On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
© Said Mesloub. 2008
Received: 31 August 2007
Accepted: 5 February 2008
Published: 18 March 2008
This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the regularity of a solution.
Well posedness of the problem is proved in a weighted Sobolev space when the problem data is a related weighted space. In , a model of a one-dimensional quasistatic contact problem in thermoelasticity with appropriate boundary conditions is given and this work is motivated by the work of Xie , where the author discussed the solvability of a class of nonlinear integrodifferential equations which arise from a one-dimensional quasistatic contact problem in thermoelasticity. The author studied the existence, uniqueness, and regularity of solutions. We refer the reader to [1, 2], and references therein for additional information. In the present paper, following the method used in , we will prove the existence and uniqueness of (see below for definition) solutions of a nonlinear parabolic integrodifferential equation with Bessel operator supplemented with a one point boundary condition and an initial condition. The proof is established by exploiting some a priori estimates and using a fixed point argument.
2. The Problem
For general references and proprieties of these spaces, the reader may consult .
Throughout this paper, the following tools will be used.
where (see [5, Lemma 1]).
(3)The well-known Gronwall lemma (see, e.g., [6, Lemma 7.1].)
The need of weighted spaces here is because of the singular term appearing in the left-hand side of (2.1) and the annihilation of inconvenient terms during integration by parts.
3. Existence and Uniqueness of the Solution
We now prove the existence theorem.
The author is grateful to the anonymous referees for their helpful suggestions and comments which allowed to correct and improve the paper. This work has been funded and supported by the Research Center Project no. Math/2008/19 at King Saud University.
- Xie WQ: A class of nonlinear parabolic integro-differential equations. Differential and Integral Equations 1993,6(3):627-642.MathSciNetMATHGoogle Scholar
- Shi P, Shillor M: A quasistatic contact problem in thermoelasticity with a radiation condition for the temperature. Journal of Mathematical Analysis and Applications 1993,172(1):147-165. 10.1006/jmaa.1993.1013MathSciNetView ArticleMATHGoogle Scholar
- Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 65. Academic Press, New York, NY, USA; 1975:xviii+268.Google Scholar
- Lions J-L: Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der mathematischen Wissenschaften. Volume 111. Springer, Berlin, Germany; 1961:ix+292.Google Scholar
- Ladyzhenskaya OA: The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences. Volume 49. Springer, New York, NY, USA; 1985:xxx+322.View ArticleGoogle Scholar
- Garding L: Cauchy Problem for Hyperbolic Equations, Lecture Notes. University of Chicago, Chicago, Ill, USA; 1957.Google Scholar
- Simon J:Compact sets in the space . Annali di Matematica Pura ed Applicata 1987,146(1):65-96.View ArticleMATHGoogle Scholar
- Aubin J-P: Un théorème de compacité. Comptes Rendus de l'Académie des Sciences 1963, 256: 5042-5044.MathSciNetMATHGoogle Scholar
- Dubinskii JuA: Weak convergence for nonlinear elliptic and parabolic equations. Matematicheskii Sbornik 1965,67(109)(4):609-642. (Russian)MathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.