- Research Article
- Open Access
On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation
Boundary Value Problems volume 2008, Article number: 814947 (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.
In this paper, we are concerned with a one-dimensional nonlinear parabolic integrodifferential equation with Bessel operator, having the form
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
We consider the following problem:
where and are given functions with assumptions that will be given later.
In this paper, denotes the usual norm of the weighted space where we use the weights and while . The respective inner products on and are given by
Let be the subspace of with finite norm
and be the subspace of whose elements satisfy . In general, a function in the space , with , nonnegative integers possesses -derivatives up to th order in the and th derivatives up to th order in We also use weighted spaces in the interval such as and whose definitions are analogous to the spaces on We set
For general references and proprieties of these spaces, the reader may consult .
Throughout this paper, the following tools will be used.
(1)Cauchy inequality with (see, e.g., ),
which holds for all and for arbitrary and
(2)An inequality of Poincaré type,
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
Let and be two solutions of the problem (2.1)–(2.3) and let , where
then the function satisfies
If we denote by
then calculating the two integrals , using conditions (3.3), (3.4), and a combining with we obtain
In light of inequalities (2.7) and (2.8), each term of the right-hand side of (3.6) is estimated as follows:
Therefore, using inequalities (3.7), we infer from (3.6)
By applying Gronwall's lemma to (3.8), we conclude that
We now prove the existence theorem.
Let and be given and satisfying
for small enough and that
Then there exists at least one solution of problem (2.1)–(2.3).
We define, for positive constants and which will be specified later, a class of functions which consists of all functions satisfying conditions (2.2), (2.3), and
Given the problem
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.
We, first, show that maps into itself. For this purpose we write in the form where is a solution of the problem
and is a solution of the problem
By multiplying (3.15), (3.18), respectively, by the operators, and , then integrating over we obtain
By using conditions (3.16), (3.17), (3.19), (3.20), an evaluation of the left-hand side of both equalities (3.21) and (3.22) gives, respectively,
and applying inequalities (2.7), (2.8), and Gronwall's lemma, we obtain the following estimates:
We also multiply by and square both sides of (3.15), integrate over , use the integral then integrate by parts and using inequality (2.7), we obtain
Direct computations yield
By choosing and small enough in the previous inequality, we obtain
Inequalities (3.21)–(3.25) then give
At this point we take and so that it follows from the last two inequalities that and from which we deduce that hence maps into itself. To show that is a continuous mapping, we consider and their corresponding images and It is straightforward to see that satisfies
Define the function by the formula
then it follows from (3.26) and (3.28) that satisfies
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
Let be a solution of problem (2.1)–(2.3), then the following a priori estimate holds
From (2.1), we have
Multiplying (2.1) by , integrating over carrying out standard integrations by parts, and using conditions (2.2) and (2.3) yields
Adding side to side equalities (3.38) and (3.39), then using inequalities (2.7) and (2.8) to estimate the involved integral terms to get
Let be the elementary inequality
Adding the quantity to both sides of (3.38), then combining the resulted inequality with (3.39), we obtain
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).
Xie WQ: A class of nonlinear parabolic integro-differential equations. Differential and Integral Equations 1993,6(3):627-642.
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.1993.1013
Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 65. Academic Press, New York, NY, USA; 1975:xviii+268.
Lions J-L: Équations Différentielles Opérationnelles et Problèmes aux Limites, Die Grundlehren der mathematischen Wissenschaften. Volume 111. Springer, Berlin, Germany; 1961:ix+292.
Ladyzhenskaya OA: The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences. Volume 49. Springer, New York, NY, USA; 1985:xxx+322.
Garding L: Cauchy Problem for Hyperbolic Equations, Lecture Notes. University of Chicago, Chicago, Ill, USA; 1957.
Simon J:Compact sets in the space . Annali di Matematica Pura ed Applicata 1987,146(1):65-96.
Aubin J-P: Un théorème de compacité. Comptes Rendus de l'Académie des Sciences 1963, 256: 5042-5044.
Dubinskii JuA: Weak convergence for nonlinear elliptic and parabolic equations. Matematicheskii Sbornik 1965,67(109)(4):609-642. (Russian)
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.