\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 108, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/108\hfil Well-posedness of boundary-value problems]
{Well-posedness of boundary-value problems for partial differential
equations of even order}

\author[D. Amanov,  A. Ashyralyev \hfil EJDE-2014/108\hfilneg]
{Djumaklych Amanov,  Allaberen Ashyralyev}  % in alphabetical order

\address{Djumaklych Amanov \newline
Institute of Mathematics at National University of Uzbekistan, 29,
Do`rmon yo`li street, Tashkent, Uzbekistan}
\email{amanov\_d@rambler.ru}

\address{Allaberen Ashyralyev \newline
Department of Mathematics Fatih University, 34500, Buyukcekmece,
Istanbul, Turkey and ITTU, Ashgabat, Turkmenistan }
\email{aashyr@fatih.edu.tr}


\thanks{Submitted October 15, 2013. Published April 15, 2014.}
\subjclass[2000]{35G15}
\keywords{Boundary value problem; well-posedness; a priori
estimate; \hfill\break\indent  regular and strong solutions; spectrum of problems}

\begin{abstract}
 In this article, we establish the well-posedness of two boundary
 value problems for $2k$-th order partial differential equations.
 It is shown that the solvability of these problems
 depends on the evenness and oddness of the number $k$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction and formulation of the problems}

There is a huge number of theoretical and applied
works devoted to the study of partial differential equations of higher
order. Solutions of some classical and nonclassical problems for specific
partial differential equations of higher order can be found, for example, in
the monographs by  Egorov and Fyodorov \cite{e1},  Gazzola and  
Evans \cite{e2},  Grunau and Sweers \cite{g1},  Mizohata \cite{m1}, 
 Peetre \cite{p1},  Polyanin \cite{p2},  Vragov \cite{v1}, and in 
the articles by  Kirane and Qafsaoui \cite{k2} and  Qafsaoui \cite{q1}. 
These problems are studied in various directions:
qualitative properties of solutions, spectral problems, various statements
of boundary value problems, and numerical investigations.

The study of well-posedness of the Cauchy problem, Goursat problem and
boundary value problem for partial differential equations and mixed type
partial differential equations have been studied extensively in a large
number of articles;  see, for example,
\cite{a1,a3,a4,d2,d3,d4,k3,k4,r1,s2,s4,t1,v1,z1}
and the references therein. As examples, we present some of them.

 In  \cite{a1,a3}, in the domain 
$\Omega =\{ (x,t) :0<x<p,\;0<t<T\}$ the boundary value problem
\[
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}-\frac{{\partial ^2u}}{{
\partial t^2}}=f(x,t)
\]
was investigated. Here $2\leqslant k$ is a fixed integer. There its solvability
was established.

 In  \cite{a4}, $p$-evolution equations in $(t,x)$ with real
characteristics were studied. Sufficient conditions for the well-posedness
of the Cauchy problem in Sobolev spaces, in terms of decay estimates of the
coefficients as the space variable $x\to \infty $ were given.

In  \cite{k3}, some correct boundary value problems for the following
differential- operator equation
\[
(-1)^{k}A\frac{d^{n}u(t)}{dt^{n}}+Bu(t)=f(t),t\in (0,T),
\]
where $k=[(n-1)/2],A$ and $B$ are linear self-adjoint operators in Hilbert
space, $B$ is a positive operator and operator $A$ can have a spectrum of
any kind, i.e. this equation can be of mixed type, were investigated. The
existence of weak solutions was proved and estimates for them were obtained.
Boundary conditions are satisfied in the sense of the existence of weak
limits of the solution and its derivatives. The results were illustrated by
three partial differential equation examples.

 In \cite{k4}, boundary value problems for some class of
higher-order mixed type partial differential equations were considered. Note
that these partial differential equations are unsolved with respect to the
highest derivative. The author establishes existence and uniqueness in
various settings including a so-called partially hyperbolic case. The
results rely on coerciveness properties.

 In \cite{t1}, a partial differential operator, parabolic in
the sense of Petrovski\u{\i}, of higher order in time, is perturbed by an
unbounded Volterra-type integral operator, the maximal order of which equals
the order of the leading part of the parabolic operator. As an example for
such general time-varying Volterra integro-linear-partial-differential
equations in sufficiently smooth bounded domains in $\mathbb{R}^{n}$, one
may consider the equation as a model for heat conduction with memory (first
order in time), while for second-order equations in $t$, the parabolic part
can be viewed as modeling the vibrations of a structurally damped
higher-order material (beams, plates, etc.) with memory effects (concerning
the deformation history) included in the Volterra operator. The
nonhomogeneous initial and boundary value (with homogeneous boundary
conditions) problem with distributed load is solved by a general "variation
of constants formula\textquotedblright : the procedure is first to find a
fundamental solution of the parabolic "reference\textquotedblright\ problem,
then formally to solve the equation by considering the integral as a
perturbation and (this appears to be the main difficulty) constructing
thereby the fundamental solution (i.e. the integral resolvent) of the
original problem.

 In \cite{y1}, a relation between parabolicity and coerciveness
in Besov spaces for a higher-order linear evolution equation in a Banach
space was investigated. It was proved that coerciveness in Besov spaces and
parabolicity of the equation are in fact equivalent.

In this article, two boundary value problems for $2k$-th order partial
differential equations are investigated. 
In the domain $\Omega =\{( {x,t}) :0<x<p,0<t<T \} $ we consider the partial
differential equation
\begin{equation}
Lu\equiv \frac{{\partial ^{2k}u}}{{\partial x^{2k}}}+\frac{{\partial ^2u}}{
{\partial t^2}}=f(x,t),  \label{e1}
\end{equation}
where $2\leqslant k$ is a fixed integer.

\subsection*{Problem 1} 
Obtain the solution $u(x,t)$ of equation \eqref{e1} in
the domain $\Omega $ satisfying conditions
\begin{gather}
\frac{{\partial ^{2m}u(0,t)}}{{\partial x^{2m}}}
=\frac{{\partial ^{2m}u(p,t)}}{{\partial x^{2m}}}=0,\quad
 m=0,1,\dots ,k-1,\; 0\leq t\leq T, \label{e2};\\
u(x,0)=0,\quad u(x,T)=0,\quad 0\leqslant x\leqslant p.  \label{e3}
\end{gather}

\subsection*{Problem 2} 
 Obtain the solution $u(x,t)$ of equation \eqref{e1} in
the domain $\Omega $ satisfying conditions \eqref{e2} and
\begin{equation}
u(x,0)=0,\quad u_{t}(x,0)=0,\quad 0\leqslant x\leqslant p\,.   \label{e4}
\end{equation}

Well-posedness of boundary value problems 1 and 2  in the cases even and
odd $k$  is investigated. It is proved that solvability results of the
present paper depend on the evenness or oddness of number $k$.

This article is organized as follows. 
Section 1 is an introduction where we provide the formulation of problems. 
In sections 2 and 3, well-posedness of problem 1 in the cases even and 
odd $k$ is established. The spectrum of problem 1 is studied. 
In section 4, well-posedness of problem 2 in the cases even and odd $k$ is established. The spectrum of
problem 2 is studied. Finally, section 5 is a conclusion.

\section{Well-posedness of problem 1 when $k$ is odd}

In this section, we present some basic definitions and preliminary facts
which are used throughout the paper. We denote
\begin{gather*}
V_1(\Omega )=\{u(x,t): u\in C_{x,t}^{2k-2,0}(\bar{\Omega})\cap
C_{x,t}^{2k,2}(\Omega ),\text{ \eqref{e2} and \eqref{e3} hold}\},\\
\begin{aligned}
W_1(\Omega )=\Big\{& f(x,t):f\in C_{x,t}^{1,0}(\bar{\Omega}),\frac{
\partial f}{\partial x}\in {\rm Lip}_{\alpha }[0,p] 
\text{uniformly with respect to $t$}, \\
 & 0<\alpha <1,\;f(0,t)=f(p,t)=0\Big\} .
\end{aligned}
\end{gather*}
We define a solution of problem 1 as follows:

\begin{definition} \label{def2.1} \rm
 A function $u(x.t)\in V_1(\Omega )$ is said to be
a regular solution of problem 1 with $f(x,t)\in C(\Omega )$, if it satisfies
\eqref{e1} in the domain $\Omega $ and conditions \eqref{e2} and \eqref{e3}.
\end{definition}

\begin{definition} \label{def2.2} \rm
A function $u(x,t)\in L_2(\Omega )$ is said to be
a strong solution of the problem 1 with $f\in L_2(\Omega )$, if there
exists a sequence $\{u_n\}\subset V_1(\Omega )$, $n=1,2,\dots$, such
that $\| {u_n-u}\| _{L_2(\Omega )}\to 0$,
$\| {Lu_n-f}\| _{L_2(\Omega )}\to 0$ as $n\to \infty$.
\end{definition}

\begin{definition} \label{def2.3} \rm
A function $u(x,t)\in W_2^{2k,2}(\Omega )$ is
called an almost everywhere solution of problem 1 with $f\in L_2(\Omega )$,
if it satisfies \eqref{e1}, \eqref{e2} and \eqref{e3} almost everywhere in
the domain $\Omega $. On $V_1(\Omega )$, we define the operator $L$
mapping the set $V_1(\Omega )$ into $C(\Omega )$ by \eqref{e1}. The domain of
definition $V_1(\Omega )$ is dense in $L_2(\Omega )$. The closure in
$L_2(\Omega )$ is denoted by $\overline{L}$.
\end{definition}

