# Discontinuous Parabolic Problems with a Nonlocal Initial Condition

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## 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## 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

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.

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 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;

(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.

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

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

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]):

(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.

Proof.

## 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.

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

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).

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

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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