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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 281, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/281\hfil 
Wave equations with data on the whole boundary]
{Boundary-value problems for  wave equations with
data on the whole boundary}

\author[M. A. Sadybekov, N. A. Yessirkegenov \hfil EJDE-2016/281\hfilneg]
{Makhmud A. Sadybekov, Nurgissa A. Yessirkegenov}

\address{Makhmud A. Sadybekov \newline
Institute of Mathematics and Mathematical Modeling,
125 Pushkin str., 050010 Almaty, Kazakhstan}
\email{sadybekov@math.kz}

\address{Nurgissa A. Yessirkegenov \newline
Institute of Mathematics and Mathematical Modeling,
125 Pushkin str., 050010 Almaty, Kazakhstan. \newline
Department of Mathematics, Imperial College London,
180 Queen's Gate, SW7 2AZ London, United Kingdom}
\email{n.yessirkegenov15@imperial.ac.uk}

\thanks{Submitted May 12, 2016. Published October 19, 2016.}
\subjclass[2010]{35L05, 35L20, 49K40, 35D35}
\keywords{Wave equation; well-posedness of problems; classical solution;
\hfill\break\indent strong solution; d'Alembert's formula}

\begin{abstract}
 In this article we propose a new formulation of boundary-value problem for a
 one-dimensional wave equation in a rectangular domain in which boundary
 conditions are given on the whole boundary. We prove the well-posedness of
 boundary-value problem in the classical and generalized senses.
 To substantiate the well-posedness of this problem it is
 necessary to have an effective representation of the general
 solution of the problem. In this direction we obtain a convenient
 representation of the general solution for the wave equation in a
 rectangular domain based on d'Alembert classical formula. The
 constructed general solution automatically satisfies the boundary
 conditions by a spatial variable. Further, by setting different boundary
 conditions according to temporary variable, we get some functional
 or functional-differential equations. Thus, the proof of the
 well-posedness of the formulated problem is reduced to question of the
 existence and uniqueness of solutions of the corresponding
 functional equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction}

Let $\Omega\subset \mathbb{R}^{2}$ be a rectangular domain, bounded by
following lines: 
$AB: t=0,\; 0\leq x\leq \ell$;
$BC: x=\ell,\; 0\leq t\leq T$;
$CD: t=T,\; 0\leq x\leq \ell$;
and $AD: x=0,\; 0\leq t \leq T$.

We consider a nonhomogeneous wave equation in $\Omega$,
\begin{equation}
u_{tt}-u_{xx}= f(x,t). \label{e1.1}
\end{equation}

It is well known that the Dirichlet problem for the wave equation
\eqref{e1.1} in a rectangular domain is ill-posed \cite{h1}.
Specifically, in case of our domain $\Omega$ it is easy to see
that the homogeneous equation \eqref{e1.1} with Dirichlet
conditions
\begin{gather}
u\big|_{AB\cup BC \cup AD}=0, \label{e1.2} \\
u\big|_{CD}=0 \label{e1.3}
\end{gather}
has countable number of nontrivial solutions of the form
$u_{mn}(x,t)=\sin\frac{m\pi x}{\ell} \sin\frac{n\pi t}{T}$,
 $m,n=1,2,\dots$, when the conditions $n \ell=m T$ hold.

The Dirichlet problem for a wave equation is one of the most
difficult models of mathematical physics. The wave equation
describes almost all types of small vibrations in distributional
mechanical systems such as longitudinal sound vibrations in gas,
fluid, solids; transverse waves in strings and etc. Components of
electromagnetic vectors and potentials, and hence many
electromagnetic phenomena (from quasistatics to optics) in some
extent are explained by properties of solutions of wave equation.

Hadamard \cite{h2}, Huber \cite{hu1} for the first time noted
nonuniqueness of solution of the Dirichlet problem for a wave
equation. Bourgin and  Duffin \cite{b1} considered Dirichlet
problem for the one-dimensional equation \eqref{e1.1} in a
rectangle $\{0\leq t \leq T,\; 0\leq x \leq \ell\}$. By using
Laplace transformation, they showed that if the number $T/\ell$ is
irrational, then there is the uniqueness of the solution of the
problem in the class of continuously differentiable functions with
the second derivatives integrable according to Lebesgue.

