Discontinuous Parabolic Problems with a Nonlocal Initial Condition
© Abdelkader Boucherif. 2011
Received: 28 February 2010
Accepted: 13 June 2010
Published: 5 July 2010
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.
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 ). For we denote its partial derivatives in the distributional sense (when they exist) by ,
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 .
In order to investigate problem (1.1), we introduce some notations, function spaces, and notions from set-valued analysis.
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 Kuratowski measure of noncompactness satisfies the following properties.
(i) if and only if is compact;
(iv) , ;
(v) where denotes the convex hull of .
Definition 1.1 (see ).
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.
The multivalued function defined by for all is called N- measurable on if both functions and are N- measurable on .
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 ]).
is N-measurable, compact and convex-valued, upper semicontinuous and maps bounded sets into precompact sets.
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 ).
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
(v) and are continuous functions of
(vi) for some positive constants (see );
(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
3. Problem with a Discontinuous Nonlinearity
In this section, we investigate the multivalued problem (1.7). We define the notion of a weak solution.
A solution of (1.7) is a function such that
(i)there exists with
We introduce the notion of lower and upper solutions of problem (1.7).
is a weak lower solution of (1.7) if
is a weak upper solution of (1.7) if
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.
Assume that (H1), (H2), and (H3) are satisfied. Then the multivalued problem (1.7) has at least one solution
where depends only on Let
is compact in . Since the function is continuous and the operator is uniformly bounded there 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
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.
that is, it is a condensing multifunction We 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  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).
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.
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
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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