Let be an open bounded domain in , , with a smooth boundary . We denote the norm (usually the Euclidean norm) of by . Let be a positive real number. Set and . For we denote its partial derivatives (when they exist) by .
Let denote the Banach space of continuous functions , endowed with the norm
For , we say that is in if is measurable and , in which case we define its norm by
Consider the linear nonhomogeneous problem
with the following nonlocal initial condition:
Here, is an elliptic operator given by
We will assume throughout this paper that the functions are Hölder continuous, , and moreover, there exist positive numbers such that
Let be continuous. For the problem (1.3), (1.4) together with initial condition
we have the following classical result.
Lemma 1.1 (see [1–4]).
Assume that the function is Hölder continuous on and is continuous on . Then problem (1.3), (1.4), (1.8) has a unique solution , which for each , is given by
where , is the Green's function corresponding to the linear homogeneous problem. This function has the following properties (see [1, 4]).
are continuous functions of .
In addition to the above, satisfies the following important estimate.
, for some positive constants (see ).
Since , it is clear that the functions and are continuous. Let
and let Also, property (vi) above shows that .
In this paper, we consider a nonlocal problem for a class of nonlinear parabolic equations with a lower semicontinuous multivalued right hand side. More specifically, we consider the following problem,
Parabolic problems with discontinuous nonlinearities arise as simplified models in the description of porous medium combustion , chemical reactor theory . Also, best response dynamics arising in game theory can be modeled by a parabolic equation with a discontinuous right hand side [7, 8]. Parabolic problems with discontinuous nonlinearities have been also investigated in the papers [9–13]. On the other hand, parabolic problems with integral boundary conditions appear in the modeling of concrete problems, such as heat conduction [14, 15] and thermoelasticity . Also, the importance of nonlocal conditions and their applications in different field has been discussed in [17, 18]. Several papers have been devoted to the study of parabolic problems with integral conditions [19, 20]. Next, we state some important facts about multivalued functions and results that will be used in the remainder of the paper.
A subset is measurable if belongs to the -algebra generated by all sets of the form where is Lebesgue measurable in and is Borel measurable in . Let and be Banach spaces. denotes the set of all nonempty subsets of . The domain of a multivalued map is the set Dom( has closed values if is a closed subset of for each and we write . Also, denotes the set of all nonempty closed and convex subsets of . is bounded if is called lower semicontinuous (lsc) on if is open in whenever is open in , or the set is closed in whenever is closed in . For more details on multivalued maps, we refer the interested reader to the books [21–24].
Let denote the Kuratowski measure of noncompactness. See  for definitions and details.
Theorem 1.2 (see [26, Theorem 3.1]).
Let be a separable Banach space. Assume the following conditions hold. There exists , independent of , with for any solution to a.e. on for each is a closed map, is a bounded subset of , and for all with strict inequality if . Then the inclusion has a solution .