There are many works that were dedicated to study Dirichlet problem for 
the string equation (see \cite{s01}).
Arnold's survey \cite{a1} and Berezanskii's \cite[Chap. IV]{b01} monograph 
give more detailed discussion of papers related to this topic.
These papers show that the homogeneous Dirichlet problem has nontrivial solutions, 
if the ratio $T/\ell$ of the sides of the rectangle 
$\{0\leq t \leq T,\; 0\leq x \leq \ell\}$ (in which the solution of the
 Dirichlet problem for the string equation is sought) is a rational number.
By using the method of separation of variables, the solution of 
inhomogeneous Dirichlet problem is constructed. In this process, small 
denominators which are hampering the series' convergence arise \cite{s1}.
If the ratio $T/\ell$ of the sides is an algebraic number of degree $n\geq2$ 
or an irrational number with bounded element, then, for sufficiently smooth 
boundary data (functions), the constructed series' convergence can be 
proved for the class of smooth solutions of the string equation.

In \cite{p1} the existence and uniqueness of generalized solution
for a second-order hyperbolic equation with integral conditions in
a rectangle are proved.

In \cite{k1} the uniqueness of solution of initial-boundary value
problem for a one-dimensional wave equation is proved and it is
shown that this solution coincides with the wave potential.

In \cite{sy1}-\cite{y1} it is proved the well-posedness of
boundary value problems for a one-dimensional wave equation in a
rectangular domain in case when boundary conditions are given on
the whole boundary of domain.

Also we note that lately interest has increased to the research of
classical initial-boundary problems for a wave equation in
rectangular domains in connection with problems of the
optimization of boundary control of string vibrations (see
\cite{i1,m1,m2}).

In this article, we prove the well-posedness of the problem for a
one-dimensional wave equation in a rectangular domain in case when
boundary conditions are given on the whole boundary of domain
which generalizes results of \cite{sy1}-\cite{y1}.

\section{Representation of solution of the first initial-boundary value problem}

Hereafter, we will assume that $\ell/T\geq2$.

\subsection*{Problem 1} Find a solution of equation \eqref{e1.1} 
in $\Omega$ with the initial conditions
\begin{gather}
u(x,0)=\tau(x), \quad 0\leq x\leq \ell, \label{e2.1} \\
u_{t}(x,0)=\nu(x), \quad 0\leq x\leq \ell \label{e2.2}
\end{gather}
and with boundary conditions
\begin{equation}
u(0,t)=0, \quad u(\ell,t)=0, \quad 0\leq t\leq \frac{\ell}{2}. \label{e2.3}
\end{equation}

Problem 1 is a classical first initial-boundary value problem.
The solution of the Cauchy problem for  \eqref{e1.1} with
initial conditions \eqref{e2.1} and \eqref{e2.2} exists and is
unique. But it is uniquely defined not in all $\Omega$, but only
in its part $\Omega_1=\{(x,t):(x,t)\in \Omega, t\leq x \leq
\ell-t\}$. And in the domain $\Omega \setminus\Omega_1$ the
solution is not uniquely defined from the data of Cauchy
\eqref{e2.1}, \eqref{e2.2}. It is uniquely defined only by using
boundary conditions of considered problems.

Let $u(x,t)$ be a solution of Problem 1. We introduce a new
function $\widetilde{u}(x,t)$ defined in $\widetilde{\Omega}$,
containing initial domain $\Omega$:
$\widetilde{\Omega}=\{(x,t):0\leq t\leq \frac{\ell}{2},
t-\frac{\ell}{2}\leq x \leq \frac{3 \ell}{2}-t\}$.

The function $\widetilde{u}(x,t)$ is given by the formula
\begin{equation}
\tilde{u}(x,t)=
\begin{cases}
-u(-x,t), & -\frac{\ell}{2}\leq x \leq 0;\\
u(x,t), & 0\leq x \leq \ell;\\
-u(2 \ell-x,t), & \ell\leq x \leq \frac{3 \ell}{2}.
\end{cases} \label{e2.4}
\end{equation}

Taking into account the boundary condition \eqref{e2.3}, it is
easy to see that the function $\widetilde{u}(x,t)$ is continuous
and continuously differentiable at the transition lines $x=0$ and
$x=\ell$. Since the function $u(x,t)$ is smooth in $\Omega$, then
the function $\widetilde{u}(x,t)$ is smooth in
$\widetilde{\Omega}$.

