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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 53, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/53\hfil 
Stability of third-order partial di¤erential equations]
{Stability of boundary-value problems for third-order partial
differential equations}

\author[A. Ashyralyev, Kh. Belakroum, A. Guezane-Lakoud \hfil EJDE-2017/53\hfilneg]
{Allaberen Ashyralyev, Kheireddine Belakroum, Assia Guezane-Lakoud}

\address{Allaberen Ashyralyev \newline
Department of Mathematics,
 Near East University, Nicosia, TRNC,
Mersin 10, Turkey.\newline
Institute of Mathematics and Mathematical Modeling,
050010, Almaty, Kazakhstan}
\email{allaberen.ashyralyev@neu.edu.tr}

\address{Kheireddine Belakroum \newline
Department of Mathematics,
Fr\`eres Mentouri University,
Constantine, Algeria}
\email{belakroumkheireddine@yahoo.com}

\address{Assia Guezane-Lakoud \newline
Laboratory of Advanced Materials,
Mathematics Department, Faculty of Sciences,
Badji Mokhtar Annaba University,
P.O. Box 12, Annaba, 23000, Algeria}
\email{a\_guezane@yahoo.fr}

\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted December 14, 2016. Published February 21, 2017.}
\subjclass[2010]{35G15, 47A62}
\keywords{Stability; boundary value problem; Hilbert space;
\hfill\break\indent third order partial differential equation;
self-adjoint positive definite operator}

\begin{abstract}
 We consider a boundary-value problem for the third-order partial differential
 equation
 \begin{gather*}
 \frac{d^3u(t)}{dt^3}+Au(t)=f(t),\quad 0<t<1, \\
 u(0)=\varphi ,\quad u(1)=\psi ,\quad u'(1)=\xi
 \end{gather*}
 in a Hilbert space $H$ with a self-adjoint positive definite operator $A$.
 Using the operator approach, we establish stability estimates for
 the solution of the boundary value problem. We study  three types
 of boundary value problems and obtain stability estimates for the solution
 of these problems.
\end{abstract}

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

\section{Introduction}

Boundary value problems for third order partial differential equation have
been considered in fields of sciences and engineering, such as modern
physics, chemical diffusion and mechanic fluids. The well-posedness of
various boundary-value problems for partial differential and difference
equations has been studied extensively by many researchers
\cite{A4,A3,A5,A16,A25,A18,A19,A20,A17,A26} and the references therein.

The following boundary-value problem for a third order partial
differential equation with three points boundary condition is studied in \cite{A17},
\begin{gather*}
\frac{\partial ^3u(x,t)}{\partial t^3}+\frac{\partial }{\partial x}
\Big( a( x,t), \frac{\partial u(x,t)}{\partial x}\Big) =f(x,t), \\
\int_c^1u( x,t) dx=0,\quad t\in [ 0,T],\; 0\leq c<1,\; T>0, \\
u(x,0)=0,\quad \frac{\partial u}{\partial t}(x,0)=0,\quad
\frac{\partial^2u}{\partial t^2}(x,T)=0,\quad x\in [ 0,1] ,
\end{gather*}
where $a(x,t)$ and its derivatives satisfy the condition
$0<a_{0}<a(x,t)<a_1$, $| ( a(x,t)) _x| \leq b$, and
$f(x,t)$ is given smooth function in $[ 0,1] \times [0,T] $,
 It was obtained the approximate solution of
the considered problem, the authors established a bounded linear operator
and an orthogonal basis to use the reproducing kernel space method,
numerical results are also given.

There are several methods for solving partial differential
equations. For instance, the method of operator as a tool for investigation
of the stability of partial differential equation in Hilbert and Banach
space has been systematically devoted by several authors (see for example
\cite{A8,A11,A2,A9,A7,A10,A6,A21, A12,A14,A15,A22,A23} and
the references therein).

In this article we consider the boundary-value problem for third-order
partial differential equation
\begin{equation}
\begin{gathered}
\frac{d^3u(t)}{dt^3}+Au(t)=f(t),\quad 0<t<1, \\
u(0)=\varphi ,\quad u(1)=\psi ,\quad u'(1)=\xi
\end{gathered}  \label{P3}
\end{equation}
in a Hilbert space $H$ with a self-adjoint positive definite operator
$A\geq \delta I$, where $\delta >0$.
We are interested in the stability of the solution of problem \eqref{P3}.

A function $u(t)$ is a solution of problem \eqref{P3} if the following
conditions are satisfied:
\begin{enumerate}
\item[(i)] $u(t)$ is thrice continuously differentiable on the interval
$( 0,1) $ and continuously differentiable on the segment
$[0,1] $. The derivatives at the end points of the segment are
understood as the appropriate unilateral derivatives.

\item[(ii)] The element $u(t)$ belongs to $D( A) $ for all
$t\in[ 0,1] $, and function $Au(t)$ is continuous on the segment
$[ 0,1] $.

