Discontinuous Parabolic Problems with a Nonlocal Initial Condition

Abstract

We study parabolic differential equations with a discontinuous nonlinearity and subjected to a nonlocal initial condition. We are concerned with the existence of solutions in the weak sense. Our technique is based on the Green's function, integral representation of solutions, the method of upper and lower solutions, and fixed point theorems for multivalued operators.

1. Introduction

Let be a an open bounded domain in , with a smooth boundary Let and where is a positive real number Then is smooth and any point on satisfies the inside (and outside) strong sphere property (see [1]). For we denote its partial derivatives in the distributional sense (when they exist) by ,

In this paper, we study the following parabolic differential equation with a nonlocal initial condition

(11)

where is not necessarily continuous, but is such that for every fixed the function is measurable and is of bounded variations over compact interval in and nondecreasing, and is continuous; is a strongly elliptic operator given by

(12)

Discontinuous parabolic problems have been studied by many authors, see for instance [2–5]. Parabolic problems with integral conditions appear in the modeling of concrete problems, such as heat conduction [6–10] and in thermoelasticity [11].

In order to investigate problem (1.1), we introduce some notations, function spaces, and notions from set-valued analysis.

Let denote the Banach space of all continuous functions , equipped with the norm Let for each and for each For we say that is in if is measurable and in which case we define its norm by

(13)

Let and let denote the Sobolev space of functions having first generalized derivatives in and let be its corresponding dual space. Then and they form an evolution triple with all embeddings being continuous, dense, and compact (see [2, 12]). The Bochner space (see [13]) is the set of functions with generalized derivative For we define its norm by

(14)

Then is a separable reflexive Banach space. The embedding of into is continuous and the embedding is compact.

Now, we introduce some facts from set-valued analysis. For complete details, we refer the reader to the following books. [14–16]. Let and be Banach spaces. We will denote the set of all subsets, of having property by For instance, denotes the set of all nonempty subsets of ; means closed in when we have the bounded subsets of for convex subsets, for compact subsets and for compact and convex subsets. The domain of a multivalued map is the set    is convex (closed) valued if is convex (closed) for each    is bounded on bounded sets if is bounded in for all (i.e.,    is called upper semicontinuous (u.s.c.) on if for each the set is nonempty, and for each open subset of containing , there exists an open neighborhood of such that In terms of sequences, is usc if for each sequence , , and is a closed subset of such that then

The set-valued map is called completely continuous if is relatively compact in for every If is completely continuous with nonempty compact values, then is usc if and only if has a closed graph (i.e., , ). When then has a fixed point if there exists such that A multivalued map is called measurable if for every , the function defined by is measurable. denotes The Kuratowski measure of noncompactness (see [15, page 113]) of is defined by

(15)

The Kuratowski measure of noncompactness satisfies the following properties.

(i) if and only if is compact;

(ii)

(iii)

(iv), ;

(v) where denotes the convex hull of .

Definition 1.1 (see [17]).

A function is called N- measurable on if for every measurable function the function is measurable.

Examples of N- measurable functions are Carathéodory functions, Baire measurable functions.

Let and Then (see [17, Proposition ]) the function is lower semicontinuous, that is, for every the set is open for any , and the function is upper semicontinuous, that is, for every the set is open for any . Moreover, the functions and are nondecreasing.

Definition 1.2.

The multivalued function defined by for all is called N- measurable on if both functions and are N- measurable on .

Definition 1.3.

The operator defined by

(16)

is called the Nemitskii operator of the multifunction

Since is an N- measurable and upper semicontinuous multivalued function with compact and convex values, we have the following properties for the operator (see [17, Corollary ]).

Lemma 1.4.

is N-measurable, compact and convex-valued, upper semicontinuous and maps bounded sets into precompact sets.

We will consider solutions of problem (1.1) as solutions of the following parabolic problem with multivalued right-hand side:

(17)

where for all As pointed out in [15, Example page 5], this is the most general upper semicontinuous set-valued map with compact and convex values in .

Theorem 1.5 (see [18]).

Let be a Banach space and a condensing map. If the set for some is bounded, then has a fixed point.

We remark that a compact map is the simplest example of a condensing map.

2. The Linear Problem

We will assume throughout this paper that the functions are Hölder continuous, and moreover, there exist positive numbers , and such that

(21)

Given a continuous function the linear parabolic problem

(22)

is well known and completely solved (see the books [1, 19, 20]).