Let us find an equation in which the function $\widetilde{u}(x,t)$
satisfies that equation in $\widetilde{\Omega}$. By direct
calculation, it is easy to see that this function satisfies the
following nonhomogeneous wave equation in $\widetilde{\Omega}$
\begin{equation}
\widetilde{u}_{tt}-\widetilde{u}_{xx}=\widetilde{f}(x,t), \label{e2.5}
\end{equation}
where
\begin{equation}  
\widetilde{f}(x,t)=
\begin{cases}
-f(-x,t), & -\frac{\ell}{2}\leq x \leq 0;\\
f(x,t), & 0\leq x \leq \ell;\\
-f(2 \ell-x,t), & \ell\leq x \leq \frac{3 \ell}{2}.
\end{cases} \label{e2.6}
\end{equation}

From the initial conditions \eqref{e2.1}, \eqref{e2.2}, taking
into account \eqref{e2.4}, we obtain initial conditions for the
function $\widetilde{u}(x,t)$ in $\widetilde{\Omega}$:
\begin{gather}
\widetilde{u}(x,0)=\widetilde{\tau}(x), ~0\leq x\leq \ell, \label{e2.7} \\
\widetilde{u}_{t}(x,0)=\widetilde{\nu}(x), ~0\leq x\leq \ell, \label{e2.8}
\end{gather}
where functions $\widetilde{\tau}(x)$ and $\widetilde{\nu}(x)$ are
given by the  equalities
\begin{equation}
\widetilde{\tau}(x)=
\begin{cases}
-\tau(-x), & -\frac{\ell}{2}\leq x \leq 0;\\
\tau(x), & 0\leq x \leq \ell;\\
-\tau(2 \ell-x), & \ell \leq x \leq \frac{3 \ell}{2}.
\end{cases} \label{e2.9}
\end{equation}
\begin{equation}
\widetilde{\nu}(x)=
\begin{cases}
-\nu(-x), & -\frac{\ell}{2}\leq x \leq 0;\\
\nu(x), & 0\leq x \leq \ell;\\
-\nu(2 \ell-x), & \ell \leq x \leq \frac{3 \ell}{2}.
\end{cases} \label{e2.10}
\end{equation}

In $\widetilde{\Omega}$ the solution of the Cauchy problem \eqref{e2.5}, 
\eqref{e2.7}, \eqref{e2.8} exists, is unique and expressed by the classical 
formula of d'Alambert:
\begin{equation}
\widetilde{u}(x,t)=\frac{\widetilde{\tau}(x+t)+\widetilde{\tau}(x-t)}{2}+
\frac{1}{2}\int_{x-t}^{x+t}\widetilde{\nu}(\xi)d\xi+\frac{1}{2}
\int_{0}^{t}\int_{x-t+\eta}^{x+t-\eta}\widetilde{f}
(\xi,\eta)d\xi\,d\eta. \label{e2.11}
\end{equation}
By direct calculation, it is easy to check that the function 
$\widetilde{u}(x,t)$ satisfies equation \eqref{e2.5} and the initial 
conditions \eqref{e2.7} and \eqref{e2.8}.

Now we show that by \eqref{e2.9}, \eqref{e2.10}, and
taking into account \eqref{e2.6}, the function
$\widetilde{u}(x,t)$ satisfies the boundary condition \eqref{e2.3}
of Problem 1.

We calculate
\begin{equation}
\widetilde{u}(0,t)=\frac{\widetilde{\tau}(t)+\widetilde{\tau}(-t)}{2}+
\frac{1}{2}\int_{-t}^{t}\widetilde{\nu}(\xi)d\xi+\frac{1}{2}
\int_{0}^{t}\int_{-t+\eta}^{t-\eta}\widetilde{f}(\xi,\eta)d\xi\,d\eta. \label{e2.12}
\end{equation}
By  \eqref{e2.9} it is easy to obtain
\begin{equation}
\frac{\widetilde{\tau}(t)+\widetilde{\tau}(-t)}{2}
=\frac{\tau(t)-\tau(t)}{2}=0. \label{e2.13}
\end{equation}

