{(1)}}U_{n,\xi \eta }F(\xi ,\eta )\,d\xi
\,d\eta -\int_{D_{\varepsilon }^{(1)}}C(\xi ,\eta )U_{n}(\xi ,\eta
)F(\xi ,\eta )\,d\xi \,d\eta \\
&=:I_1-I_2.
\end{aligned} \label{eq:6p91}
\end{equation}
Then
\begin{equation}
\begin{aligned}
I_1=&\int_0^{1-\varepsilon }\int_{\xi }^{1}(U_{n,\xi \eta
}F)(\xi ,\eta )\,d\eta \,d\xi \\
=&-\int_0^{1-\varepsilon }U_{n,\xi }(\xi ,\xi )F(\xi
,\xi )\,d\xi -\int_{D_{\varepsilon }^{(1)}}(U_{n,\xi }F_{\eta
})(\xi ,\eta )\,d\xi \,d\eta ,
\end{aligned}\label{eq:6p12}
\end{equation}
and, by \eqref{eq:6p90}, $F(\xi ,1)=0$. Since
\begin{equation}
\begin{aligned}
&\int_{D_{\varepsilon }^{(1)}}(U_{n,\xi }F_{\eta })(\xi ,\eta)\,d\xi \,d\eta \\
&=\int_0^{1-\varepsilon }(U_{n}F_{\eta })(\eta ,\eta )\,d\eta
+\int_{1-\varepsilon }^{1}(U_{n}F_{\eta })(1-\varepsilon ,\eta
)\,d\eta -\int_{{D}_{\varepsilon }^{(1)}}(U_{n}F_{\xi \eta })(\xi ,\eta
)\,d\xi \,d\eta ,
\end{aligned}
\end{equation}
equation \eqref{eq:6p12} becomes
\begin{equation}
\begin{split}
I_1& =-\int_0^{1-\varepsilon }[U_{n,\xi }(\xi ,\xi )F(\xi ,\xi
)+U_{n}(\xi ,\xi )F_{\eta }(\xi ,\xi )]\,d\xi \\
& \quad -\int_{1-\varepsilon }^{1}(U_{n}F_{\eta })(1-\varepsilon ,\eta
)\,d\eta +\int_{D_{\varepsilon }^{(1)}}(U_{n}F_{\xi \eta })(\xi ,\eta
)\,d\xi \,d\eta .
\end{split} \label{eq:6p13}
\end{equation}
 From \eqref{eq:6p13} and \eqref{eq:6p91} it follows that
\begin{equation}
\begin{split}
K_1\geq& I_1-I_2=-\int_0^{1-\varepsilon }[U_{n,\xi }(\xi
,\xi )F(\xi ,\xi )+U_{n}(\xi ,\xi )F_{\xi }(\xi ,\xi )]\,d\xi \\
&-\int_{1-\varepsilon }^{1}(U_{n}F_{\eta })(1-\varepsilon ,\eta
)\,d\eta +\int_{{D}_{\varepsilon }^{(1)}}U_{n}[F_{\xi \eta }-CF](\xi ,\eta
)\,d\xi \,d\eta .
\end{split}\label{eq:6p14}
\end{equation}
Because of \eqref{eq:5p92}, the last integral vanishes. Thus, using the
boundary conditions for the functions $U_{n}$ and $F$, when $\eta =\xi $, we
see that
\begin{equation}
\begin{aligned}
K_1&\geq I_1-I_2\\
&=-\int_0^{1-\varepsilon }[U_{n,\xi }(\xi ,\xi)F(\xi,\xi )
+U_{n}(\xi ,\xi )F_{\xi }(\xi ,\xi )]\,d\xi
 -\int_{1-\varepsilon }^{1}(U_{n}F_{\eta })(1-\varepsilon
,\eta )\,d\eta \\
&=-\frac{1}{2}(FU_{n})(1-\varepsilon ,1-\varepsilon )
-\frac{1}{2}\int_0^{1-\varepsilon }\alpha (1-\xi
)U_{n}(\xi ,\xi )F(\xi ,\xi )\,d\xi \\
&\quad -\int_{1-\varepsilon
}^{1}(U_{n}F_{\eta })(1-\varepsilon ,\eta )\,d\eta .
\end{aligned} \label{eq:6p22}
\end{equation}
Since $\alpha (\xi )$, \quad $F_{\eta }\leq 0$ and
$F,U_{n},U_{n,\eta }\geq 0$, for $0<\delta <\varepsilon <1$, we have
\begin{equation}
\begin{split}
K_1& \geq I_1-I_2 \\
& \geq -\frac{1}{2}(U_{n}F)(1-\varepsilon ,1-\varepsilon
)+\int_{1-\varepsilon }^{1}U_{n}(1-\varepsilon ,\eta )\lvert F_{\eta
}(1-\varepsilon ,\eta )\rvert \,d\eta \\
& \geq -\frac{1}{2}(U_{n}F)(1-\varepsilon ,1-\varepsilon )+\int_{1-\delta
}^{1}U_{n}(1-\varepsilon ,\eta )\lvert F_{\eta }(1-\varepsilon ,\eta )\rvert
\,d\eta \\
& \geq -\frac{1}{2}(U_{n}F)(1-\varepsilon ,1-\varepsilon )+\int_{1-\delta
