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Boundary Value Problems

, 2011:965759 | Cite as

Discontinuous Parabolic Problems with a Nonlocal Initial Condition

  • Abdelkader Boucherif
Open Access
Research Article
  • 946 Downloads
Part of the following topical collections:
  1. Nonlocal Boundary Value Problems

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.

Keywords

Weak Solution Multivalued Function Parabolic Problem Unique Weak Solution Multivalued Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

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

In this paper, we study the following parabolic differential equation with a nonlocal initial condition
where Open image in new window is not necessarily continuous, but is such that for every fixed Open image in new window the function Open image in new window is measurable and Open image in new window is of bounded variations over compact interval in Open image in new window and nondecreasing, and Open image in new window is continuous; Open image in new window is a strongly elliptic operator given by

Discontinuous parabolic problems have been studied by many authors, see for instance [2, 3, 4, 5]. Parabolic problems with integral conditions appear in the modeling of concrete problems, such as heat conduction [6, 7, 8, 9, 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 Open image in new window and let Open image in new window denote the Sobolev space of functions Open image in new window having first generalized derivatives in Open image in new window and let Open image in new window be its corresponding dual space. Then Open image in new window and they form an evolution triple with all embeddings being continuous, dense, and compact (see [2, 12]). The Bochner space Open image in new window (see [13]) is the set of functions Open image in new window with generalized derivative Open image in new window For Open image in new window we define its norm by

Then Open image in new window is a separable reflexive Banach space. The embedding of Open image in new window into Open image in new window is continuous and the embedding Open image in new window is compact.

Now, we introduce some facts from set-valued analysis. For complete details, we refer the reader to the following books. [14, 15, 16]. Let Open image in new window and Open image in new window be Banach spaces. We will denote the set of all subsets, of Open image in new window having property Open image in new window by Open image in new window For instance, Open image in new window denotes the set of all nonempty subsets of Open image in new window ; Open image in new window means Open image in new window closed in Open image in new window when Open image in new window we have the bounded subsets of Open image in new window Open image in new window for convex subsets, Open image in new window for compact subsets and Open image in new window for compact and convex subsets. The domain of a multivalued map Open image in new window is the set Open image in new window    Open image in new window is convex (closed) valued if Open image in new window is convex (closed) for each Open image in new window    Open image in new window is bounded on bounded sets if Open image in new window is bounded in Open image in new window for all Open image in new window (i.e., Open image in new window    Open image in new window is called upper semicontinuous (u.s.c.) on Open image in new window if for each Open image in new window the set Open image in new window is nonempty, and for each open subset Open image in new window of Open image in new window containing Open image in new window , there exists an open neighborhood Open image in new window of Open image in new window such that Open image in new window In terms of sequences, Open image in new window is usc if for each sequence Open image in new window , Open image in new window , and Open image in new window is a closed subset of Open image in new window such that Open image in new window then Open image in new window

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

The Kuratowski measure of noncompactness satisfies the following properties.

(i) Open image in new window if and only if Open image in new window is compact;

(ii) Open image in new window

(iii) Open image in new window

(iv) Open image in new window , Open image in new window ;

(v) Open image in new window where Open image in new window denotes the convex hull of Open image in new window .

Definition 1.1 (see [17]).

A function Open image in new window is called N- measurable on Open image in new window if for every measurable function Open image in new window the function Open image in new window is measurable.

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

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

Definition 1.2.

The multivalued function Open image in new window defined by Open image in new window for all Open image in new window is called N- measurable on Open image in new window if both functions Open image in new window and Open image in new window are N- measurable on Open image in new window .

Definition 1.3.

is called the Nemitskii operator of the multifunction Open image in new window

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

Lemma 1.4.

Open image in new window 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:

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

Theorem 1.5 (see [18]).