Let $k$ be an odd number. First, we will establish a priori estimate for the
regular solution of problem 1.

\begin{lemma} \label{lem2.1} 
If $u\in V_1(\Omega )$ and $\frac{{\partial ^{m+1}u}}{{
\partial x^{m}\partial t}}\in C(\bar{\Omega})$, $m=1,2,\dots,k$, 
$\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}\in L_2(\Omega )$, 
$\frac{{\partial^2u}}{{\partial t^2}}\in L_2(\Omega )$, 
then there exists a constant $C_1>0$ that depends only on numbers $k$ and $T$, 
and does not depend on the function $u(x,t)$ such that the following a 
priori estimate holds:
\begin{equation}
\| u\| _{W_2^{2k,2}(\Omega )} \leqslant C_1\| {Lu}\| _{L_2(\Omega )},  \label{e5}
\end{equation}
where
\[
\| u\| _{W_2^{2k,2}(\Omega )}=\sum_{m=0}^{2k}{
\| {\frac{{\partial ^{m}u}}{{\partial x^{m}}}}\|
_{L_2(\Omega )}^2+}\sum_{m=1}^2{\| {\frac{{\partial
^{m}u}}{{\partial t^{m}}}}\| _{L_2(\Omega )}^2+}
\sum_{m=2}^{k+1} \| {\frac{{\partial ^{m}u}}{{\partial
x^{m-1}\partial t}}}\| _{L_2(\Omega )}^2.
\]
\end{lemma}

\begin{proof}
 Multiplying by $u(x,t)$ both sides of the equation
\begin{equation}
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}+\frac{{\partial ^2u}}{{
\partial t^2}}=Lu  \label{e6}
\end{equation}
and taking the integral over the domain $\Omega $, we obtain
\begin{equation}
\int_0^{p}{\int_0^{T}{u( {\frac{{\partial ^{2k}u}}{{
\partial x^{2k}}}+\frac{{\partial ^2u}}{{\partial t^2}}}) }}\,dx\,dt
=\int_0^{p}{\int_0^{T}u}Lu\,dt\,dx.  \label{e7}
\end{equation}
Since $k$ is odd number, using equality \eqref{e7}, identities
\begin{gather*}
u\frac{{\partial ^2u}}{{\partial t^2}}=\frac{\partial }{{\partial t}}
( {u\frac{{\partial u}}{{\partial t}}}) -( {\frac{{\partial u
}}{{\partial t}}}) ^2,
\\
u\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}=\sum_{m=0}^{k-1}{(-1)^{m}
\frac{\partial }{{\partial x}}( {\frac{{\partial ^{m}u}}{{\partial x^{m}
}}\frac{{\partial ^{2k-m-1}u}}{{\partial x^{2k-m-1}}}})
+(-1)^{k}( {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}) ^2,}
\end{gather*}
and conditions \eqref{e2} and \eqref{e3}, we obain
\[
\| {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}
\| _{L_2(\Omega )}^2+\| {\frac{
{\partial u}}{{\partial t}}}\| _{L_2(\Omega
)}^2=\int_0^{p}{\int_0^{T}{( {-u}) Lu\,dt\,dx.}}
\]
From this equality and the H\"{o}lder inequality it follows the 
estimate
\[
\| {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}
\| _{L_2(\Omega )}^2+\| {\frac{
{\partial u}}{{\partial t}}}\| _{L_2(\Omega
)}^2\leq \| u\| _{L_2(\Omega )}
\text{ }\| {Lu}\| _{L_2(\Omega )}.
\]
Applying the inequality
\[
ab\leqslant \frac{\varepsilon }{2}a^2+\frac{1}{{2\varepsilon }}b^2
\]
for $a>0$, $b>0$ and $\varepsilon >0$,  we obtain
\begin{equation}
\| {u_{t}}\| _{L_2(\Omega
)}^2+\| {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}
\| _{L_2(\Omega )}^2\leqslant \frac{\varepsilon }{2
}\| u\| _{L_2(\Omega )}^2+ \frac{1}{{2\varepsilon }}\| {Lu}\|
_{L_2(\Omega )}^2.  \label{e8}
\end{equation}
Now,  we obtain the estimate for $\| u\| _{L_2(\Omega )}^2$.
Using condition \eqref{e3},  we obtain
\[
u^2(x,t)=\int_0^{t}{\frac{\partial }{{\partial t}}}( {
u^2(x,\tau )}) d\tau .
\]
Therefore,
\[
u^2(x,t)\leqslant 2\int_0^{T}{| {u(x,t)u_{t}(x,t)}| dt.}
\]
Taking the integral over the domain $\Omega $  on both sides of the
inequality and applying the H\"{o}lder inequality, we obtain
\[
\| u\| _{L_2(\Omega)}^2\leqslant 2T\| u\|
_{L_2(\Omega )}\| {u_{t}}\| _{L_2(\Omega )}.
\]
From that, it follows
\[
\| u\| _{L_2(\Omega)}^2\leqslant 4T^2 \| {u_{t}}\| _{L_2(\Omega )}^2.
\]
The last estimate and inequality \eqref{e8} yield that
\begin{equation}
\| u\| _{L_2(\Omega
)}^2\leqslant 4T^2( {\frac{\varepsilon }{2}\|
u\| _{L_2(\Omega )}^2+\frac{1}{{2\varepsilon }}\| Lu\| _{L_2(\Omega )}^2}) .
  \label{e9}
\end{equation}
Adding the both sides of inequalities \eqref{e8} and \eqref{e9},  we obtain
\[
\| u\| _{L_2(\Omega)}^2+\| {u_{t}}\| _{L_2(\Omega)}^2
+\| \frac{{\partial ^{k}u}}{{\partial x^{k}}}\| _{L_2(\Omega )}^2
\leqslant ( {4T^2+1}) ( \frac{\varepsilon }{2}\| u\| _{L_2(\Omega )}^2
+\frac{1}{{2\varepsilon }}\| {Lu}\| _{L_2(\Omega )}^2)
\]
for any $\varepsilon >0$. Choosing $\varepsilon =1/(4T^2+1)$, we
can write
\begin{equation}
\| u\| _{L_2(\Omega)}^2+\| {u_{t}}\| _{L_2(\Omega
)}^2+\| {\frac{{\partial {}^{k}u}}{{\partial x^{k}}}}
\| _{L_2(\Omega )}^2\leqslant ( {4T^2+1}) ^2\| {Lu}\|
_{L_2(\Omega )}^2.  \label{e10a}
\end{equation}
Taking the square of both sides of \eqref{e6} and integrating the obtained
equality over the domain $\Omega $,  we obtain
\begin{equation}
\| {u_{tt}}\| _{L_2(\Omega)}^2+\| {\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}}
\| _{L_2(\Omega )}^2+2\int_{\Omega }{\frac{{
\partial ^2u}}{{\partial t^2}}\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}
\,dx\,dt}=\| {Lu}\| _{L_2(\Omega)}^2.  \label{e11}
\end{equation}
We consider the integral
\[
\int_{\Omega }{\frac{{\partial ^2u}}{{\partial t^2}}\frac{{
\partial ^{2k}u}}{{\partial x^{2k}}}\,dx\,dt.}
\]
We have that
\begin{align*}
\frac{\partial ^2u}{\partial t^2}\frac{\partial ^{2k}u}{\partial x^{2k}}
&=\sum_{m=0}^{k-1}( {-1}) ^{m}\frac{\partial }{{\partial x}}
\Big( {\frac{{\partial ^{2+m}u}}{{\partial t^2\partial x^{m}}}\frac{{
\partial ^{2k-m-1}u}}{{\partial x^{2k-m-1}}}}\Big)\\
&\quad +( {-1}) ^{k}\frac{\partial }{{\partial t}}\Big( {\frac{{
\partial ^{1+k}u}}{{\partial t\partial x^{k}}}\frac{{\partial ^{k}u}}{{
\partial x^{k}}}}\Big) +( {-1}) ^{k+1}\Big( {\frac{{\partial
^{k+1}u}}{{\partial t\partial x^{k}}}}\Big) ^2.
\end{align*}
If $m$ is odd, then $2k-m-1$ is even. Therefore, from \eqref{e2} it follows
that  $\frac{{\partial ^{2k-m-1}u}}{{\partial x^{2k-m-1}}}=0$ at $x=0$ and $x=p$.
If $m$ is even, then from \eqref{e2} it follows that 
$\frac{{\partial ^{m+2}u}}{{\partial t^2\partial x^{m}}}=0$ at $x=0$ and $x=p$. 
Moreover, from \eqref{e3}
 it follows $\frac{{\partial ^{k}u}}{{\partial x^{k}}}=0$ at $t=0$ and 
$t=T$. Therefore, integrating both sides of the last equality over the
domain $\Omega $, we obtain
\[
2\iint_{\Omega }{\frac{{\partial ^2u}}{{\partial t^2}}\frac{{
\partial ^{2k}u}}{{\partial x^{2k}}}\,dx\,dt}=2\| {\frac{{
\partial ^{k+1}u}}{{\partial x^{k}\partial t}}}\|
_{L_2(\Omega )}^2.
\]
Using this equality and \eqref{e11}, we obtain
\[
\| {u_{tt}}\| _{L_2(\Omega
)}^2+2\| {\frac{{\partial ^{k+1}u}}{{\partial
x^{k}\partial t}}}\| _{L_2(\Omega )}^2
+\| {\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}}\| _{L_2(\Omega )}^2
=\| {Lu}\| _{L_2(\Omega )}^2.
\]
From that it follows
\begin{equation}
\| {u_{tt}}\| _{L_2(\Omega
)}^2+\| {\frac{{\partial ^{k+1}u}}{{\partial
x^{k}\partial t}}}\| _{L_2(\Omega )}^2
+\| {\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}}\| _{L_2(\Omega )}^2
\leqslant \| {Lu} \| _{L_2(\Omega )}^2.  \label{e12}
\end{equation}
Adding both sides of inequalities \eqref{e10a} and \eqref{e12}, we obtain
\begin{equation}
\begin{aligned}
&\| u\| _{L_2(\Omega)}^2+\| {u_{t}}\|
_{L{}_2(\Omega )}^2+\| {u_{tt}}\| _{L_2(\Omega )}^2+\| {\frac{{\partial
^{k+1}u}}{{\partial x^{k}\partial t}}}\|_{L_2(\Omega )}^2
 +\| {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}
\| _{L_2(\Omega )}^2+\| {\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}}\|
_{L_2(\Omega )}^2\\
&\leqslant C_1\| {Lu}\| _{L_2(\Omega )}^2,
\end{aligned} \label{e13}
\end{equation}
where $C_1=( {4T^2+1}) ^2+1$.