\item[(iii)] $u(t)$ satisfies the equation and boundary conditions \eqref{P3}.
\end{enumerate}

The outline of this article is as follows.
In Section 2 the main theorem on
stability of problem \eqref{P3} is established.
Section 3 proves the stability estimates for the solution of three
problems for partial differential equations of third order in $t$.
Conclusion presents in Section 4.

\section{Main theorem on stability}

Let us prove some lemmas needed in the sequel.

\begin{lemma}[\cite{A14}] \label{L1}
For $t\geq 0$ the following estimate holds
\begin{equation}
\| \exp \{ \pm itA^{1/3}\} \| _{H\to H}\leq 1.  \label{E1}
\end{equation}
\end{lemma}

\begin{lemma}[\cite{A2}] \label{L2}
The operator $\Delta $ defined by the  formula
\begin{equation*}
\Delta =\frac{1}{3}\{ I-( ae^{-( 1+a) B}+\bar{a}
e^{-( 1+\bar{a}) B}) \}
\end{equation*}
has a bounded inverse $T=\Delta ^{-1}$ and
\begin{equation}
\| T\| _{H\to H}\leq \frac{3}{1-2e^{-(3/2) \delta ^{1/3}}}.  \label{E2}
\end{equation}
Here $a=\frac{1}{2}+i\frac{\sqrt{3}}{2}$,
$\bar{a}=\frac{1}{2}-i\frac{\sqrt{3}}{2}$, $B=A^{1/3}$.
\end{lemma}

\begin{lemma}\label{L3}
 Suppose that $\varphi \in D( A) $, $\psi \in D(A) $, $\xi \in D( A) $ and
$f(t)$ is continuously differentiable on $[ 0,1]$. Then there is unique
solution of problem \eqref{P3} and
\begin{equation}
\begin{aligned}
u(t)
&=e^{-Bt}u(0)+\frac{1}{1+a}B^{-1}( e^{-( 1-t)B}-e^{-( a+t) B}) ( u'(1)+Bu(1))\\
&\quad +\frac{1}{a-\bar{a}}B^{-2}\{ \frac{1}{1+a}( e^{-( 1-t)aB}
-e^{-( a+t) B}) \\
&\quad -\frac{1}{1+\overline{a}}(
e^{-( 1-t) \bar{a}B}-e^{-( \bar{a}+t) B})\} 
 ( u''(1)+\bar{a}Bu'(1)-aB^2u(1))\\
&\quad -\frac{1}{a-\bar{a}}B^{-2} 
 \int_{0}^{t}\Big[ \frac{1}{1+a}( e^{-( t-s)B}-e^{-( t+sa) B}) \\
&\quad -\frac{1}{1+\overline{a}}(
e^{-( t-s) B}-e^{-( t+s\overline{a}) B}) \Big]
f(s)ds,
\end{aligned}\label{E3}
\end{equation}
where
\begin{equation}
\begin{aligned}
u''(1)
&=T\Big\{ B^2e^{-B}u(0)+\frac{1}{1+a}B(I-e^{-( a+1) B}) ( u'(1)+Bu(1)) \\
&\quad +\frac{1}{a-\bar{a}}\big\{ \frac{1}{1+a}( a^2I-e^{-( a+1)
B}) -\frac{1}{1+\overline{a}}( \overline{a}^2I-e^{-( \bar{a
}+1) B}) \big\} \\
&\quad\times ( \bar{a}Bu'(1)-aB^2u(1))
 -\frac{1}{a-\bar{a}}B^{-1}\Big[ e^{-( 1+a) B}-e^{-( 1+
\overline{a}) B}\\
&\quad -\frac{1}{1+a}( e^{-( a+1) B}-I)
+\frac{1}{1+\overline{a}}( e^{-( \overline{a}+1) B}-I)\Big] f(1)
\\
&\quad -\frac{1}{a-\bar{a}}B^{-2}\Big[ \frac{1}{1+a}( I-e^{-( a+1)
B}) -\frac{1}{1+\overline{a}}( I-e^{-( \overline{a}+1)
aB}) \Big] f'(1) \\
&\quad  -\frac{1}{a-\bar{a}}\int_{0}^1\Big[ \frac{1}{1+a}(
e^{-( 1-s) B}-e^{-( sa+1) B}) \\
&\quad -\frac{1}{1+
\overline{a}}( e^{-( 1-s) B}-e^{-( s\overline{a}
+1) B}) \Big] f(s)ds\Big\} .
\end{aligned}\label{E4aa}
\end{equation}
\end{lemma}

