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.