From \eqref{e2.10} by a simple change of variables in the integral we obtain
\begin{equation}
\begin{aligned}
\frac{1}{2}\int_{-t}^{t}\widetilde{\nu}(\xi)d\xi
& =\frac{1}{2}\int_{-t}^{0}\widetilde{\nu}(\xi)d\xi+
\frac{1}{2}\int_{0}^{t}\widetilde{\nu}(\xi)d\xi \\
&=\frac{1}{2}\int_{t}^{0}\nu(\xi)d\xi
 +\frac{1}{2}\int_{0}^{t}\nu(\xi)d\xi=0. 
\end{aligned} \label{e2.14}
\end{equation}
In the third summand in \eqref{e2.12} we shall make obvious change of variables.
Since $0\leq t-\eta \leq \frac{\ell}{2}, -\frac{\ell}{2}\leq \eta-t\leq 0$, 
then we obtain
\begin{equation}
\begin{aligned}
&\frac{1}{2}\int_{0}^{t}\int_{-t+\eta}^{t-\eta}
 \widetilde{f}(\xi,\eta)d\xi\,d\eta \\
&=\frac{1}{2}\int_{0}^{t}\int_{-t+\eta}^{0}\widetilde{f}
 (\xi,\eta)d\xi\,d\eta
+\frac{1}{2}\int_{0}^{t}\int_{0}^{t-\eta}\widetilde{f}(\xi,\eta)d\xi\,d\eta \\
&=\frac{1}{2}\int_{0}^{t}\int_{t-\eta}^{0}f(\xi,\eta)d\xi\,d\eta
+\frac{1}{2}\int_{0}^{t}\int_{0}^{t-\eta}f(\xi,\eta)d\xi\,d\eta=0.
\end{aligned} \label{e2.15}
\end{equation}
Summing  in \eqref{e2.13}-\eqref{e2.15}, we obtain from \eqref{e2.12}, that
$\widetilde{u}(0,t)=0$. That is, the first boundary condition of \eqref{e2.3}
is fulfilled.

Similarly we check the fulfilling of the second boundary condition 
from \eqref{e2.3}.

Hence, the formula \eqref{e2.11} gives the solution of Problem 1. 
Let us write its solution in $\Omega$ by functions $f, \tau, \nu$. 
For that, we substitute values $\widetilde{f},
\widetilde{\tau}, \widetilde{\nu}$ into formula \eqref{e2.11} 
expressed by formulas \eqref{e2.6},\eqref{e2.9} and \eqref{e2.10}.

Let us introduce notation: 
$\Omega_1=\{(x,t):(x,t)\in \Omega, t<x<\ell-t\}$,
$\Omega_2= \{(x,t):(x,t)\in \Omega, x<t\}$ and
$\Omega_3=\{(x,t):(x,t)\in \Omega, x+t>\ell\}$.
Then by direct calculation we obtain representation of solution of Problem 1.

In $\Omega_2$:
\begin{equation}
\begin{aligned}
\widetilde{u}(x,t)
&=\frac{\tau(x+t)-\tau(t-x)}{2}+
\frac{1}{2}\int_{t-x}^{x+t}\nu(\xi)d\xi\\
&\quad +\frac{1}{2}\int_{0}^{t-x} \int_{t-x-\eta}^{x+t-\eta}f(\xi,\eta)d\xi\,d\eta 
 +\frac{1}{2}\int_{t-x}^{t}\int_{x-t+\eta}^{x+t-\eta}
f(\xi,\eta)d\xi\,d\eta.
\end{aligned} \label{e2.16}
\end{equation}

In $\Omega_1$:
\begin{equation}
\widetilde{u}(x,t)
=\frac{\tau(x+t)+\tau(x-t)}{2}+
\frac{1}{2}\int_{x-t}^{x+t}\nu(\xi)d\xi 
 +\frac{1}{2}\int_{0}^{t}\int_{x-t+\eta}^{x+t-\eta}
f(\xi,\eta)d\xi\,d\eta.
 \label{e2.17}
\end{equation}

In $\Omega_3$:
\begin{equation}
\begin{aligned}
&\widetilde{u}(x,t)\\
&=\frac{\tau(x-t)-\tau(2 \ell-x-t)}{2}+
\frac{1}{2}\int_{x-t}^{2 \ell-x-t}\nu(\xi)d\xi \\
&\quad +\frac{1}{2}\int_{0}^{x+t-\ell}\int_{x-t+\eta}^{2 \ell-x-t+\eta}
 f(\xi,\eta)d\xi\,d\eta 
 +\frac{1}{2}\int_{x+t-\ell}^{t}\int_{x-t+\eta}^{x+t-\eta}
 f(\xi,\eta)d\xi\,d\eta.
\end{aligned} \label{e2.18}
\end{equation}

\section{Main result and its proof}

Let $E=(T,T)$, $F=(\ell-T,T)$ be points on a boundary $CD$.