}^{1}U_{n}(1-\varepsilon ,1-\delta )\lvert F_{\eta }(1-\varepsilon ,\eta
)\rvert \,d\eta \\
& \geq -\frac{1}{2}(U_{n}F)(1-\varepsilon ,1-\varepsilon
)+(U_{n}F)(1-\varepsilon ,1-\delta ) \\
& \geq U_{n}(1-\varepsilon ,1-\delta )\left[ F(1-\varepsilon ,1-\delta )-%
\frac{1}{2}F(1-\varepsilon ,1-\varepsilon )\right] \\
& \geq \nu (U_{n}F)(1-\varepsilon ,1-\delta ),
\end{split} \label{eq:6p27}
\end{equation}
provided taht the constant $v>0$ satisfies
\begin{equation}
2(1-\nu )F(1-\varepsilon ,1-\delta )\geq F(1-\varepsilon ,1-\varepsilon ).
\label{eq:6p50}
\end{equation}
Using the explicit formula \eqref{eq:6p90} for the function
$F(\xi ,\eta )$, we see that the above inequality is equivalent to
\begin{equation}
2(1-\nu )\big( \frac{\delta }{\varepsilon +\delta }\big) ^{n-\frac{1}{2}%
}\geq 2^{-n+\frac{1}{2}},  \label{eq:6p51}
\end{equation}
which implies
\begin{equation}
0<\nu \leq 1-\frac{1}{2}\big( \frac{\varepsilon +\delta }{2\delta }\big)
^{n-\frac{1}{2}}.  \label{eq:6p52}
\end{equation}
A necessary condition, for \eqref{eq:6p52} to be satisfied is that
\begin{equation}
1\leq \frac{\varepsilon }{\delta }<2^{\frac{2n+1}{2n-1}}-1.  \label{eq:6p53}
\end{equation}
In this concrete case, using \eqref{eq:6p53}, we can find an upper estimate
for the generalized solution $u_{n}$. To do this, we consider the domain
\begin{equation}
D^{\mu }:=\{(\xi ,\eta ):1-\eta \leq 1-\xi \leq \mu (1-\eta )\},
\label{eq:6p54}
\end{equation}
where $1\leq \mu <2^{\frac{2n+1}{2n-1}}-1$. Observe that
\begin{equation*}
\inf_{D^{\mu }}\Big\{ 1-\frac{1}{2}\Big( \frac{1-\xi +1-\eta }{2(1-\eta )}%
\Big) ^{n-\frac{1}{2}}\Big\} =1-\frac{1}{2}\Big( \frac{1+\mu }{2}%
\Big) ^{n-\frac{1}{2}}=:C_{\mu }>0.
\end{equation*}
For $\nu =C_{\mu }$ the inequalities \eqref{eq:6p51} and \eqref{eq:6p50} are
satisfied and so, by \eqref{eq:6p27}, we see that
\begin{equation}
U(\xi ,\eta )\leq 2^{-n+5/2}K_1C_{\mu }^{-1}\Big( \frac{2-\xi -\eta }{%
(1-\xi )(1-\eta )}\Big) ^{n-\frac{1}{2}},\quad (\xi ,\eta )\in D^{\mu }.
\label{eq:6p54a}
\end{equation}
By \eqref{eq:5p7} and \eqref{eq:5p6}, the inequality \eqref{eq:6p54a}
transforms to
\begin{equation}
u_{n}^{(2)}(\varrho ,t)\leq 4K_1C_{\mu }^{-1}\Big( \frac{\varrho }{
\varrho ^2-t^2}\Big) ^{n-\frac{1}{2}},  \label{eq:6p55}
\end{equation}
which is satisfied for
$(\varrho ,t)\in D_1^{\mu }:=\big\{ 0<\varrho -t\leq \varrho +t\leq \mu
(\varrho -t)\big\}$.
Finally, \eqref{eq:6p55} implies
\begin{equation}
u_{n}^{(1)}(\varrho ,t)\leq 4K_1C_{\mu }^{-1}\varrho ^{-1/2}
\Big( \frac{\varrho }{\varrho ^2-t^2}\Big) ^{n-\frac{1}{2}}
\quad\text{for }(\varrho ,t)\in D_1^{\mu },  \label{eq:6p56}
\end{equation}
which coincides with the estimate \eqref{eq:6p4a}.
Note that $C_{\mu }=1/2$ on $\{ t=0\} $ and so
\begin{equation}
u_{n}^{(1)}(\varrho ,0)\leq 8K_1\varrho ^{-n},\quad 0<\varrho <1,
\label{eq:6p57}
\end{equation}
which is the upper estimate in \eqref{eq:6p4}. The proof of Theorem
\ref{thm7.1} is complete.\hfill $\square$