\begin{proof}
Obviously, it can be written as the equivalent boundary-value problem for
the system of first order differential equations
\begin{equation}
\begin{gathered}
\frac{du(t)}{dt}+Bu(t)=v(t),u(0)=\varphi ,u(1)=\psi , \\
\frac{dv(t)}{dt}-aBv(t)=w(t),u'(1)=\xi , \\
\frac{dw(t)}{dt}-\bar{a}Bw(t)=f(t),\quad 0<t<1.
\end{gathered}   \label{*}
\end{equation}
Integrating these equations, we can write
\begin{equation}
\begin{gathered}
w(t)=e^{-( 1-t) \bar{a}B}w(1)-\int_{t}^1e^{-(
s-t) \bar{a}B}f( s) ds, \\
v(t)=e^{-( 1-t) aB}v(1)-\int_{t}^1e^{-( p-t) aB}w(p)dp, \\
u(t)=e^{-Bt}u(0)+\int_{0}^{t}e^{-( t-p) B}v(p)dp.
\end{gathered}  \label{**}
\end{equation}
Applying system of equations \eqref{*}, we obtain
\begin{gather*}
v(1)=u'(1)+Bu(1), \\
w(1)=v'(1)-aBv(1)=u''(1)+\bar{a}Bu'(1)-aB^2u(1).
\end{gather*}
Then, we have
\begin{equation}
w(t)=e^{-( 1-t) \bar{a}B}[ u''(1)+\bar{a}
Bu'(1)-aB^2u(1)] -\int_{t}^1e^{-( s-t)
\bar{a}B}f( s) ds.  \label{E4a}
\end{equation}
Using formulas \eqref{**}, \eqref{E4a}, we obtain
\begin{equation} \label{E4b}
\begin{aligned}
v(t)&=\frac{1}{a-\bar{a}}B^{-1}\Big( e^{-( 1-t) aB}-e^{-(
1-t) \bar{a}B}\Big) ( u''(1)+\bar{a}Bu'(1)-aB^2u(1))  \\
&\quad +e^{-( 1-t) aB}( u'(1)+Bu(1)) \\\
&\quad -\frac{1}{a-\bar{a}}B^{-1}\int_{t}^1( e^{-( s-t) aB}
 -e^{-(s-t) \bar{a}B}) f( s) ds.
\end{aligned}
\end{equation}
Using formulas \eqref{**}, \eqref{E4b}, we obtain formula \eqref{E3}.
Taking the second order derivative and putting $t=1$, we obtain the
following operator equation with respect to $u''(1)$.
\begin{equation}
\begin{aligned}
u''(1)&=B^2e^{-B}u(0)+\frac{1}{1+a}B( I-e^{-(a+1) B}) ( u'(1)+Bu(1))  \\
&\quad +\frac{1}{a-\bar{a}}\big\{ \frac{1}{1+a}( a^2I-e^{-( a+1)B})
  -\frac{1}{1+\overline{a}}( \overline{a}^2I-e^{-( \bar{a}+1) B}) \big\}  \\
&\times ( u''(1)+\bar{a}Bu'(1)-aB^2u(1)) \\
&-\frac{1}{a-\bar{a}}B^{-1}\Big[ e^{-( 1+a) B}-e^{-( 1+
\overline{a}) B}-\frac{1}{1+a}( e^{-( a+1) B}-I) \\
&\quad +\frac{1}{1+\overline{a}}( e^{-( \overline{a}+1) B}-I)
\Big] f(1) \\
&\quad -\frac{1}{a-\bar{a}}B^{-2}\Big[ \frac{1}{1+a}( I-e^{-( a+1)
B}) -\frac{1}{1+\overline{a}}( I-e^{-( \overline{a}+1)
aB}) \Big] f'(1) \\
&\quad -\frac{1}{a-\bar{a}}\int_{0}^1
\Big[ \frac{1}{1+a}( e^{-(1-s) B}-e^{-( sa+1) B}) \\
&\quad -\frac{1}{1+\overline{a}}
( e^{-( 1-s) B}-e^{-( s\overline{a}+1) B})\Big] f(s)ds.
\end{aligned}\label{E5}
\end{equation}
Since
\begin{align*}
\Delta &=I\frac{1}{a-\bar{a}}\big\{ \frac{1}{1+a}( a^2I-e^{-(
a+1) B}) -\frac{1}{1+\overline{a}}( \overline{a}
^2I-e^{-( \bar{a}+1) B}) \big\} \\
&=-\frac{1}{3}\{ I-( ae^{-( 1+a) B}+\bar{a}e^{-(1+\bar{a}) B}) \}
\end{align*}
has a bounded inverse $T=\Delta ^{-1}$, using lemma \ref{L2}, we obtain formula
\eqref{E4aa}.
The proof is complete.
\end{proof}

Now, we formulate the  main theorem.