\subsection*{Problem 2} Find a solution of equation \eqref{e1.1}, 
satisfying the boundary condition \eqref{e1.2} and conditions on the boundary $CD$:
\begin{gather}
u_{t}\big|_{DE}=0,\label{e3.1} \\
\alpha u_{x}+\beta u_{t}\big|_{EF}=0,\label{e3.2} \\
u\big|_{CF}=0, \label{e3.3}
\end{gather}
where $\alpha$ and $\beta$ are real numbers.

As usual, we say the function $u\in L_2(\Omega)$ is a {\sl strong solution} 
of Problem 2, if there exists the sequence of functions $u_{n}\in W_2^{2}(\Omega)$, 
satisfying boundary conditions of Problem 2 such that $u_{n}$ and $Lu_{n}$ 
converge in $L_2(\Omega)$ to $u$ and $f$ respectively.

Let us denote by $A$ the matrix
\[
\begin{pmatrix}
    \beta-\alpha & 0 & \alpha+\beta & 0& \dots& \dots& \dots& 0& 0& 0 \\
    0 & \beta-\alpha& 0& \alpha+\beta& 0& \dots& \dots& 0& 0& 0 \\
    0& 0& \beta-\alpha& 0& \alpha+\beta& 0& \dots& 0& 0& 0\\
    \vdots& \dots& \vdots& \vdots& \vdots& \vdots& \vdots& \dots& \vdots& \vdots\\
    0& \dots& \dots& 0& \dots& \dots& \dots& 0& \dots& 0\\
    0& 0& 0& \dots& 0& \dots& \dots& .& \dots& 0\\
    0& 0& \dots& \dots& \dots& 0& \beta-\alpha& 0& \alpha+\beta& 0\\
    0& 0& \dots& \dots& \dots& & 0& \beta-\alpha& 0& \alpha+\beta\\
    0& 0& \dots& \dots& \dots& \dots& \dots& 0& -1& -1\\
    -1& 1& 0& \dots& \dots& \dots& \dots& \dots& 0& 0
  \end{pmatrix}
\]
and assume that $\det A\neq0$. We denote the inverse of a matrix $A$ by $B=A^{-1}$ 
and elements of a matrix B by $\{b_{ij}\}_{i,j=\overline{1,n}}$.

The closest even and odd numbers to $n$ (including $n$) are denoted by $p$ 
and $q$ respectively.
Let $K$ be a broken line in $\Omega$, consisting of segments of the
 characteristics $x=2iT+t$, $i=0,1,\dots,[\frac{n-1}{2}]$ and 
$x=2jT-t$, $j=1,\dots,[\frac{n}{2}]$. Here $[z]$ denotes the integer part of $z$.

Also, let us introduce some notation:
\begin{gather*}
\Phi_1(x)=-F'_{1t}(x,T), \quad 0\leq x\leq T,\\
\Phi_{2j}(x)=-\alpha F'_{2x}(2jT-x,T)-\beta F'_{2t}(2jT-x,T),
\\\ j=1,2,\dots,[\frac{n-2}{2}]+1, 2j<n, 0\leq x\leq T,\\
\Phi_{2j+1}(x)=-\alpha F'_{2x}(2jT+x,T)-\beta F'_{2t}(2jT+x,T), \\
j=1,2,\dots,[\frac{n-2}{2}], 0\leq x\leq T,\\
\Phi_{n}(x)=\begin{cases}
-F'_{3x}(nT-x,T), & \text{if $n$ is even} \\
-F'_{3x}((n-1)T+x,T), & \text{if $n$ is odd},
\end{cases} \quad
0\leq x\leq T, \\
F_1(x,t)=\int_{0}^{t-x}\int_{t-x-\eta}^{x+t-\eta}f(\xi,\eta)d\xi\,d\eta
+\int_{t-x}^{t}\int_{x-t+\eta}^{x+t-\eta}f(\xi,\eta)d\xi\,d\eta,\\
F_2(x,t)=\int_{0}^{t}\int_{x-t+\eta}^{x+t-\eta}f(\xi,\eta)d\xi,d\eta, \\
F_3(x,t)=\int_{0}^{x+t-\ell}\int_{x-t+\eta}^{2 \ell-x-t+\eta}
f(\xi,\eta)d\xi\,d\eta
+\int_{x+t-\ell}^{t}\int_{x-t+\eta}^{x+t-\eta}f(\xi,\eta)d\xi\,d\eta.
\end{gather*}

\begin{theorem} \label{thm3.1} 
Let $\ell/T=n\geq2$ be a positive integer. A solution of the Problem 2 is unique, 
if and only if 
\begin{equation}
\alpha(\alpha+\beta)(\alpha-\beta)\neq 0\,. \label{e3.4}
\end{equation}
 If this \eqref{e3.4} holds, then:

