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
where and are given functions with assumptions that will be given later.
For general references and proprieties of these spaces, the reader may consult .
Throughout this paper, the following tools will be used.
which holds for all and for arbitrary and
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 are now ready to establish the existence and uniqueness of solutions of problem (2.1)–(2.3). We first start with a uniqueness result.
Let and Then problem (2.1)–(2.3), has at most one solution in
We now prove the existence theorem.
Then there exists at least one solution of problem (2.1)–(2.3).
has a unique solution . We define a mapping such that
Once it is proved that the mapping has a fixed point in the closed bounded convex subset then is the desired solution.
hence the continuity of the mapping The compactness of the set is due to the following.
is compactly embedded in , that is, the bounded sets are relatively compact in
Note that , By the Schauder fixed point theorem the mapping has a fixed point in
The following theorem gives an a priori estimate which may be used in establishing a regularity result for the solution of (2.1)–(2.3). More precisely, one should expect the solution to be in with
Applying Gronwall's lemma to (3.40) and then taking the supremum with respect to over the interval we obtain the desired a priori bound (3.37).
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.
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