Let Open image in new window be a Banach space and Open image in new window a condensing map. If the set Open image in new window for some Open image in new window is bounded, then Open image in new window 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 Open image in new window are Hölder continuous, Open image in new window and moreover, there exist positive numbers Open image in new window , and Open image in new window such that
Given a continuous function Open image in new window the linear parabolic problem

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

The linear homogeneous problem

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

(i) Open image in new window

(ii) Open image in new window

(iii) Open image in new window , Open image in new window

(iv) Open image in new window for Open image in new window

(v) Open image in new window and Open image in new window are continuous functions of Open image in new window

(vi) Open image in new window for some positive constants Open image in new window (see [19]);

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

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

Lemma 2.1.

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

Proof.

Consider the following representation (see property (vii) above):
Then Open image in new window is a bounded linear operator with
This implies that for each Open image in new window
Minkowski's inequality leads to

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 Open image in new window such that

(i)there exists Open image in new window with Open image in new window    Open image in new window

(ii) Open image in new window    Open image in new window

(iii) Open image in new window    Open image in new window

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

Definition 3.2.

Open image in new window is a weak lower solution of (1.7) if

(i) Open image in new window    Open image in new window

(ii) Open image in new window    Open image in new window

(iii) Open image in new window    Open image in new window

Definition 3.3.

Open image in new window Open image in new window is a weak upper solution of (1.7) if

(j) Open image in new window    Open image in new window

(jj) Open image in new window Open image in new window    Open image in new window

(jjj) Open image in new window Open image in new window    Open image in new window

We will assume that the function Open image in new window , generating the multivalued function Open image in new window , is N- measurable on Open image in new window , which implies that Open image in new window 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 Open image in new window such that Open image in new window    Open image in new window

(H2)there exist a lower solution Open image in new window and an upper solution Open image in new window of (1.7) such that Open image in new window ;

(H3) Open image in new window is continuous, and Open image in new window is nondecreasing with Open image in new window

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 Open image in new window

Proof.

First, it is clear that the operator Open image in new window defined by
is continuous and uniformly bounded. Consider the modified problem
We show that possible solutions of (3.2) are a priori bounded. Let Open image in new window be a solution of (3.2). It follows from the definition and the representation (2.5) that for each Open image in new window
where Open image in new window with Open image in new window Since Open image in new window is continuous and Open image in new window is uniformly bounded there exists Open image in new window such that Open image in new window Also, assumption (H1) implies that Open image in new window The relation (3.3) together with Lemma 2.1 yields

where Open image in new window depends only on Open image in new window Let Open image in new window

It is clear that solutions of (3.2) are fixed point of the multivalued operator Open image in new window , defined by
Here, Open image in new window is a single-valued operator defined by
and Open image in new window is a multivalued operator defined by

Claim 1.

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

Claim 2.

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

Claim 3.

Open image in new window that is, it is a condensing multifunction Open image in new window We have Open image in new window

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

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

We, now, show that Open image in new window We prove that Open image in new window It follows from the definition of a solution of (3.2) that there exists Open image in new window with Open image in new window Open image in new window , such that
On the other hand, Open image in new window satisfies

Since Open image in new window and the functions Open image in new window and Open image in new window are nondecreasing, it follows that Open image in new window so that Open image in new window for a.e. Open image in new window We can show in a similar way that Open image in new window for a.e. Open image in new window In this case Open image in new window , 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
Let Open image in new window It is clear that Open image in new window is a classical solution of the problem
and Open image in new window is a classical solution of the problem

Let Open image in new window where Open image in new window is a solution of the problem Open image in new window on Open image in new window and Open image in new window Then Open image in new window and Open image in new window is an upper solution of problem (4.1) provided that Open image in new window

Similarly, let Open image in new window be a solution of Open image in new window on Open image in new window and Open image in new window Then Open image in new window and Open image in new window is a lower solution of problem (4.1) provided that Open image in new window

Notes

Acknowledgments

This work is a part of a research project FT-090001. The author is grateful to King Fahd University of Petroleum and Minerals for its constant support. Also, he would like to thank the reviewers for comments that led to the improvement of the original manuscript.

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Copyright information

© Abdelkader Boucherif. 2011

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.

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia

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