(a) For all functions $f\in L_2(\Omega)$ Problem 2 have a unique
strong solution. This solution belongs to the class
 $u\in W_2^{1}(\Omega)\bigcap C(\overline{\Omega})$ and satisfies the
estimate
\begin{equation}\|u\|_{W_2^{1}(\Omega)}\leq
C\|f\|_{L_2(\Omega)}. \label{e3.5}
\end{equation}

(b) If $f\in C^{1}(\overline{\Omega})$, then the strong solution of
Problem 2 belongs to the class $u\in C^{2}(\overline{\Omega}
\backslash K) \bigcap C(\overline{\Omega})$.

(c) If $f\in C^{1}(\overline{\Omega})$, then the strong solution of
Problem 2 is classical, i.e. $u\in C^{2}(\Omega)\bigcap
C^{1}(\overline{\Omega})$, if and only if conditions \eqref{e3.6} and
\eqref{e3.7} hold:
\begin{equation}
\begin{aligned}
&\begin{pmatrix}
    b_{21}-(-1)^{i}b_{11} & b_{22}-(-1)^{i}b_{12} & \dots 
& b_{2n}-(-1)^{i}b_{1n} \\
    b_{41}-(-1)^{i}b_{31} & b_{42}-(-1)^{i}b_{32} & \dots 
& b_{4n}-(-1)^{i}b_{3n} \\
    \vdots & \vdots & \vdots & \vdots\\
    b_{p-21}-(-1)^{i}b_{p-31} & b_{p-22}-(-1)^{i}b_{p-32} & \dots 
& b_{p-2n}-(-1)^{i}b_{p-3n}\\
    b_{p1}-(-1)^{i}b_{p-11} & b_{p2}-(-1)^{i}b_{p-12} & \dots
& b_{pn}-(-1)^{i}b_{p-1n}
      \end{pmatrix} \\
& \times  \begin{pmatrix}
    \Phi_1^{(i)}(0) \\
    \Phi_2^{(i)}(0)  \\
    \vdots \\
    \Phi_{n}^{(i)}(0) 
      \end{pmatrix}
=\begin{pmatrix}
    0 \\
    0\\
    \vdots \\
    0 
\end{pmatrix}, \quad i=0,1; 
\end{aligned}\label{e3.6}
\end{equation}

  \begin{equation}
\begin{aligned}
&\begin{pmatrix}
    b_{31}-(-1)^{i}b_{21} & b_{32}-(-1)^{i}b_{22} & \dots & b_{3n}-(-1)^{i}b_{2n} \\
    b_{51}-(-1)^{i}b_{41} & b_{52}-(-1)^{i}b_{42} & \dots & b_{5n}-(-1)^{i}b_{4n} \\
    \vdots & \vdots & \vdots & \vdots\\
    b_{q-21}-(-1)^{i}b_{q-31} & b_{q-22}-(-1)^{i}b_{q-32} & \dots & b_{q-2n}-(-1)^{i}b_{q-3n}\\
    b_{q1}-(-1)^{i}b_{q-11} & b_{q2}-(-1)^{i}b_{q-12} & \dots & b_{qn}-(-1)^{i}b_{q-1n}\\
      \end{pmatrix} \\
&\times
\begin{pmatrix} 
    \Phi_1^{(i)}(T) \\
    \Phi_2^{(i)}(T)  \\
    \vdots \\
    \Phi_{n}^{(i)}(T) \\
      \end{pmatrix}
=\begin{pmatrix}
    0 \\
    0\\
    \vdots \\
    0 
\end{pmatrix},  \quad i=0,1.
\end{aligned} \label{e3.7}
\end{equation}
This solution is stable in the norm of $C^{1}(\overline{\Omega})$.
\end{theorem}