To obtain estimates for the norm
 $\| {\frac{{\partial^{m}u}}{{\partial x^{m}}}}\| _{L_2(\Omega )}^2$, 
$m=1,2,\dots,2k-1$, we use the inequality
\begin{equation}
\| {\frac{{\partial ^{n}u}}{{\partial x^{n}}}}
\| _{L_2(\Omega )}^2\leqslant \frac{1}{2}
\| {\frac{{\partial ^{n-1}u}}{{\partial x^{n-1}}}}
\| _{L_2(\Omega )}^2+\frac{1}{2}
\| {\frac{{\partial ^{n+1}u}}{{\partial x^{n+1}}}}\| _{L_2(\Omega )}^2  \label{e14}
\end{equation}
that can easily be checked.

Taking the sum of both sides of inequality \eqref{e14} with respect to $n$
from $1$ to $2k-1$, and using inequality \eqref{e13}, we obtain
\begin{equation}
\| {\frac{{\partial u}}{{\partial x}}}\| _{L_2(\Omega )}^2
+\| {\frac{{\partial ^{2k-1}u}}{{\partial x^{2k-1}}}}\| _{L_2(\Omega
)}^2\leqslant C_1\| {Lu}\|_{L_2(\Omega )}^2.  \label{e14a}
\end{equation}
In general, taking the sum of both sides of inequality \eqref{e14} with
respect to $n$ from $m$ to $k-1$, and using inequality \eqref{e13},  we obtain
\begin{equation}
\| {\frac{{\partial ^{m}u}}{{\partial x^{m}}}}
\| _{L_2(\Omega )}^2+\| {\frac{
{\partial ^{2k-m}u}}{{\partial x^{2k-m}}}}\|
_{L_2(\Omega )}^2\leqslant C_1\| {Lu}\| _{L_2(\Omega )}^2.  \label{e14b}
\end{equation}
Taking the sum of both sides of inequalities \eqref{e13} and \eqref{e14b} with
respect to $m$ from $1$ to $k-1$, we obtain
\begin{equation}
\sum_{m=0}^2{\| {\frac{{\partial ^{m}u}}{{
\partial t^{m}}}}\| }_{L_2(\Omega
)}^2+\sum_{m=1}^{2k}{\| {\frac{{\partial ^{m}u}
}{{\partial x^{m}}}}\| }_{L_2(\Omega
)}^2+\| {\frac{{\partial ^{k+1}u}}{{\partial
x^{k}\partial t}}}\| _{L_2(\Omega )}^2\leqslant
kC_1\| {Lu}\| _{L_2(\Omega)}^2.  \label{e15}
\end{equation}
Next, we use the inequality
\[
\| {\frac{{\partial ^{m}u}}{{\partial x^{m-1}\partial t}}
}\| _{L_2(\Omega )}^2\leqslant | {
| {\frac{{\partial ^2u}}{{\partial t^2}}}| }
| _{L_2(\Omega )}\| {\frac{{\partial ^{2m-2}u
}}{{\partial x^{2m-2}}}}\| _{L_2(\Omega )}
\]
that can easily be checked. Summing with respect to $m$ from $2$ to $k$ and
using inequality \eqref{e15},  we obtain
\begin{equation}
\sum_{m=2}^{k}{\| {\frac{{\partial ^{m}u}}{{
\partial x^{m-1}\partial t}}}\| }_{L_2(\Omega
)}^2\leqslant k(k-1)C_1\| {Lu}\| _{L_2(\Omega )}^2.  \label{e16}
\end{equation}
Adding both sides of inequalities \eqref{e15} and \eqref{e16}, we obtain estimate
\eqref{e5}. 
The proof is complete.
\end{proof}

\begin{corollary} \label{coro2.1} 
From estimate \eqref{e5} it follows that:
\begin{itemize}
\item[(i)] The regular solution of problem 1 is unique and continuously depends on 
$f(x,t)$.

\item[(ii)] The inverse operator $L^{-1}$ exists and it is bounded.

\item[(iii)] $\| {L^{-1}}\| \leqslant C_2$,
$C_2=k^2[ {( {4T^2+1}) ^2+1}]$.

\item[(iv)]  $\ker(L)=\{0\}$.

\item[(v)] The adjoint problem to problem 1 is well-posed.
\end{itemize}
\end{corollary}

Second, we will  study the regular solvability of problem 1. We seek a
regular solution of problem 1 in the form of a Fourier series
\begin{equation}
u(x,t)=\sum_{n=1}^{\infty }{u_n(t)X_n(x)}  \label{e17}
\end{equation}
expanded along the complete orthonormal system 
$X_n(x)=\sqrt{\frac{2}{p}}\sin \lambda _nx$ in $L_2(0,p)$, where
$\lambda _n=\frac{{n\pi }}{p}$,
$n\in N$. It is clear that $u(x,t)$ satisfies conditions \eqref{e2}. We expand the
given function $f\in W_1(\Omega )$ in the form of a Fourier series along
the functions $X_n(x)$, $n\in N$
\begin{equation}
f(x,t)=\sum_{n=1}^{\infty }f_n(t)X_n(x),  \label{e18}
\end{equation}
where
\begin{equation}
f_n(t)=\int_0^{P}f(x,t)X_n(x)dx.  \label{e19}
\end{equation}
Substituting \eqref{e17} and \eqref{e18} into \eqref{e1}, we obtain
\begin{equation}
u_n''(t)-\lambda _n^{2k}u_n(t)=f_n(t),\quad 0<t<T.
\label{e20}
\end{equation}
The solution of equation \eqref{e20} satisfying the conditions
\begin{equation}
u_n(0)=0,\quad u_n(T)=0  \label{e21}
\end{equation}
has the form
\begin{equation}
u_n(t)=-\frac{1}{{\lambda _n^{k}}}\int_0^{T}{K_n(t,\tau )}
f_n(\tau )d\tau ,  \label{e22}
\end{equation}
where
\begin{equation}
K_n(t,\tau )=\begin{cases}
\frac{{sh\lambda _n^{k}\tau sh\lambda _n^{k}(T-t)}}{{
sh\lambda _n^{k}T}}, & 0\leqslant \tau \leqslant t, \\[4pt]
\frac{{sh\lambda _n^{k}tsh\lambda _n^{k}(T-\tau )}}{{
sh\lambda _n^{k}T}}, & t\leqslant \tau \leqslant T.
\end{cases}
 \label{e23a}
\end{equation}
From \eqref{e23a} it follows $K_n(t,\tau )=K_n(\tau ,t)$ and the
following estimate holds:
\begin{equation}
K_n(t,\tau )\leqslant C{}_0e^{-\lambda _n^{k}| {t-\tau }| },C_0={\rm const.}>0.
  \label{e24}
\end{equation}