\begin{theorem}\label{T1}
$\varphi \in D( A) $, $\psi \in D( A) $,
$\xi \in D( A^{2/3}) $ and $f(t)$ is continuously differentiable
on $[ 0,1] $. Then there is a unique solution of problem \eqref{P3}
 and the following inequalities hold
\begin{gather}
\begin{aligned}
\max_{0\leq t\leq 1} \| u(t)\| _H
&\leq M\Big\{\| \varphi \| _H+\| B^{-1}\xi \|_H+\| B^{-4}f'(1)\| _H
 +\| \psi\| _H \\
&\quad +\max_{0\leq t\leq 1} \|B^{-2}f(t)\| _H\Big\} ,
\end{aligned}  \label{E6} \\
\begin{aligned}
&\max_{0\leq t\leq 1} \| \frac{d^3u(t)}{dt^3}\| _H
+\max_{0\leq t\leq 1} \| Au\| _H \\
&\leq M\big\{ \| A\varphi \| _H+\| A\psi\| _H
  +\| A^{2/3}\xi \| _H+\| f(0)\|_H+\max_{0\leq t\leq 1} \| f'(t)\|_H\big\} ,
\end{aligned}
\label{E7}
\end{gather}
where $M$ does not depend on $f(t)$, $\varphi $, $\psi $, $\xi $.
\end{theorem}

\begin{proof}
First, we estimate $\| u(t)\| _H$ for $t\in [ 0,1] $. Applying \eqref{E3},
\eqref{E1} and triangle inequality, we obtain
\begin{equation}
\begin{aligned}
&\| u(t)\| _H \\
&\leq \| e^{-Bt}\|_{H\to H}\| \varphi \| _H+\frac{1}{|1+a| }
\| e^{-( 1-t) B}-e^{-( a+t)B}\| _{H\to H}\| B^{-1}\xi +\psi \| _H \\
&\quad +\frac{1}{| a-\bar{a}| }\frac{1}{|
1+a| }\Big\{ \| e^{-( 1-t) aB}-e^{-(a+t) B}\| _{H\to H}
+\| e^{-( 1-t)\overline{a}B} \\
&\quad -e^{-( \overline{a}+t) B}\| _{H\to H}\Big\} 
 ( \| B^{-2}u''(1)\|_H+| \bar{a}| \| B^{-1}\xi \|_H
+| a| \| \varphi \| _H) \\
&\quad + \frac{1}{| a-\bar{a}| }\frac{1}{| 1+a|} 
 \int_{0}^{t}\big[ \| e^{-( t-s) B}-e^{-( sa+t) B}\| _{H\to H}
 +\|e^{-( t-s) B}\\
&\quad -e^{-( s\overline{a}+t) B}\|_{H\to H}\big] \| B^{-2}f(s)\| _Hds\\
&\leq M\{ \| \varphi \| _H+\| B^{-1}\xi \| _H +\| \psi \| _H \\
&\quad +\max_{0\leq t\leq 1} \| B^{-2}f(t)\| _H+\| B^{-2}u''(1)\| _H\}
\end{aligned} \label{al}
\end{equation}
for any $t\in [ 0,1] $. Applying formula \eqref{E4aa}, estimate
\eqref{E2}, and the triangle inequality, we obtain
\begin{equation}
\begin{aligned}
&\| B^{-2}u''(1)\| _H \\
&\leq \| T\| _{H\to H}\Big\{ \| e^{-B}\|_{H\to H}\| \varphi \| _H+\frac{1}{|
1+a| }\| I-e^{-( a+1) B}\|_{H\to H}\| B^{-1}\xi +\psi \| _H\\
&\quad +\frac{1}{| a-\bar{a}| }\frac{1}{|1+a| }\{ \| a^2I-e^{-( a+1)
B}\| _{H\to H}+\| \overline{a}^2I-e^{-(
\overline{a}+1) B}\| _{H\to H}\}
\\
&\quad \times ( | \bar{a}| \| B^{-1}\xi \|_H+| a| \| \varphi \| _H)
+ \frac{1}{| a-\bar{a}| }\frac{1}{| 1+a|} 
 \int_{0}^1\big[ \| e^{-( 1-s)B}\\
&\quad -e^{-( sa+1) B}\| _{H\to H} 
 +\|e^{-( 1-s) B}-e^{-( s\overline{a}+1) B}\|_{H\to H}\big] \| B^{-2}f(s)\| _Hds \\
&\quad +\frac{1}{| a-\bar{a}| }\Big[ \| e^{-(a+1) B}
 -e^{-( 1+\overline{a}) B}\| _{H\to H} 
  +\frac{1}{| 1+a| }\big[ \| I-e^{-(a+1) B}\| _{H\to H} \\
&\quad +\| I -e^{-( \overline{a}+1) B}\| _{H\to H}\big] \Big]
 \|B^{-3}f(1)\| _H 
 +\frac{1}{| a-\bar{a}| }\frac{1}{|
1+a| }\big\{ \| I-e^{-( a+1) B}\|_{H\to H}\\
&\quad +\| I-e^{-( \overline{a}+1)
B}\| _{H\to H}\big\} \| B^{-4}f'(1)\| _H\Big\}  
\\
&\leq M\{ \| \varphi \| _H+\| B^{-1}\xi
\| _H+\| \psi \| _H+\| B^{-4}f'(1)\| _H+\max_{0\leq t\leq 1}\|
B^{-2}f(t)\| _H\} .
\end{aligned}\label{al1}
\end{equation}

