On a Mixed Nonlinear One Point Boundary Value Problem for an Integrodifferential Equation

Abstract

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.

1. Introduction

In this paper, we are concerned with a one-dimensional nonlinear parabolic integrodifferential equation with Bessel operator, having the form

(1.1)

where

Well posedness of the problem is proved in a weighted Sobolev space when the problem data is a related weighted space. In [1], 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 [1], 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 [1], 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:

(2.1)

(2.2)

(2.3)

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

(2.4)

Let be the subspace of with finite norm

(2.5)

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

(2.6)

For general references and proprieties of these spaces, the reader may consult [3].

Throughout this paper, the following tools will be used.

(1)Cauchy inequality with (see, e.g., [4]),

(2.7)

which holds for all and for arbitrary and

(2)An inequality of Poincaré type,

(2.8)

where (see [5, Lemma 1]).

(3)The well-known Gronwall lemma (see, e.g., [6, Lemma 7.1].)

Remark 2.1.

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.

Theorem 3.1.

Let and Then problem (2.1)–(2.3), has at most one solution in

Proof.

Let and be two solutions of the problem (2.1)–(2.3) and let , where

(3.1)

then the function satisfies

(3.2)

(3.3)

(3.4)

If we denote by

(3.5)

then calculating the two integrals , using conditions (3.3), (3.4), and a combining with we obtain

(3.6)

In light of inequalities (2.7) and (2.8), each term of the right-hand side of (3.6) is estimated as follows:

(3.7)

Therefore, using inequalities (3.7), we infer from (3.6)

(3.8)

By applying Gronwall's lemma to (3.8), we conclude that

(3.9)

Hence

We now prove the existence theorem.

Theorem 3.2.

Let and be given and satisfying

(3.10)

for small enough and that

(3.11)

Then there exists at least one solution of problem (2.1)–(2.3).

Proof.

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

(3.12)

Given the problem

(3.13)

where

(3.14)

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

(3.15)

(3.16)

(3.17)

and is a solution of the problem

(3.18)

(3.19)

(3.20)

By multiplying (3.15), (3.18), respectively, by the operators, and , then integrating over we obtain

(3.21)

(3.22)

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,

(3.23)

and applying inequalities (2.7), (2.8), and Gronwall's lemma, we obtain the following estimates:

(3.24)

(3.25)

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

(3.26)

Direct computations yield

(3.27)

By choosing and small enough in the previous inequality, we obtain

(3.28)

Inequalities (3.21)–(3.25) then give

(3.29)

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

(3.30)

Define the function by the formula

(3.31)

then it follows from (3.26) and (3.28) that satisfies

(3.32)

Since

(3.33)

then

(3.34)

or

(3.35)

hence the continuity of the mapping The compactness of the set is due to the following.

Theorem 3.3.

Let with compact embedding (reflexive Banach spaces) (see [4, 7]) . Suppose that and Then

(3.36)

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

Remark 3.4.

For compactness of the set , see also [8, 9].

Remark 3.5.

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

Theorem 3.6.

Let be a solution of problem (2.1)–(2.3), then the following a priori estimate holds

(3.37)

Proof.

From (2.1), we have

(3.38)

Multiplying (2.1) by , integrating over carrying out standard integrations by parts, and using conditions (2.2) and (2.3) yields

(3.39)

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

(3.40)

Let be the elementary inequality

(3.41)

Adding the quantity to both sides of (3.38), then combining the resulted inequality with (3.39), we obtain

(3.42)

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).