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Discontinuous Parabolic Problems with a Nonlocal Initial Condition
Boundary Value Problems volume 2011, Article number: 965759 (2011)
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

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

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

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

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

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

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:

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

Given a continuous function 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, 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

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

Define an operator by

Then is a bounded linear operator with

Then for each

This implies that for each

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 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

is continuous and uniformly bounded. Consider the modified problem

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

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

where depends only on
Let
It is clear that solutions of (3.2) are fixed point of the multivalued operator , defined by

Here, is a single-valued operator defined by

and is a multivalued operator defined by

Claim 1.
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
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 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 [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

On the other hand, satisfies

Let =
for each
Then

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

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

and is a classical solution of the problem

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
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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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Boucherif, A. Discontinuous Parabolic Problems with a Nonlocal Initial Condition. Bound Value Probl 2011, 965759 (2011). https://doi.org/10.1155/2011/965759
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DOI: https://doi.org/10.1155/2011/965759