From estimates \eqref{al} and \eqref{al1} it follows estimate \eqref{E6}.
Second, we estimate $\| Au(t)\| _H$ for $t\in [ 0,1] $.  Since
\begin{equation*}
\int_{0}^{t}\big[ \frac{1}{1+a}\big( e^{-( t-s)
B}-e^{-( t+sa) B}\big) -\frac{1}{1+\overline{a}}\big(
e^{-( t-s) B}-e^{-( t+s\overline{a}) B}\big) \big] f(s)ds,
\end{equation*}
from formula \eqref{E5} it follows that
\begin{equation}
\begin{aligned}
u(t)&=e^{-Bt}u(0)+\frac{1}{1+a}B^{-1}( e^{-( 1-t)
B}-e^{-( a+t) B}) ( u'(1)+Bu(1))
 \\
&\quad +\frac{1}{a-\bar{a}}B^{-2}\big\{ \frac{1}{1+a}( e^{-( 1-t)
aB}-e^{-( a+t) B}) -\frac{1}{1+\overline{a}}(
e^{-( 1-t) \bar{a}B} \\
&\quad -e^{-( \bar{a}+t) B})\big\}  
 ( u''(1)+\bar{a}Bu'(1)-aB^2u(1)) \\
&\quad -\frac{1}{a-\bar{a}}B^{-3} 
 \Big[ [ \frac{1}{1+a}( I+\overline{a}e^{-( 1+a)
tB}) -\frac{1}{1+\overline{a}}( I+ae^{-( 1+\overline{a}
) tB}) ] f(t) \\
&\quad -[ \frac{1+\overline{a}}{1+a}-\frac{1+a}{1+\overline{a}}]
e^{-tB}f(0) 
 -\int_{0}^{t}\Big[ \frac{1}{1+a}( e^{-( t-s)
B}+\overline{a}e^{-( t+sa) B}) \\
&\quad -\frac{1}{1+\overline{a}}
( e^{-( t-s) B}+ae^{-( t+s\overline{a}) B})
\Big] f'(s)ds\Big] .
\end{aligned} \label{alg1}
\end{equation}

In the similarly manner, applying \eqref{alg1}, \eqref{E1} and the triangle
inequality, we obtain
\begin{equation}
\begin{aligned}
\| Au(t)\| _H
&\leq M\{ \| A\varphi \| _H+\| B^2\xi \| _H+\| A\psi\| _H\\
&\quad +\| f(0)\| _H+\max_{0\leq t\leq 1}
\| f'(t)\| _H+\| Bu''(1)\| _H\}
\end{aligned}  \label{al2}
\end{equation}
for any $t\in [ 0,1] $. Applying formula \eqref{E5}, estimate
\eqref{E2}, and the triangle inequality, we obtain
\begin{equation}
\| Bu''(1)\| _H\leq M\{ \|A\varphi \| _H+\| B^2\xi \| _H
+\|A\psi \| _H +\| f(0)\| _H+\max_{0\leq t\leq 1}
\| f'(t)\| _H\} .  \label{al3}
\end{equation}

Applying estimates \eqref{al2} and \eqref{al3}, we obtain
\begin{align*}
\max_{0\leq t\leq 1}\| Au(t)\| _H
&\leq M\big\{\| A\varphi \| _H+\| B^2\xi \|_H+\| A\psi \| _H
  +\| f(0)\| _H+\max_{0\leq t\leq 1}\| f'(t)\| _H\} .
\end{align*}
From this, \eqref{P3} and the triangle inequality it follows that
\begin{align*}
\max_{0\leq t\leq 1}\| \frac{d^3u(t)}{dt^3}\| _H
&\leq \max_{0\leq t\leq 1}\| Au(t)\| _H
 +\max_{0\leq t\leq 1}\|f(t)\| _H \\
&\leq M_1\{ \| A\varphi \| _H+\| B^2\xi\| _H+\| A\psi \| _H
 +\|f(0)\| _H+\max_{0\leq t\leq 1}\| f'(t)\| _H\} .
\end{align*}
The proof is complete.
\end{proof}

\section{Applications}

In this section we consider three applications of Theorem \ref{T1}.
First application. We consider the nonlocal boundary-value problem
for a third-order partial di¤erential equation,
\begin{equation}
\begin{gathered}
\frac{\partial ^3u(t,x)}{\partial t^3}-( a(x)u_x(t,x))
_x+\delta u(t,x)=f(t,x),\quad 0<t,x<1, \\
u(0,x)=\varphi (x),\quad u(1,x)=\psi (x), \quad u_{t}(1,x)=\xi(x), \quad
0\leq x\leq 1, \\
u(t,0)=u(t,1),\quad u_x(t,0)=u_x(t,1),\quad 0\leq t\leq 1
\end{gathered}.   \label{PR3}
\end{equation}
This problem has a unique smooth solution $u(t,x)$ for smooth
$a(x)\geq a>0$, $x\in ( 0,1) $, $\delta >0$, $a(1)=a(0)$,
$\varphi(x)$, $\psi (x)$, $\xi (x)$ $(x\in [ 0,1] )$ and
$f(t,x)$ $(t\in( 0,1) ,x\in ( 0,1) )$ functions.
This allows us to reduce problem \eqref{P3} in a Hilbert space
$H=L_2[ 0,1] $ with a self-adjoint positive definite operator
$A^{x}$ defined by \eqref{PR3}.
Let us give a number of results from  the abstract Theorem \ref{T1}.