\begin{proof}
Since $T\leq \ell/2$, we use the representation of solution 
\eqref{e2.16}--\eqref{e2.18} in this proof.
Taking into account \eqref{e1.2}, we obtain
$u\big|_{AB}=\tau(x)=0$, then substitute the representation of
solution \eqref{e2.16} in the boundary condition \eqref{e3.1}:
\begin{equation}
\nu(x+T)-\nu(T-x)+F_{1t}'(x,T)=0, \quad 0\leq x \leq T. \label{e3.9}
\end{equation}
Now substitute the representation of solution \eqref{e2.17} into 
the boundary condition \eqref{e3.2}. Then
\begin{equation}
(\alpha+\beta)\nu(x+T)+(\beta-\alpha)\nu(x-T)+\alpha F'_{2x}(x,T)
+\beta F'_{2t}(x,T)=0,  \label{e3.10}
\end{equation}
for $T\leq x \leq T(n-1)$.
In equation \eqref{e3.10} for each $n-2$ segments $[Ti,T(i+1)], i=\overline{1,n-2}$ 
we make change of $x=T(i+1)-y$, $0\leq y\leq T$,
\begin{equation}
\begin{aligned}
&(\alpha+\beta)\nu(T(i+2)-y)+(\beta-\alpha)\nu(Ti-y)+
\alpha F_{2x}'(T(i+1)-y,T) \\
&+\beta F_{2t}'(T(i+1)-y,T)=0, \quad 0\leq y \leq T, \;
i=\overline{1,n-2}.
\end{aligned} \label{e3.11}
\end{equation}

Now we substitute the representation of solution \eqref{e2.18} 
into the boundary condition \eqref{e3.3}. Then
$$
\int_{x-T}^{2 \ell-x-T}\nu(\xi)d\xi+ F_3(x,T)=0, \quad
T(n-1)\leq x \leq Tn.
$$
We take a derivative with respect to $x$, then we have
\begin{equation}
-\nu(2\ell-x-T)-\nu(x-T)+F_{3x}'(x,T)=0,\quad T(n-1)\leq x \leq Tn. \label{e3.12}
\end{equation}
Then we make change of variable $x=\ell-y$, $0\leq y\leq T$,
\begin{equation}
-\nu(\ell+y-T)-\nu(\ell-y-T)+F_{3x}'(\ell-y,T)=0,
\quad 0\leq y\leq T. \label{e3.13}
\end{equation}

We have $n$ nonhomogeneous equations. Now we show that the number
of unknown functions in equations \eqref{e3.9}, \eqref{e3.11} and
\eqref{e3.13} equals $n$. For that we consider 2 cases:
\smallskip

\noindent\textbf{Case 1.}
 Let $n=2m, m\in \mathbb{Z^{+}}$. Then in \eqref{e3.11} for even numbers 
$i=2k$, $k=\overline{1,m-1}$ we make change of variable $y=T-z$,
\begin{equation}
\begin{aligned}
&(\alpha+\beta)\nu((2k+1)T+z)+(\beta-\alpha)\nu((2k-1)T+z)+
\alpha F_{2x}'(2kT+z,T)\\
&+\beta F_{2t}'(2kT+z,T)=0, \quad 0\leq z \leq T, \;
k=\overline{1,m-1}.
\end{aligned} \label{e3.14}
\end{equation}
Here it is easy to see that the number of unknown functions in
equations \eqref{e3.9}, \eqref{e3.11} (for odd numbers $i=2k-1$,
$k=\overline{1,m-1}$), \eqref{e3.13} and \eqref{e3.14} equals $n$.
\smallskip

\noindent\textbf{Case 2.} 
Let $n=2m+1, m\in \mathbb{Z^{+}}$. Then in \eqref{e3.11} for even 
$i=2k$, $k=\overline{1,m-1}$ we make change of $y=T-z$,
\begin{equation}
\begin{aligned}
&(\alpha+\beta)\nu((2k+1)T+z)+(\beta-\alpha)\nu((2k-1)T+z)+
\alpha F_{2x}'(2kT+z,T) \\
&+\beta F_{2t}'(2kT+z,T)=0,\quad 0\leq z \leq T,\;  k=\overline{1,m-1}. 
\end{aligned}\label{e3.15}
\end{equation}
Then in \eqref{e3.13} we make change of of variable $y=T-z$,
\begin{equation}
-\nu((2m+1)T-z)-\nu((2m-1)T+z)+F_{3x}'(2mT+z,T)=0,quad
0\leq z\leq T. \label{e3.16}
\end{equation}
Here it is easy to see that the number of unknown functions in
equations \eqref{e3.9}, \eqref{e3.11} (for odd numbers $i=2k-1$,
$k=\overline{1,m}$), \eqref{e3.15} and \eqref{e3.16} equals $n$.