\begin{lemma} \label{lem2.2} 
If $f\in W_1(\Omega )$, then for any $t\in [ 0,T] $ the following estimate 
is valid:
\begin{equation}
| {u_n(t)}| \leqslant \frac{{C_0C_2}}{{\lambda _n^{2k+1+\alpha }}}.  \label{e25}
\end{equation}
\end{lemma}

\begin{proof} 
Integrating by parts with respect to $x$ in \eqref{e19}, we obtain
\[
f_n(t)=\frac{1}{{\lambda _n}}f_n^{(1,0)}(t),
\]
where
\[
f_n^{(1,0)}(t)=\int_0^{p}{\frac{{\partial f}}{{\partial x}}}\sqrt{
\frac{2}{p}}\cos \lambda _nx\,dx.
\]
Since $\frac{{\partial f}}{{\partial x}}\in {\rm Lip}_{\alpha }[ 0,p]$ 
is uniform with respect to $t$, then (see \cite{d2})
\[
| {f_n^{(1,0)}(t)}| \leqslant \frac{{C_2}}{{\lambda_n^{\alpha }}}.
\]
So,
\begin{equation}
| {f_n(t)}| \leqslant \frac{{C_2}}{{\lambda_n^{1+\alpha }}}.  \label{e26}
\end{equation}
Now, we will estimate $| {u_n(t)}| $. Applying equality
\eqref{e22} and estimates \eqref{e24} and \eqref{e26}, we obtain
\begin{align*}
| {u_n(t)}| &\leqslant \frac{{C_0C_2}}{{\lambda
_n^{1+k+\alpha }}}\Big[ {\int_0^{t}{e^{-\lambda _n^{k}(t-\tau
)}}d\tau +\int_{t}^{T}{e^{-\lambda _n^{k}(\tau -t)}d\tau }}\Big]
\\
&\leqslant \frac{{C_0C_2}}{{\lambda _n^{2k+1+\alpha }}}\Big( {
e^{-\lambda _n^{k}(T-t)}-e^{-\lambda _n^{k}t}}\Big) \\
&\leqslant \frac{{C_0C_2}}{{\lambda _n^{2k+1+\alpha }}}{e^{-\lambda _n^{k}(T-t)}}
\leqslant \frac{{C_0C_2}}{{\lambda _n^{2k+1+\alpha }}}.
\end{align*}
The proof is complete.
\end{proof}

\begin{theorem} \label{thm2.1} 
If $f(x,t)\in W_1(\Omega )$, then there exists a
regular solution of problem 1.
\end{theorem}

\begin{proof} 
We will prove uniform and absolute convergence of series \eqref{e17} and
\begin{equation}
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}=-\sum_{n=1}^{\infty
}\lambda _n^{2k}u_n(t)X_n(x),  \label{e27}
\end{equation}
\begin{equation}
\frac{{\partial ^2u}}{{\partial t^2}}=\sum_{n=1}^{\infty
}f_n(t)X_n(x)+\sum_{n=1}^{\infty }\lambda
_n^{2k}u_n(t)X_n(x).  \label{e28}
\end{equation}
From \eqref{e26} it follows that the series \eqref{e17} and the first series
in \eqref{e28} uniformly and absolutely converge. In the same manner, from 
\eqref{e25} it follows that the second series in \eqref{e28} and the 
series \eqref{e27} uniformly and absolutely converge.
 Adding equality \eqref{e27} and \eqref{e28}, we note that solution
 \eqref{e17} satisfies equation \eqref{e1}. Solution \eqref{e17} satisfies
boundary conditions \eqref{e2} owing to properties of the
function $X_n(x)$ and conditions \eqref{e3} owing to properties of the
kernel \eqref{e23a}. 
The proof is complete.
\end{proof}

Third, we will study an existence and uniqueness of the strong solution of
problem 1.

\begin{theorem} \label{thm2.2} 
For any $f\in L_2(\Omega )$ there exists a unique
strong solution of problem 1 and it satisfies estimate \eqref{e5}, it continuously
depends on $f(x,t)$ and it can be represented in the form of Fourier series
\[
u(x,t)=\sum_{n=1}^{\infty }{u_n(t)X_n(x)}
\]
expanded in full orthonormal system 
$X_n(x)=\sqrt{\frac{2}{p}}\sin \lambda_nx$ in $L_2(0,p)$, where 
$\lambda _n=\frac{{n\pi }}{p}$, $n\in N$.
\end{theorem}

\begin{proof} 
It is possible to represent solution \eqref{e17} in the form
\begin{equation}
u(x,t)=\int_0^{T}{\int_0^{p}{K(x,t;\xi ,\tau )}}f(\xi,\tau )d\xi \partial \tau ,  \label{e29}
\end{equation}
where
\begin{equation}
K(x,t;\xi ,\tau )=-\sum_{n=1}^{\infty }{\frac{{X_n(x)X_n(\xi )}}{{
\lambda _n^{k}}}}K_n(t,\tau ).  \label{e30}
\end{equation}
From estimate \eqref{e24} and $k\geqslant 2$ it follows that series \eqref{e30}
 converges uniformly and absolutely. Consequently
\begin{equation}
| {K(x,t;\xi ,\tau )}| \leqslant C_{3},\,C_{3}={\rm const}>0.
\label{e31}
\end{equation}
From relations $C_0^{\infty }(\Omega )\subset W_1(\Omega )\subset
L_2(\Omega )$ it follows that $W_1(\Omega )$ is dense in 
$L_2(\Omega)$. Then for any $f\in L_2(\Omega )$ there exists a sequence
 $\{ {f_{m}}\} \subset W_1(\Omega )$, $m=1,2,\dots$ such that
 $\| {f_{m}-f}\| _{L_2(\Omega )}\to 0$
as $m\to \infty $. Consequently, $\{ {f_{m}}\} $ is a
Cauchy sequence in $L_2(\Omega )$. We denote by 
$u_{m}(x,t)\in V_1(\Omega )$ the solution of \eqref{e1} with the right side term 
$f_{m}(x,t)$. According to Theorem \ref{thm2.1} and Corollary 
\ref{coro2.1}, there exists a unique solution of the form
\begin{equation}
u_{m}(x,t)=\int_0^{T}\int_0^{p}K(x,t;\xi ,\tau )f_{m}(\xi ,\tau )\,d\xi\,d\tau ,\quad m=1,2,\dots  \label{e32}
\end{equation}
of the equation
\begin{equation}
Lu_{m}(x,t)=f_{m}(x.t),\quad m=1,2,\dots.  \label{e33}
\end{equation}
Since $u_{m}(x,t)\in V_1(\Omega )$, according to \eqref{e5} we have
\[
\| {u_{m}-u_{l}}\|_{W_2^{2k,2}}\leqslant C_1\| {Lu_{m}-Lu_{l}}
\| _{L_2(\Omega )}
=C_1\| {f_{m}-f_{l}}\| _{L_2(\Omega )}\to 0
\]
as $m\to \infty $, $l\to \infty $, that $\{ {u_{m}}\} $ is a Cauchy sequence 
in $W_2^{2k,2}(\Omega )$. According to completeness of the $W_2^{2k,2}(\Omega )$, 
there exists a unique limit $u(x,t)=\lim_{m\to \infty }u_{m}(x,t)\in
W_2^{2k,2}(\Omega )$. Passing to limit in \eqref{e33} as $m\to \infty $,
 we obtain
\begin{equation}
\overline{L}u=\lim_{m\to \infty }Lu_{m}(x,t)=
\mathop{\lim }_{m\to \infty }f_{m}(x,t)=f,\,\,\,\,\,\,f\in
L_2(\Omega );  \label{e34}
\end{equation}
i.e.,
\[
\| {Lu_{m}-f}\| _{L_2(\Omega)}\to 0\quad \text{as }m\to \infty .
\]
Consequently, according to Definition 2.2, the function 
$u(x,t)\in W_2^{2k,2}(\Omega )$ is the strong solution of problem 1. 
Hence, it follows that the domain $D(\overline{L})$ of definition of operator 
$\overline{L}$ consists of all strong solutions of problem 1 and 
$R(\overline{L})=L_2(\Omega )$. So, $u_{m}(x,t)\in V_1(\Omega )$ 
from \eqref{e5} we have that
\[
(u_{m},u_{m})_{W_2^{2k,2}(\Omega )}\leqslant C_1^2( {Lu_{m},Lu_{m}}) _{L_2(\Omega )}.
\]
Passing to the limit in the above inequality as $m\to \infty $,  we obtain
\begin{equation}
\| u\|_{W_2^{2k,2}}^2\leqslant C_1^2\| {\overline{L}u}
\| _{L_2(\Omega )}^2,  \label{e35}
\end{equation}
where $u\in W_2^{2k,2}(\Omega )$, $\overline{L}u=f\in L_2(\Omega )$. We
conclude that estimate \eqref{e5} is also true for the strong solution $u(x,t)$.
Passing to limit in \eqref{e33} as $m\to \infty $, we obtain
\[
u(x,t)=\int_0^{T}{\int_0^{P}{K(x,t;\xi ,\tau )}}f(\xi,\tau )d\xi \partial \tau ,
\]
where $u(x,t)\in W_2^{2k,2}(\Omega )$, $f\in L_2(\Omega )$. From
estimate \eqref{e35} it follows that the strong solution of problem 1 is
unique and it continuously depends on $f(x,t)$. 
The proof of Theorem \ref{thm2.2} is complete.
\end{proof}