\begin{theorem}\label{T4}
For the solution of \eqref{PR3}, the following two stability
inequalities hold:
\begin{gather}
\begin{aligned}
\max_{0\leq t\leq 1}\| u(t,\cdot)\| _{L_2[ 0,1] }
&\leq M\Big[ \max_{0\leq t\leq 1}\| f(t,\cdot)\|_{L_2[ 0,1] }
+\| f_{t}(1,\cdot)\| _{L_2[ 0,1] } \\
&\quad+\| \varphi \| _{L_2[ 0,1]}+\| \psi \| _{L_2[ 0,1] }
  +\| \xi\| _{L_2[ 0,1] }\Big]  ,
\end{aligned} \label{OP6} \\
\begin{aligned}
&\max_{0\leq t\leq 1}\| u(t,\cdot)\| _{W_2^2[0,1] }
+\max_{0\leq t\leq 1}\| \frac{\partial ^3u}{
\partial t^3}(t,\cdot)\| _{L_2[ 0,1] } \\
&\leq M\Big[ \max_{0\leq t\leq 1}\| f_{t}(t,\cdot)\|_{L_2[ 0,1] }
 +\| f(0,\cdot)\| _{L_2[ 0,1] }+\| \varphi \| _{W_2^2[ 0,1]} \\
&\quad +\| \psi \| _{W_2^2[ 0,1] }+\| \xi\| _{W_2^2[ 0,1] }\Big],
\end{aligned} \label{OP7}
\end{gather}
where $M$ does not depend on $f(t,x)$ and $\varphi (x)$, $\psi (x)$,
$\xi (x)$.
\end{theorem}

\begin{proof}
Problem \eqref{PR3} can be written in the abstract form
\begin{equation}
\begin{gathered}
\frac{d^3u(t)}{dt^3}+Au(t)=f(t),\quad  0\leq t\leq 1, \\
u(0)=\varphi ,\quad u(1)=\psi ,\quad u'(1)=\xi
\end{gathered} \label{OP70}
\end{equation}
in the Hilbert space $L_2[ 0,1] $, for all square integrable
functions defined on $[ 0,1] $. Here the self-adjoint positive
definite operator $A=A^{x}$ defined by
\begin{equation}
A^{x}u(x)=-( a(x)u_x) _x+\delta u(x)  \label{OP8}
\end{equation}
with domain
\begin{equation*}
D(A^{x})=\{ u(x):u,u_x,( a(x)u_x) _x\in L_2[ 0,1
] ,u(0)=u(1),u'(0)=u'(1)\} .
\end{equation*}
where $f(t)=f(t,x)$ and $u(t)=u(t,x)$ are respectively known and unknown
abstract functions defined on $[ 0,1] $ with  $H=L_2[ 0,1] $.
Therefore, estimates \eqref{OP6}-\eqref{OP7} follow
from estimates \eqref{E6}-\eqref{E7}. The proof is complete.
\end{proof}


Second application. Let $\Omega \subset\mathbb{R}^{n}$ be a bounded open domain with
 smooth boundary $S$, $\bar{\Omega}=\Omega \cup S$.
In $[ 0,1] \times \Omega $, we consider the
boundary-value problem for a third-order partial differential equation
\begin{equation}
\begin{gathered}
\frac{\partial ^3u(t,x)}{\partial t^3}-\sum_{r=1}^{n}(
a_{r}(x)u_{x_{r}}(t,x)) _{x_{r}}=f(t,x), \\
x=(x_1,\dots ,x_{n})\in \Omega ,\ 0<t<1, \\
u(0,x)=\varphi (x),\quad u(1,x)=\psi (x),\quad u_{t}(1,x)=\xi (x),\quad x\in
\bar{\Omega}, \\
u(t,x)=0,\quad x\in S,\quad  0\leq t\leq 1,
\end{gathered}   \label{OP9}
\end{equation}
where $a_{r}(x)$, $x\in \Omega $, $\varphi (x)$, $\psi (x)$, $\xi (x)$,
$x\in \bar{\Omega}$ and $f(t,x)$ $(x\in [ 0,1] )$, $x\in \Omega $
are given smooth functions and $a_{r}(x)>0$. We introduce the Hilbert space
 $L_2(\bar{\Omega})$, the space of integrable functions defined on
$\bar{\Omega}$ equipped with norm
\begin{equation*}
\| f\| _{L_2(\bar{\Omega})}=\Big(
\idotsint_{x\in \bar{\Omega}}| f(x)|
^2dx_1\dots dx_{n}\Big) ^{1/2}.
\end{equation*}