Thus, we have $n$ nonhomogeneous equations for $n$ unknown
functions $\nu(x_{i})$, $T(i-1)\leq x_{i}\leq Ti$,
$i=\overline{1,n}$. The existence and uniqueness of the solution
of Problem 2 are equivalent to the existence and uniqueness of the
functions $\nu(x_{i})$, $T(i-1)\leq x_{i}\leq Ti$,
$i=\overline{1,n}$, satisfying equations \eqref{e3.9},
\eqref{e3.11} and \eqref{e3.13}.

Also the existence and uniqueness of the functions $\nu(x_{i})$, 
$T(i-1)\leq x_{i}\leq Ti$, $i=\overline{1,n}$ are equivalent 
to the following:
$$
0 \neq \det A
=\begin{cases}
2(\beta-\alpha)^{2k-1}(\alpha+\beta)^{2k-1}, &\text{if } n=4k;\\
-2\alpha(\beta-\alpha)^{2k-2}(\alpha+\beta)^{2k-2}, &\text{if }  n=4k-1;\\
-2(\beta-\alpha)^{2k-2}(\alpha+\beta)^{2k-2}, & \text{if }  n=4k-2;\\
2\alpha(\beta-\alpha)^{2k-3}(\alpha+\beta)^{2k-3}, & \text{if }  n=4k-3
\end{cases}
\quad  \; k=1,2,\dots.
$$

In case the $\alpha(\alpha+\beta)(\alpha-\beta)\neq 0$ we can see that
$\det A\neq 0$.
Thus, we have proved the existence and uniqueness of the solution of 
Problem 2 when  condition \eqref{e3.4} holds.

Now we show the stability according to the norm of $C^{1}(\overline{\Omega})$. 
By \eqref{e3.6}, \eqref{e3.7} and equations \eqref{e3.9}, \eqref{e3.11}, 
\eqref{e3.13}, we obtain
\begin{gather}
\nu(iT-0)=\nu(iT+0), \quad i=1,2,\dots,n-1; \\
\nu'(iT-0)=\nu'(iT+0), \quad i=1,2,\dots,n-1.
\end{gather}

Therefore the solution of Problem 2 is stable according to the norm 
of $C^{1}(\overline{\Omega})$.

From the existence and uniqueness of the classical solution of
Problem 2 by standard methods we obtain existence and uniqueness
of the strong solution of Problem 2.

From the representation of the solution of the problem it is easy to note 
that the strong solution depends only on $\nu(x)$ and $f(x,t)$. 
Since $\det A \neq 0$, then
from equations \eqref{e3.9}, \eqref{e3.11} and \eqref{e3.13} it is 
seen that the function $\nu(x)$ depends only on $f(x,t)$.
Then
\begin{align*}
\|u\|_{W_2^{1}(\Omega)}
&\leq C_1\|\nu(x)\|_{L_2(0,\ell)}+C_2\|f\|_{L_2(\Omega)}
= C_1\|B f\|_{L_2(0,\ell)}+C_2\|f\|_{L_2(\Omega)} \\
&\leq C_3 |B|\times\|f\|_{L_2(\Omega)} \leq C \|f\|_{L_2(\Omega)}.
\end{align*}
\end{proof}

We note that in [12] it is proved the well-posedness of problem for  \eqref{e1.1} 
with boundary condition \eqref{e1.2} and with conditions on the boundary $CD$:
\begin{equation}
u_{t}\big|_{DE}=0,\quad  u\big|_{CE}=0. \label{e3.18}
\end{equation}
From the condition \eqref{e3.4} follows that the case $\alpha=0$
in \eqref{e3.2} leads to ill-posedness of Problem 2. Thus, unlike
problem \eqref{e1.1}, \eqref{e1.2}, \eqref{e3.18} the problem with
boundary conditions 
$$
u_{t}\big|_{DF}=0, \quad u\big|_{CF}=0
$$ 
is ill-posed.

\subsection*{Acknowledgements}
This research is financially supported by a grant from the
Ministry of Science and Education of the Republic of Kazakhstan
(Grant No. 0824/GF4). This publication is supported by the target
program 0085/PTSF-14 from the Ministry of Science and Education of
the Republic of Kazakhstan.

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\end{document}