\begin{remark} \label{rmk7.1} \rm
Since in Theorems \ref{thm6.1}, \ref{thm6.2} and \ref{thm7.1} the conditions imposed upon lower terms
of \eqref{eq:5p1} are not invariant with respect to substitution of the
independent variables
\begin{equation}
v(\varrho ,\varphi ,t)=u(\varrho ,\varphi ,t)\ \exp \lambda (\varrho ,t),
\label{eq:6p1000}
\end{equation}
for various functions $\lambda (\varrho ,t)$, we can find a series of
singular solutions of Problem $P_{\alpha }$ for different classes of
equations of the form \eqref{eq:5p1}. This procedure is interesting by
itself and is demonstrated by the following simple example
\end{remark}

\noindent\textbf{Example.}
Consider the special form of the equation \eqref{eq:5p1}, with constant
lower order terms, that is
\begin{equation}
Lu\equiv
u_{x_1x_1}+u_{x_2x_2}-u_{tt}+b_1u_{x_1}+b_2u_{x_2}+bu_{t}+%
\frac{1}{4}(b_1^2+b_2^2-b^2)u=f,\text{ in }\Omega_0,
\label{eq:6p1a}
\end{equation}
with the boundary conditions \eqref{eq:5p2}. Obviously, the equation
\eqref{eq:6p1a} satisfies the conditions of Theorems \ref{thm6.3} for
$\alpha \in C^{1}([0,1])$ and we obtain a singular solution $u_{n}$
By using the transform \eqref{eq:6p305}, the equation
\eqref{eq:6p1a} becomes the wave equation \eqref{eq:6p1}.
Then Theorem \ref{thm7.1} becomes useful and in the case, when
$\alpha (|x|)\leq -b/2$, in addition, we have two-sided estimates
\eqref{eq:6p4} of the
generalized solution $u_{n}(x_1,x_2,t)$, whose exact
behavior around the point $O\ $ is
$(x_1^2+x_2^2+t^2)^{-n/2}\cos n(\arctan \frac{x_2}{x_1})$.
\smallskip

\noindent \textbf{Acknowledgments.}
The authors would like to thank the anonymous referee for
making several helpful suggestions for improving the presentation of this
paper.

The research of Hristov was partially supported by the Bulgarian NSC
under Grant MM-904/99. The essential part of the present work was finished
and prepublished in \cite{GHPTR}, while Popivanov was visiting the
University of Ioannina during 2001. Popivanov would like to thank the
Ministry of National Economy of Helenic Republic for providing the NATO
Science Fellowship (Ref. No.169/DOP/01) and the University of Ioannina for
its hospitality




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\end{document}