\begin{theorem} \label{T2}
 For the solution of  \eqref{OP9} the following two stability
inequalities hold:
\begin{gather}
\begin{aligned}
\max_{0\leq t\leq 1}\| u(t,\cdot)\| _{L_2(\bar{\Omega}) }
&\leq M_2\Big[ \max_{0\leq t\leq 1}\| f(t,\cdot)\|
_{L_2( \bar{\Omega}) }+\| f_{t}(1,\cdot)\|_{L_2( \bar{\Omega}) } \\
&\quad +\| \varphi \|_{L_2( \bar{\Omega}) }
+\| \psi \|_{L_2( \bar{\Omega}) } +\| \xi \| _{L_2(\bar{\Omega}) }\Big],
\end{aligned} \label{AP8} \\
\begin{aligned}
&\max_{0\leq t\leq 1}\| u(t,\cdot)\| _{W_2^2[
0,1] }+\max_{0\leq t\leq 1}\| \frac{\partial ^3u}{
\partial t^3}(t,\cdot)\| _{L_2( \bar{\Omega}) } \\
&\leq M_2\Big[ \max_{0\leq t\leq 1}\|
f_{t}(t,\cdot)\| _{L_2( \bar{\Omega}) }
+\|f(0,\cdot)\| _{L_2( \bar{\Omega}) }
+\| \varphi\| _{W_2^2( \bar{\Omega}) } \\
&\quad +\| \psi\| _{W_2^2( \bar{\Omega}) }
+\| \xi\| _{W_2^2( \bar{\Omega}) }\Big],
\end{aligned} \label{AP9}
\end{gather}
where $M_2$ does not depend on $f(t,x)$ and $\varphi (x)$, $\psi(x)$, $\xi (x)$.
\end{theorem}

\begin{proof}
Problem \eqref{OP9} can be written in abstract form \eqref{OP70} in Hilbert
space $L_2( \bar{\Omega}) $ with self-adjoint positive definite
operator $A=A^{x}$ defined by the formula
\begin{equation}
A^{x}u(x)=-\sum_{r=1}^{n}( a_{r}(x)u_{x_{r}}) _{x_{r}}
\label{OP1}
\end{equation}
with domain
\begin{align*}
D(A^{x})=\big\{& u(x):u(x),u_{x_{r}}(x),( a_{r}(x)u_{x_{r}})
_{x_{r}}\in L_2( \bar{\Omega}) ,\; 1\leq r\leq n, \\
&u(x)=0,x\in S\big\} .
\end{align*}
Here $f(t)=f(t,x)$ and $u(t)=u(t,x)$ are known and unknown respectively
abstract functions defined on $[ 0,1] $ with the value in $
H=L_2( \bar{\Omega}) $. So, estimates \eqref{AP8}-\eqref{AP9}
follow from estimates \eqref{E6}-\eqref{E7} and from the coercivity
inequality for the solution of the elliptic differential problem in $
L_2( \bar{\Omega}) $.
\end{proof}

\begin{theorem}\label{T21}
For the solution of the elliptic differential problem
\begin{equation*}
-\sum_{r=1}^{n}(a_{r}(x)u_{x_{r}})_{x_{r}}=w(x),\quad x\in \Omega ,\quad
u(x)=0, \;x\in S
\end{equation*}
the  coercivity inequality
\begin{equation*}
\sum_{r=1}^{n}\| u_{x_{r}x_{r}}\| _{L_2(
\overline{\Omega }) }\leq M\| w\| _{L_2(
\overline{\Omega }) }
\end{equation*}
is valid \cite{A24}.
Here $M$ does not depend on $w(x)$.
\end{theorem}


Third application. We consider the boundary-value problem for a
third-order partial differential equation
\begin{equation}
\begin{gathered}
\frac{\partial ^3u(t,x)}{\partial t^3}-\sum_{r=1}^{n}(
a_{r}(x)u_{x_{r}}(t,x)) _{x_{r}}+\delta u(t,x)=f(t,x), \\
x=(x_1,\dots ,x_{n})\in \Omega ,\quad 0<t<1, \\
u(0,x)=\varphi (x),\quad u(1,x)=\psi (x),\quad u_{t}(1,x)=\xi (x),\quad x\in
\bar{\Omega}, \\
\frac{\partial u}{\partial \vec{n}}(t,x)=0,\quad x\in S,\; 0\leq t\leq 1,
\end{gathered}   \label{OP2}
\end{equation}
where $a_{r}(x)$, $x\in \Omega $, $\varphi (x)$, $\psi (x)$, $\xi (x)$,
$x\in \bar{\Omega}$ and $f(t,x)$ $(x\in [ 0,1] )$, $x\in \Omega $
are given smooth functions and $a_{r}(x)>0$, $\vec{n}$ is the
normal vector to $S$.