\begin{corollary} \label{coro2.2} 
The strong solution $u(x,t)$ is almost everywhere
solution in $\Omega $. This follows from \eqref{e34} and 
$u(x,t)\in W_2^{2k,2}(\Omega )$.
\end{corollary}

 Fourth, we will study the spectrum of problem 1.

\begin{definition} \label{def2.4} 
The spectrum of a problem is the set of eigenvalues
of the operator of the problem.
\end{definition}

According to Definition 2.4, the spectrum of problem 1 is the set of
eigenvalues of operator $\overline{L}$.

\begin{theorem} \label{thm2.3} 
The spectrum of problem 1 consists of real eigenvalues
of finite multiplicity of the operator $\overline{L}$.
\end{theorem}

\begin{proof} 
From \eqref{e5} and \eqref{e29} we conclude that it is defined a
bounded symmetric operator $L^{-1}$ on $W_1(\Omega )$ which is the inverse
of the operator $L$ and acts from $W_1(\Omega )$ to $V_1(\Omega )$ by
the rule
\[
(L^{-1}f)(x,t)=\int_0^{T}{\int_0^{p}{K(x,t;\xi ,\tau )}}
f(\xi ,\tau )\,d\xi\,d\tau .
\]
It can be extended to whole space $L_2(\Omega )$. This extension will be
denoted by $\overline{L^{-1}}$, the closure of $L^{-1}$, 
$D(\overline{L^{-1})}=L_2(\Omega )$. The operator $\overline{L^{-1}}$ 
is symmetric, bounded and defined on the whole space $L_2(\Omega )$,
 so it is self-adjoint. It follows from \eqref{e31} that 
$K(x,t;\xi ,\tau ))\in L_2( {\Omega \times \Omega }) $; therefore 
$\overline{L^{-1}}$ is a compact operator in $L_2(\Omega )$. 
Then the spectrum of the operator $\overline{L^{-1}}$
is discrete and consists of real eigenvalues of finite multiplicity. 
The relation between eigenvalues of the operator $\overline{L^{-1}}$ and 
$\overline{L}$ is as follows of monograph \cite{d3}: if $\mu _n\neq 0$ is an
eigenvalue of the operator $\overline{L^{-1}}$, then $\mu _n^{-1}$ is
eigenvalue of the operator $\overline{L}$. 
The proof is complete.
\end{proof}

\section{Well-posedness of problem 1 when $k$ is even}

Let $k$ be even number. In the same manner, we seek a regular solution of
problem 1 in the form of Fourier series \eqref{e17} and we obtain the 
equation
\[
u_n''+\lambda _n^{2k}u_n(t)=f_n(t),\quad 0<t<T
\]
for the function $u_n(t)$. The solution of this equation satisfying the
 condition
\[
u_n(0)=0,\quad u_n(T)=0
\]
which has the form
\[
u_n(t)=-\frac{1}{{\lambda _n^{k}}}\int_0^{T}{K_n(t,\tau )}f_n(\tau )d\tau ,
\]
where
\[
K_n(t,\tau )=\begin{cases}
\frac{{\sin \lambda _n^{k}\tau \sin \lambda _n^{k}(T-t)}}{{
\sin \lambda _n^{k}T}}, & 0\leqslant \tau \leqslant t, \\[4pt]
\frac{{\sin \lambda _n^{k}t\sin \lambda _n^{k}(T-\tau )}}{{
\sin \lambda _n^{k}T}}, & t\leqslant \tau \leqslant T.
\end{cases}
\]
Therefore, the solution of problem 1 exists, if the following condition
holds
\[
\big| {\sin \lambda _n\big( {\frac{{\pi n}}{p}}\big) ^{k}T}
\big| \geqslant \delta >0.
\]
It is difficult to show the numbers $p$ and $T$ satisfying the last
condition. Therefore, we give an other variant of solution of problem 1 in
the case of even $k$. In the case of even $k$, solvability of problem 1
depends on domain geometry. We consider the following spectral problem.

\subsection*{Problem 3}
Find the solution $u(x,t)$ of the equation
\begin{equation}
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}+\frac{{\partial ^2u}}{{
\partial t^2}}=\lambda u  \label{e36}
\end{equation}
satisfying \eqref{e2} and \eqref{e3}, where $\lambda $ is spectral parameter.
 This problem has the following eigenvalues
\[
\lambda _{nm}=( {\frac{{n\pi }}{p}}) ^{2k}-( {\frac{{n\pi }}{
T}}) ^2
\]
and eigenfunctions
\[
u_{nm}(x,t)=\frac{2}{\sqrt{pT}}\sin \frac{{n\pi }}{p}x\sin \frac{{n\pi }}{T}
t,\quad m,n=1,2,\dots
\]
and they form a complete orthonormal system in $L_2(\Omega )$.

We expand the functions $u(x,t)$ and $f(x,t)$ into the Fourier series in
functions $u_{nm}(x,t)$,
\begin{gather}
u(x,t)=\sum_{n,m=1}^{\infty }{\alpha _{nm}u_{nm}(x,t)},  \label{e37}\\
f(x,t)=\sum_{n,m}^{\infty }{f_{nm}u_{nm}(x,t)},  \label{e38}
\end{gather}
where
\begin{equation}
f_{nm}=\int_0^{P}\int_0^{T}{f(x,t)u_{nm}(x,t)}\,dt\,dx
\label{e39}
\end{equation}
and $\alpha _{nm}$ are unknown Fourier coefficients of the function $u(x,t)$.
Substituting \eqref{e37} and \eqref{e38} into \eqref{e1},  we obtain the
solution of problem 1 in the form
\begin{equation}
u(x,t)=\sum_{n,m=1}^{\infty }{\frac{{f_{nm}}}{{\lambda _{nm}}}
u_{nm}(x,t)}.  \label{e40}
\end{equation}
Suppose that $m=n^{k}$, $T=p^{k}/ \pi ^{k-1}$. Then 
$\lambda _{nm}=\lambda _{nn^{k}}=0$ and problem 1 has infinitely many of linearly
independent solutions in the form
\[
u_{nn^{k}}(x,t)=\sin \big(\frac{{n\pi }}{p}x \big)
\sin\big( \frac{{n^{k}\pi ^{k}}}{{p^{k}}}t\big), \quad n=1,2,\dots
\]
at $f(x,t)=0$. Number $\lambda _{nn^{k}}=0$ is an eigenvalue of infinite
multiplicity of the spectral problem \eqref{e36}, \eqref{e2}, \eqref{e3}.

