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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 17, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/17\hfil Continuous dependence of solutions]
{Continuous dependence of solutions to mixed boundary value
problems for a parabolic equation}
\author[M. Z. Djibibe, K. Tcharie, N. I. Yurchuk\hfil EJDE-2008/17\hfilneg]
{Moussa Zakari Djibibe, Kokou Tcharie, Nikolay Iossifovich Yurchuk } % in alphabetical order
\address{Moussa Zakari Djibibe\newline
University of Lom\'e - Togo\\
Department of Mathematics \\
PO Box 1515 Lom\'e, Togo}
\email{mdjibibe@tg.refer.org\; zakari.djibibe@gmail.com\; tel +228 924 45 21}
\address{Kokou Tcharie \newline
University of Lom\'e - Togo\\
Department of Mathematics\\
PO Box 1515 Lom\'e, Togo}
\email{tkokou@yahoo.fr}
\address{Nikolay Iossifovich Yurchuk \newline
Dept. of Mechanics and Mathematics, Belarussian State University,
220050, Minsk, Belarus}
\email{yurchuk@bsu.by}
\thanks{Submitted October 16, 2007. Published February 5, 2008.}
\subjclass[2000]{35K20, 35K25, 35K30}
\keywords{Priori estimate; mixed problem; continuous dependence;
\hfill\break\indent boundary conditions}
\begin{abstract}
We prove the continuous dependence, upon
the data, of solutions to second-order parabolic equations.
We study two boundary-value problems: One has a nonlocal (integral)
condition and the another has a local boundary condition.
The proofs are based on a priori estimate for the difference
of solutions.
\end{abstract}
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\section{Introduction}
This paper is devoted to the proof of the continuous dependence, upon
the data, of generalized solutions of a second order parabolic equation.
The boundary conditions are of mixed type.
This article contributes to the development of the a priori estimates
method for solving such problems.
The questions related to these problems are so miscellaneous that the
elaboration of a general theory is still premature. Therefore,
the investigation of these problems requires at every time a
separate study.
The importance of problems with integral condition has been pointed out
by Samarskii \cite{ref24}. Mathematical modelling by evolution
problems with a nonlocal constraint of the form
${\frac{1}{1 -\alpha}\int_{\alpha}^1 u(x, t)\,dx = E(t)}$
is encountered in heat transmission theory, thermoelasticity,
chemical engineering, underground water flow, and plasma physic.
See for instance Benouar-Yurchuk \cite{ref1}, Benouar-Bouziani
\cite{ref2}-\cite{ref3}, Bouziani \cite{ref4}-\cite{ref7},
Cannon et al \cite{ref14}-\cite{ref16}, Ionkin
\cite{ref18}-\cite{ref19}, Kamynin \cite{ref20} and Yurchuk
\cite{ref26}-\cite{ref28}.
Mixed problems with nonlocal boundary
conditions or with nonlocal initial conditions were studied in
Bouziani \cite{ref7}-\cite{ref9}, Byszewski et al
\cite{ref11}-\cite{ref13}, Gasymov \cite{ref17}, Ionkin
\cite{ref18}-\cite{ref19}, Lazhar \cite{ref22},
Mouravey-Philipovski \cite{ref23} and Said-Nadia
\cite{ref25}.
The results and the method used here are a further elaboration of
those in \cite{ref1}. We should
mention here that the presence of integral term in the boundary
condition can greatly complicate the application of standard
functional and numerical techniques.
This work can be considered as a continuation of the results
in \cite{ref7}, \cite{ref26} and \cite{ref28}.
The remainder of the paper is divided into four section.
In Section 2, we give the statement of the problem.
Then in Section 3, we establish a priori estimate. Finally, in section
4, we show the continuous dependence of a solution upon the data.
\section{Statement of the problem}
In the rectangle $G = \{(x,t) : 0