The linear homogeneous problem

(23)

has only the trivial solution. There exists a unique function, called Green's function corresponding to the linear homogeneous problem. This function satisfies the following (see [1, 20]):

(i)

(ii)

(iii),

(iv) for

(v) and are continuous functions of

(vi) for some positive constants (see [19]);

(vii)for any Hölder continuous function : , the function , given for by is the unique classical solution, that is, of the nonhomogeneous problem (2.2).

It is clear from property (vi) above that Also, the integral representation in (vii) implies that the function is continuous. Let

Lemma 2.1.

If then (2.2) has a unique weak solution Moreover, there exists a positive constant , depending only on and such that

(24)

Proof.

Consider the following representation (see property (vii) above):

(25)

Define an operator by

(26)

Then is a bounded linear operator with

(27)

Then for each

(28)

This implies that for each

(29)

Minkowski's inequality leads to

(210)

3. Problem with a Discontinuous Nonlinearity

In this section, we investigate the multivalued problem (1.7). We define the notion of a weak solution.

Definition 3.1.

A solution of (1.7) is a function such that

(i)there exists with   

(ii)   

(iii)   

We introduce the notion of lower and upper solutions of problem (1.7).

Definition 3.2.

is a weak lower solution of (1.7) if

(i)   

(ii)   

(iii)   

Definition 3.3.

is a weak upper solution of (1.7) if

(j)   

(jj)   

(jjj)   

We will assume that the function , generating the multivalued function , is N- measurable on , which implies that is an N- measurable, upper semicontinuous multivalued function with nonempty, compact, and convex values. In addition, we will need the following assumptions:

(H1)there exists such that   

(H2)there exist a lower solution and an upper solution of (1.7) such that ;

(H3) is continuous, and is nondecreasing with

We state and prove our main result.

Theorem 3.4.

Assume that (H1), (H2), and (H3) are satisfied. Then the multivalued problem (1.7) has at least one solution

Proof.

First, it is clear that the operator defined by

(31)

is continuous and uniformly bounded. Consider the modified problem

(32)

We show that possible solutions of (3.2) are a priori bounded. Let be a solution of (3.2). It follows from the definition and the representation (2.5) that for each

(33)

where with Since is continuous and is uniformly bounded there exists such that Also, assumption (H1) implies that The relation (3.3) together with Lemma 2.1 yields

(34)

where depends only on Let

It is clear that solutions of (3.2) are fixed point of the multivalued operator , defined by

(35)

Here, is a single-valued operator defined by

(36)

and is a multivalued operator defined by

(37)

Claim 1.

is compact in . Since the function is continuous and the operator is uniformly boundedthere exists such that Also, is continuous and has no singularity for . It follows that the operator is continuous and there exists  depending only on and such that so that is uniformly bounded in Since the embedding is compact it follows that is compact in

Claim 2.

is also compact in . This follows from the continuity of the Green's function and the properties of the Nemitski operator See Lemma 1.4.

Claim 3.

that is, it is a condensing multifunctionWe have

Also Lemma 1.4 implies that has nonempty, compact, convex values. Since is single-valued, the operator has nonempty compact and convex values. We show that has a closed graph. Let and We show that Now, implies that It is clear that in We can use the last part of Lemma in [13] to conclude that which, in turn, implies that This will imply that is upper semicontinuous.

Therefore, is condensing. t remains to show that the set for some is bounded; but this is a consequence of inequality (3.4). Theorem 1.5 implies that the operator has a fixed point which is a solution of (3.2).

We, now, show that We prove that It follows from the definition of a solution of (3.2) that there exists with , such that

(38)

On the other hand, satisfies

(39)

Let = for each Then

(310)

Since and the functions and are nondecreasing, it follows that so that for a.e. We can show in a similar way that for a.e. In this case , and (3.2) reduces to (1.7). Therefore, problem (1.7) has a solution, and consequently, (1.1) has a solution.

4. Example

Consider the problem

(41)

Let It is clear that is a classical solution of the problem

(42)

and is a classical solution of the problem

(43)

Let where is a solution of the problem on and Then and is an upper solution of problem (4.1) provided that

Similarly, let be a solution of on and Then and is a lower solution of problem (4.1) provided that