In this case, the solution of problem 1 exists, if the following
orthogonality conditions hold
\[
\int_0^{p} \int_0^{T} f(x,t)\sin \big(\frac{{n\pi }}{p}x\big)
\sin \big(\frac{{n^{k}\pi ^{k}}}{p^{k}}t\big) \,dt\,dx=0,\quad n=1,2,\dots.
\]
Now, suppose that $p$ and $T$ such that $\frac{{P^{k}}}{{T\pi ^{k-1}}}$ is
irrational algebraic number of second degree (see \cite{s3}). Then according to
Liouville's theorem \cite{s3} there exists number $\varepsilon _0>0$ such that
\[
\big| {\frac{{p^{k}}}{{T\pi ^{k-1}}}-\frac{{n^{k}}}{m}}\big|
\geqslant \frac{{\varepsilon _0}}{{m^2}}.
\]
In this case
\begin{equation}
\lambda _{nm}\geqslant \frac{{\pi ^{k+1}}}{{Tp^{k}}}\varepsilon _0.
\label{e41}
\end{equation}

 \begin{theorem} \label{thm3.1} 
Let $f\in L_2(\Omega )$ and number $\frac{{P^{k}}}{{T\pi ^{k-1}}}$ 
be irrational algebraic number of second degree.
Then there exists the solution of problem 1, it belongs to $L_2(\Omega )$,
and satisfies estimate
\begin{equation}
\| u\| _{L_2(\Omega )}\leqslant C\| f\| _{L_2(\Omega )},
\label{e42}
\end{equation}
where $C=\frac{{Tp^{k}}}{{\pi ^{k+1}\varepsilon _0}}$ and it continuously
depends on $f(x,t)$.
\end{theorem}

\begin{proof} 
Since $f\in L_2(\Omega )$. Then for $f$, the Parseval
equality is true
\[
\sum_{n,m=1}^{\infty }{| {f_{nm}}| ^2=\| f\| }_{L_2(\Omega )}^2.
\]
Owing to it, for any $S>0$ and natural number $N$, we have
\begin{align*}
\| u{_{N+S,N+S}-u_{NN}}\|_{L_2(\Omega )}^2
&=\sum_{n=N+1}^{N+S}{\sum_{m=N+1}^{N+S}{\frac{{| {f_{nm}}| ^2}}{{\lambda _{nm}^2}}}}\\
&\leqslant C\sum_{n=N+1}^{N+S}{\sum_{m=N+1}^{N+S}{| {f_{nm}}| }}^2\to 0
\quad \text{as } N\to \infty.
\end{align*}
Consequently, the series \eqref{e40} converges in $L_2(\Omega )$ and 
$u(x,t)\in L_2(\Omega )$. It is easy to show that 
$\|u\| _{L_2(\Omega )}\leqslant C\| f\| _{L_2(\Omega )}$. From that it follows
continuously dependence of the solution of problem 1 on $f(x,t)$. 
The proof is complete.
\end{proof}

Now, we  study the regular solvability of the problem 1 for  $k$ even.
We denote
\begin{align*}
W_2(\Omega )=\Big\{& f: f\in C_{x,t}^{2k,2}(\overline{\Omega }),\,
\frac{{ \partial ^{2k+2}f}}{{\partial x^{k+1}\partial t}}\in L_2(\Omega ),\,
\frac{{\partial ^{4}f}}{{\partial x\partial t^{3}}}\in L_2(\Omega ),\\
& \frac{{\partial ^{2l}f(0,t)}}{{\partial x^{2l}}}=\frac{{\partial
^{2l}f(p,t)}}{{\partial x^{2l}}}=0,\,l=0,1,2,\dots,k;\\
&\frac{{\partial^{2s}f(x,0)}}{{\partial t^{2s}}}
=\frac{{\partial ^{2s}f(x,T)}}{{\partial t^{2s}}}=0,\,s=0,1\Big\} .
\end{align*}

\begin{theorem} \label{thm3.2} 
If $P^{k}/(T\pi ^{k-1})$ is irrational
algebraic number of second degree and $f\in W_2(\Omega )$, then there
exists regular solution of problem 1 and satisfies estimate \eqref{e42}.
\end{theorem}

\begin{proof} We will prove uniform and absolute convergence of series 
\eqref{e37} and that
\begin{gather}
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}=\sum_{n,m=1}^{\infty }{
\frac{{f_{nm}}}{{\lambda _{nm}}}( {\frac{{n\pi }}{p}}) }
^{2k}u_{nm}(x,t),  \label{e43}
\\
\frac{{\partial ^2u}}{{\partial t^2}}=-\sum_{n,m=1}^{\infty }{
\frac{{f_{nm}}}{{\lambda _{nm}}}( {\frac{{m\pi }}{T}}) }
^2u_{nm}(x,t).  \label{e44}
\end{gather}
The series
\begin{equation}
\sum_{n,m=1}^{\infty }{n^{2k}}| {f_{nm}}|
\label{e45}
\end{equation}
is majorant for series \eqref{e43} and the series
\begin{equation}
\sum_{n,m=1}^{\infty }{m^2}| {f_{nm}}|
\label{e46}
\end{equation}
is majorant for series \eqref{e44} .

Integrating \eqref{e39} by parts $2k+1$ times with respect to $x$ and one
time with respect to $t$, we obtain
\begin{gather}
|f_{nm}|=\frac{{Tp^{2k+1}}}{{\pi ^{2k+2}}}\frac{1}{{mn^{2k+1}}}
|f_{nm}^{(2k+1,1)}|,  \label{e47}
\\
f_{nm}=-\frac{{pT^{3}}}{{\pi ^{4}nm^{3}}}f_{nm}^{(1,3)},  \label{e48}
\end{gather}
where
\begin{gather*}
f_{nm}^{(2k+1,1)}=\int_0^{p} \int_0^{T} 
\frac{{\partial^{2k+2}f}}{{\partial x^{2k+1}\partial t}}\frac{2}{\sqrt{pT}}
\cos \big(\frac{{n\pi}}{p}x\big)\cos \big(\frac{{m\pi }}{T}t\big)\,dt\,dx,
\\
f_{nm}^{(1,3)}=\int_0^{P}{\int_0^{T}{\frac{{\partial ^{4}f}
}{{\partial x\partial t^{3}}}\frac{2}{\sqrt{PT}}\cos\big( \frac{{n\pi }}{P}x\big)
\cos\big(\frac{{m\pi }}{T}t\big)\,dt\,dx}}.
\end{gather*}
Using \eqref{e47}, the H\"{o}lder inequality and the Bessel inequality from 
\eqref{e45}, we obtain
\[
\sum_{n,m=1}^{\infty }{n^{2k}}| {f_{nm}}|
\leqslant \frac{{Tp^{2k+1}}}{{6\pi ^{2k}}}\| {\frac{{
\partial ^{2k+2}f}}{{\partial x^{2k+1}\partial t}}}\|
_{L_2(\Omega )}.
\]
Consequently, series\ \eqref{e43} uniformly and absolutely converges. In
\ref{e46} using \eqref{e48}, the H\"{o}lder inequality and the Bessel
inequality,  we obtain
\[
\sum_{n,m=1}^{\infty }{m^2}| {f_{nm}}| \leqslant
\frac{{pT^{3}}}{{6\pi ^2}}\| {\frac{{\partial ^{4}f}}{{
\partial x\partial t^{3}}}}\| _{L_2(\Omega )}.
\]
Hence, absolutely and uniformly convergence of series\ \eqref{e44} follows.

Adding equality \eqref{e43} and \eqref{e44}, we note that solution \eqref{e37}
satisfies equation \eqref{e1}. Solution \eqref{e37} satisfies boundary
conditions \eqref{e2} and \eqref{e3} owing to properties of functions $
u_{nm}(x,t)$. From $W_2(\Omega )\subset L_2(\Omega )$ for $f\in
W_2(\Omega )$ it follows estimate (42). 
The proof is complete.
\end{proof}

\begin{corollary} \label{coro3.1} 
In the case even $k$, solvability of problem 1
depends on domain geometry. A minor change of numbers $p$ and $T$ can lead
to an ill-posed problem.
\end{corollary}


\section{Well-posedness of problem 2}

 First, we will study the regular solvability of problem 2.
Let $k$ be odd number, then problem 2 is not correct. Indeed, if we seek the
solution of problem 2 in the form \eqref{e17}, we obtain
\[
u_n(t)=\frac{1}{{\lambda _n^{k}}}\int_0^{t}{f_n(\tau )}
sh\lambda _n^{k}(t-\tau )d\tau
\]
and $|u_n(t)|\to \infty $ as $n\to \infty $.
 Therefore, we will consider the case when $k$ is even. We denote
\begin{gather*}
V_2(\Omega )=\{U: U\in C_{x,t}^{2k-2,1}(\overline{\Omega })\cap
C_{x,t}^{2k,2}(\Omega );\text{ \eqref{e2} and \eqref{e4} are satisfied}\}\\
\begin{aligned}
W_{3}(\Omega )=\Big\{& f: f\in C_{x,t}^{k,0}(\overline{\Omega }),\,
\frac{{\partial ^{k+1}f}}{{\partial x^{k+1}}}\in L_2(\Omega );\,
\frac{{\partial ^{2m}f}}{{\partial x^{2m}}}=0\\
& \text{at $x=0$  and $x=p$, $m=0,1,\dots,\frac{{k-2}}{2}$}\Big\}.
\end{aligned}
\end{gather*}
We define the operator $A$ mapping $V_2 (\Omega )$ into $C(\Omega )$ by the
rule $Au = Lu$. Let $\overline A$ be the closure of $A$ in $L_2 (\Omega )$.