\begin{theorem} \label{T3}
For the solution of  \eqref{OP2}, the following two stability inequalities
hold:
\begin{gather}
\begin{aligned}
\max_{0\leq t\leq 1}\| u(t,\cdot)\| _{L_2(\bar{\Omega}) }
&\leq M_3\Big[ \max_{0\leq t\leq 1}\| f(t,\cdot)\|_{L_2( \bar{\Omega}) }
+\| f_{t}(1,\cdot)\|_{L_2( \bar{\Omega}) } \\
&\quad +\| \varphi \|_{L_2( \bar{\Omega}) }
 +\| \psi \|_{L_2( \bar{\Omega}) }+\| \xi \| _{L_2(\bar{\Omega}) }\Big],
\end{aligned}  \label{OP3} \\
\begin{aligned}
&\max_{0\leq t\leq 1}\| u(t,\cdot)\| _{W_2^2[
0,1] }+\max_{0\leq t\leq 1}\| \frac{\partial ^3u}{
\partial t^3}(t,\cdot)\| _{L_2( \bar{\Omega}) } \\
&\leq M_3\Big[ \max_{0\leq t\leq 1}\|
f_{t}(t,\cdot)\| _{L_2( \bar{\Omega}) }
+\|f(0,\cdot)\| _{L_2( \bar{\Omega}) }
+\| \varphi\| _{W_2^2( \bar{\Omega}) } \\
&\quad +\| \psi\| _{W_2^2( \bar{\Omega}) }
+\| \xi\| _{W_2^2( \bar{\Omega}) }\Big],
\end{aligned} \label{OP4}
\end{gather}
where $M_3$ does not depend on $f(t,x)$ and $\varphi (x)$, $\psi(x)$, $\xi (x)$.
\end{theorem}

\begin{proof}
Problem \eqref{OP2} can be written in the abstract form \eqref{OP70} in
the Hilbert space $L_2( \bar{\Omega}) $ with a self-adjoint positive definite
operator $A=A^{x}$ defined by the formula
\begin{equation}
A^{x}u(x)=-\sum_{r=1}^{m}( a_{r}(x)u_{x_{r}})_{x_{r}}+\delta u(x)  \label{OP5}
\end{equation}
with domain
\begin{align*}
D(A^{x})=\Big\{ &u(x):u(x),u_{x_{r}}(x),( a_{r}(x)u_{x_{r}})
_{x_{r}}\in L_2( \bar{\Omega}) ,\; 1\leq r\leq m,\\
&\frac{\partial u}{\partial \vec{n}}=0,\; x\in S\Big\} .
\end{align*}
Here $f(t)=f(t,x)$ and $u(t)=u(t,x)$ are respectively known and unknown
abstract functions defined on $[ 0,1] $ with the value in
$H=L_2( \bar{\Omega}) $. So, estimates \eqref{OP3}-\eqref{OP4}
follow from estimates \eqref{E6}-\eqref{E7} and from the coercivity
inequality for the solution of the elliptic differential problem in
$L_2( \bar{\Omega}) $.
\end{proof}

\begin{theorem} \label{T22}
For the solution of the elliptic differential problem
\begin{equation*}
-\sum_{r=1}^{n}(a_{r}(x)u_{x_{r}})_{x_{r}}+\delta u(x)=w(x),x\in
\Omega ,\frac{\partial }{\partial \vec{n}}u(x)=0,\text{ }x\in S
\end{equation*}
the  coercivity inequality
\begin{equation*}
\sum_{r=1}^{n}\| u_{x_{r}x_{r}}\| _{L_2(
\overline{\Omega }) }\leq M\| w\| _{L_2(
\overline{\Omega }) }
\end{equation*}
is valid \cite{A24}. Here $M$ does not depend on $w(x)$
\end{theorem}

\subsection*{Conclusions}

This article is devoted to the stability of the boundary value problem for a
third order partial differential equation. Theorem on stability estimates
for the solution of this problem is established. Three applications of the
main theorem to a third order partial differential equations are given.
Theorems on stability estimates for solutions of these partial differential
equations are obtained.

In papers \cite{A1}, \cite{A2}, three step difference schemes generated by
Taylor's decomposition on three points for the numerical solution of local
and nonlocal boundary value problems of the linear ordinary differential
equation of third order were investigated.

Note that Taylor's decomposition on four points is applicable for the
construction of difference schemes of problem \eqref{P3}. Operator method of
\cite{A10} permits to establish the stability of these difference problem
for the approximation problem of \eqref{P3}.\bigskip

\subsection*{Acknowledgements}

The research was supported by the Ministry of Education and Science of
the Russian Federation (Agreement number 02.a03.21.0008).

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\end{document}