Note that the definition of regular solution $u(x,t)\in V_2(\Omega )$ of
problem 2 is same as Definition 2.1. Moreover, the definition of strong
solution of problem 2 is same as Definition 2.2, but in this case
 $\{u_n\} \subset V_2(\Omega )$.

We seek a regular solution of problem 2 in the form \eqref{e17}. Then, for
unknown functions $u_n(t)$ we have
\[
u_n''+\lambda _n^{2k}u_n(t)=f_n(t),0<t<T.
\]
The solution of this equation satisfying following two conditions $u_n(0)=0$, 
$u_n'(0)=0$ has the form
\begin{equation}
u_n(t)=\frac{1}{{\lambda _n^{k}}}\int_0^{t}{f_n(\tau )}\sin
\lambda _n^{k}(t-\tau )d\tau .  \label{e49}
\end{equation}

\begin{lemma} \label{lem4.1}
 If $f\in W_{3}(\Omega )$, then for any $t\in [0,T] $ the following 
inequalities hold:
\begin{gather}
| {u_n(t)}| \leqslant \frac{\sqrt{T}}{{\lambda _n^{k}}}\| {f_n}\| _{L_2(0,T)},
\label{e50} \\
| {u_n'(t)}| \leqslant \frac{\sqrt{T}}{{\lambda _n}}\| {f_n^{(1,0)}}\|_{L_2(0,T)},  \label{e51}
\\
| {u_n''(t)}| \leqslant \frac{1}{{\lambda _n^2}}| {f_n^{(2,0)}(t)}| 
+\frac{\sqrt{T}}{{\lambda _n}}\| {f_n^{(k+1,0)}}\| _{L_2(0,T)}.
\label{e52}
\end{gather}
\end{lemma}

\begin{proof} 
Integrating \eqref{e19} by parts $k+1$ times with respect to $x$,  we obtain
\begin{equation}
|f_n(t)|=\frac{1}{{\lambda _n^{k+1}}}|f_n^{(k+1,0)}(t)|,  \label{e53}
\end{equation}
where
\[
f_n^{(k+1,0)}(t)=\int_0^{p}{\frac{{\partial ^{k+1}f}}{{\partial
x^{k+1}}}\sqrt{\frac{2}{p}}\cos \lambda _nxdx}.
\]
Differentiating \eqref{e49} twice with respect to $t$, we obtain
\begin{gather}
u_n'(t)=\int_0^{t}{f_n(\tau )}\cos \lambda_n^{k}(t-\tau )d\tau ,  \label{e54} \\
u_n''(t)=f_n(t)-\lambda _n^{2k}u_n(t).  \label{e55}
\end{gather}
Applying the H\"{o}lder inequality to \eqref{e49}, we obtain 
inequality \eqref{e50}. Using \eqref{e53}, when $k=0$ and applying H\"{o}lder 
inequality to \eqref{e54}, we obtain inequality \eqref{e51}. Taking into 
account \eqref{e50},  we obtain
\[
\lambda _n^{2k}| {u_n(t)}| \leqslant \sqrt{T}\lambda_n^{k}\| {f_n}\| _{L_2(0,T)}\,.
\]
Furthermore, using \eqref{e53},  we obtain
\[
\lambda _n^{2k}|u_n(t)|\leq \frac{\sqrt{T}}{\lambda _n}\Vert f_n^{(k+1,0)}\Vert _{L_2(0,T)}.
\]
Using \eqref{e53} for $k=1$ and owing to \eqref{e55}, we obtain inequality 
\eqref{e52}. 
The proof is complete.
\end{proof}

\begin{theorem} \label{thm4.1} 
If $f\in W_{3}(\Omega )$, then there exists the regular solution of problem 2.
\end{theorem}

\begin{proof} 
We will prove uniform and absolute convergence of series \eqref{e17} and
\begin{gather}
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}=\sum_{n=1}^{\infty
}\lambda _n^{2k}u_n(t)X_n(x),  \label{e56} \\
\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}=\sum_{n=1}^{\infty}
f_n(t)X_n(x)-\sum_{n=1}^{\infty }\lambda_n^{2k}u_n(t)X_n(x).  \label{e57}
\end{gather}
From Lemma \ref{thm4.1} it follows that series \eqref{e17}, \eqref{e56}, and \eqref{e57}
uniformly and absolutely converges. Adding equalities \eqref{e56} and 
\eqref{e57}, we note that solution \eqref{e17} satisfies equation \eqref{e1}.
Solution \eqref{e17} satisfies boundary conditions \eqref{e2} owing to
properties of the function $X_n(x)$. From \eqref{e49} and  \eqref{e54} it
follows that solution \eqref{e17} satisfies conditions \eqref{e4}. 
The proof is complete.
\end{proof}

Second, we will establish a priori estimate for a solution of problem 2.

\begin{lemma} \label{lem4.2} 
If $u(x,t)\in V_2(\Omega )$, $\frac{{\partial ^2u}}{{\partial t^2}}\in L_2(\Omega )$, 
$\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}\in L_2(\Omega )$, then there exists 
a constant $C>0$ that depends only on numbers $T$ and $k$, and does not depend 
on the function $u(x,t)$ such that
\begin{equation}
\| u\| _{W_2^{(k,1)}(\Omega )}\leqslant C\| {Lu}\| _{L_2(\Omega )},  \label{e58}
\end{equation}
where
\[
\| u\| _{W_2^{k,1}(\Omega )}^2=\sum_{m=0}^{k}{
\| {\frac{{\partial ^{m}u}}{{\partial x^{m}}}}\| }_{L_2(\Omega )}^2
+\| {\frac{{\partial u}}{{\partial t}}} \| _{L_2(\Omega )}^2.
\]
\end{lemma}

\begin{proof} 
Multiplying by $\frac{{\partial u}}{{\partial t}}$ both
sides of the identity \eqref{e6} and integrating it over the domain 
$\Omega _{\tau }=\{ (x,t):0<x<p,0<t<\tau ;\tau <T\} $, we obtain
\begin{equation}
\int_0^{P}\int_0^{\tau }\frac{{\partial u}}{{\partial t}}
( {\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}+\frac{{\partial ^2u}}{{
\partial t^2}}}) \,dt\,dx
=\int_0^{P}\int_0^{\tau }\frac{{\partial u}}{{\partial t}}Lu\,dt\,dx.  \label{e59}
\end{equation}
Using the identities
\begin{gather*}
\frac{{\partial u}}{{\partial t}}\frac{{\partial ^{2k}u}}{{\partial x^{2k}}}
=\sum_{m=0}^{k-1}{(-1)^{m}\frac{\partial }{{\partial x}}( {\frac{
{\partial ^{m+1}u}}{{\partial t\partial x^{m}}}\frac{{\partial ^{2k-1-m}u}}{{
\partial x^{2k-1-m}}}}) +(-1)^{k}\frac{1}{2}\frac{\partial }{{\partial
t}}( {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}) ^2,}
\\
\frac{{\partial u}}{{\partial t}}\frac{{\partial ^2u}}{{\partial t^2}}=
\frac{1}{2}\frac{\partial }{{\partial t}}( {\frac{{\partial u}}{{
\partial t}}}) ^2
\end{gather*}
and conditions \eqref{e2}, \eqref{e4} and equality \eqref{e59},  we obtain
\begin{equation}
\frac{1}{2}\int_0^{p}{\big[ {\frac{{\partial ^{k}u(x,\tau )}}{{
\partial x^{k}}}}\big] ^2dx
+\frac{1}{2}}\int_0^{p}\big[ {\frac{{\partial u(x,\tau )}}{{\partial \tau }}}
\big] ^2dx=\int_0^{p} \int_0^{\tau }u_{t}Lu\,dt\,dx.  \label{e60}
\end{equation}
From equality \eqref{e60} it follows the inequality
\[
\int_0^{p}{\big[ {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}\big]
^2dx+}\int_0^{p}{\big[ {\frac{{\partial u}}{{\partial \tau }}}
\big] ^2dx\leqslant \int_0^{p}{\int_0^{T}{| {
u_{t}Lu}| \,dt\,dx.}}}
\]
Integrating it with respect to $\tau $ from $0$ to $T$, we obtain
\[
\| {\frac{{\partial ^{k}u}}{{\partial x^{k}}}}\|
_{L_2(\Omega )}^2+\| {\frac{{\partial u}}{{\partial t}}}
\| _{L_2(\Omega )}^2\leqslant 2T\int_0^{p}{
\int_0^{T}{| {u_{t}Lu}| \,dt\,dx.}}
\]
Then, in the same manner as the proof of Lemma \ref{thm2.1}, we complete the 
proof of Lemma \ref{thm4.2}.
\end{proof}

\begin{corollary} \label{coro4.1} 
Applying estimate \eqref{e58}, we can obtain the following facts:
\begin{itemize}
\item[(i)] The regular solution of problem 2 is unique and continuously depends 
on $f(x,t)$.

\item[(ii)] The inverse operator $A^{-1}$ exists and it is bounded.

\item[(iii)] $\| {A^{-1}}\| \leqslant C$.

\item[(iv)] $Ker(A)=\{0\} $.

\item[(v)] The adjoint problem to problem 2 is well-posed.
\end{itemize}
\end{corollary}

Third, we will study an existence and uniqueness of the strong solution of
the problem 1.  It is possible to represent the regular solution 
\eqref{e17} in the form 
\begin{equation}
u(x,t)=\int_0^{t}\int_0^{p}K(x,t;\xi ,\tau )f(\xi ,\tau
)\,d\xi\,d\tau ,  \label{e61}
\end{equation}
where
\begin{equation}
K(x,t;\xi ,\tau )=\sum_{n=1}^{\infty } \frac{{X_n(x)X_n(\xi )}}{{
\lambda _n^{k}}}\sin \big(\lambda _n^{k}(t-\tau )\big).  \label{e62}
\end{equation}
Obviously 
\begin{equation}
| {K(x,t;\xi ,\tau )}| \leqslant C,\quad C={\rm const.}>0.
\label{e63}
\end{equation}

The method of proof of Theorem \ref{thm2.2} enables us to establish the
following fact.

\begin{theorem} \label{thm4.2} 
For any $f\in L_2(\Omega )$ there exists unique
strong solution of problem 2 and it satisfies estimate \eqref{e58}, it
continuously depends on $f(x,t)$ and it can be represented in the form 
\eqref{e61}.
\end{theorem}

Fourth, we will study the spectrum of problem 2.

\begin{definition} \label{def4.1} \rm
We say that a problem has the Volterra property
(see \cite{k1}) if the inverse operator of the problem has the Volterra property.
\end{definition}

\begin{theorem} \label{thm4.3} 
If the number $k$ is even, then the spectrum of the
problem 2 is empty.
\end{theorem}

\begin{proof} 
From \eqref{e58}, \eqref{e61} and \eqref{e63} we conclude that
it is defined operator $A^{-1}$ on $W_{3}(\Omega )$ which is inverse of the
operator $A$ and acts from $W_{3}(\Omega )$ to $V_2(\Omega )$ by the rule
\begin{equation}
( {A^{-1}f}) (x,t)=\int_0^{p}{\int_0^{t}{
K(x,t;\xi ,\tau )}}f(\xi ,\tau )\,d\tau\,d\xi .  \label{e64}
\end{equation}
It follows from \eqref{e63} that
 $K(x,t;\xi ,\tau )\in L_2(\Omega \times \Omega )$, 
therefore $A^{-1}$ is a compact operator in $L_2(\Omega )$. As $
D(A^{-1})\equiv W_{3}(\Omega )$ is dense in $L_2(\Omega )$, the operator 
$ A^{-1}$ can be extended to whole space $L_2(\Omega )$. This extension, we
denote it by $\overline{A^{-1}}$, the closure of $A^{-1}$,
 $D(\overline{ A^{-1}})=L_2(\Omega )$. $\overline{A^{-1}}$ is a compact 
operator in $L_2(\Omega )$. Now we show that 
$\sigma (\overline{A^{-1}})=\{0\} $, where $\sigma (\overline{A^{-1}})$ 
is the spectrum of the operator $\overline{A^{-1}}$ (see \cite{s1}).
 For this purpose we calculate the
spectral radius of the operator $\overline{A^{-1}}$ (see \cite{s1}).
\[
r(\overline{A^{-1}})=\lim_{n\to \infty }( {
\| {\overline{A^{-n}}}\| }) ^{1/n}.
\]
It is easy to show that
\[
( {\overline{A^{-n}}f}) (x,t)=\int_0^{p}{
\int_0^{t}{K_n(x,t;\xi ,\tau )}}f(\xi ,\tau )\,d\tau\,d\xi ,
\]
where
\begin{gather*}
K_n(x,t;\xi ,\tau )=\int_0^{p}{\int_0^{T}{K(x,t;\xi',\tau ')K_{n-1}
(\xi ',\tau ';\xi ,\tau )}}d\tau 'd\xi ',\quad  n=2,3,\ldots , \\
K_1(x,t;\xi ,\tau )=K(x,t;\xi ,\tau ), \\
| {K_n(x,t;\xi ,\tau )}| \leqslant \frac{{C^{n}p^{n-1}(t-\tau )^{n-1}}}{{(n-1)!}}.
\end{gather*}
By direct computation,  we obtain
\[
\| {\overline{A^{-n}}}\| \leqslant \frac{{(CpT)^{n}}}{{n!}}.
\]

Using Stirling formula (see \cite{i1}) for $n!$ we convince that 
$r(\overline{A^{-1}})=\{0\} $. Since the spectrum of $\overline{A^{-1}}$ lays
in the circle $| \lambda | \leqslant r(\overline{A^{-1}})$,
 the spectrum consists of one point a zero. Consequently, operator 
$\overline{A^{-1}}$ has the Volterra property. The relation between
eigenvalues of the operator $\overline{A^{-1}}$ and $\overline{A}$ is as
follows \cite{d1}:

If $\mu _n\neq 0$ is eigenvalue of the operator $\overline{A^{-1}}$, then 
$\mu _n^{-1}$ is eigenvalue of operator $\overline{A}$. Since zero is not
eigenvalue of the operator $\overline{A^{-1}}$, then the spectrum of the
problem 2 is empty set. 
The proof of Theorem \ref{thm4.3} is complete.
\end{proof}

\begin{corollary} \label{coro4.2} 
For even $k$ the problem 2 has the Volterra property.
\end{corollary}

Note that problem 2 is not self-adjoint.

\subsection*{Problem $2^{\ast}$}
 Obtain the solution $v(x,t)$ of \eqref{e1} in
the domain $\Omega $ satisfying conditions \eqref{e2} and $v(x,T)=0$, 
$v_{t}(x,T)=0$, $0<x<p$.

The method of investigation of problem 2 is applicable also for problem
$2^{\ast}$. Therefore, all statements concerning to problem 2 are
true and for the problem $2^{\ast }$.

\begin{corollary} \label{coro4.3} 
If $k$ is odd, then from estimate \eqref{e35}
follows that the solution of \eqref{e1} belongs to $C^{\infty }(\Omega )$.
Therefore, for $k$ odd, \eqref{e1} is hypoelliptic. If $k$ is even,
then \eqref{e1} is not hypoelliptic.
\end{corollary}

\section{Conclusion}

In this article we investigated two boundary value problems for the
equation of the even order in a rectangular domain.
Problem 1 for $k$ odd  has a unique solution.
For the solution of problem 1, a priori estimate in
the norm of space $W_2^{2k,2}(\Omega )$ is obtained. 
For $k$ even, the solvability of problem 1 depends on geometry of the domain. 
In this case, there is a condition for the sizes of the domain. 
If this condition is satisfied, then problem 1 is well-posed. 
Problem 2 for $k$ odd is not correct. 
For $k$ even, this problem has the unique solution. 
For the solution of this problem, a priori estimate in the norm of space 
$W_2^{k,1}(\Omega) $ is obtained. 
The spectrum of problems 1 and 2 is studied. These
statements without proof are formulated in \cite{a2}. 
 Moreover, applying the result in \cite{a5} the two-step difference schemes 
of a high order of accuracy schemes for the numerical solution of 
well-posed problems 1 and 2 can be presented. 
Of course, the stability inequalities for the solution of
these difference schemes have been established without any assumptions about
the grid steps $\tau $ in $t$ and $h$ in the space variable $x$.

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\end{document}