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Nonlocal Conditions for Lower Semicontinuous Parabolic Inclusions
Advances in Difference Equations volume 2011, Article number: 109570 (2011)
Abstract
We discuss conditions for the existence of at least one solution of a discontinuous parabolic equation with lower semicontinuous right hand side and a nonlocal initial condition of integral type. Our technique is based on fixed point theorems for multivalued maps.
1. Introduction
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ1_HTML.gif)
For , we say that
is in
if
is measurable and
, in which case we define its norm by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ2_HTML.gif)
Consider the linear nonhomogeneous problem
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ4_HTML.gif)
with the following nonlocal initial condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ5_HTML.gif)
Here, is an elliptic operator given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ6_HTML.gif)
We will assume throughout this paper that the functions are Hölder continuous,
, and moreover, there exist positive numbers
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ7_HTML.gif)
Let be continuous. For the problem (1.3), (1.4) together with initial condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ8_HTML.gif)
we have the following classical result.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ9_HTML.gif)
where , is the Green's function corresponding to the linear homogeneous problem. This function has the following properties (see [1, 4]).
-
(i)
.
-
(ii)
.
-
(iii)
.
-
(iv)
for
.
-
(v)
are continuous functions of
.
In addition to the above,
satisfies the following important estimate.
-
(vi)
, for some positive constants
(see [2]).
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,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ10_HTML.gif)
Parabolic problems with discontinuous nonlinearities arise as simplified models in the description of porous medium combustion [5], chemical reactor theory [6]. 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 [16]. 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 [25] 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
.
2. Main Result
By a solution of problem (1.10), (7), (8) we mean a function such that there exists a function
with
for each
and (1.3), (1.4), (1.5) hold.
Theorem 2.1.
Assume that the following conditions are satisfied.
(HF) is
measurable,
is lsc for a.e.
, there exist
such that
with 2Vol
and there exists
such that
for any bounded set
,
(Hk) is continuous, bounded and there exists
such that
.
Then problem (1.10), (7), (8) has a solution provided that  .
Proof.
We shall follow the ideas developed in [27]. It follows from the integral representation (1.9) that any solution of (1.10), (7), (8) is a solution of the operator inclusion
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ11_HTML.gif)
for , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ12_HTML.gif)
where is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ13_HTML.gif)
while is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ14_HTML.gif)
First, we show that solutions of (2.1) are a priori bounded. We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ15_HTML.gif)
where , that is
for each
. Since
is bounded there exists
such that
. It follows from the properties of the Green's function and the assumption (HF) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ16_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ17_HTML.gif)
Equation (2.7) implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ18_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ19_HTML.gif)
Therefore, there exists , independent of
, but depending on
and the Green's function such that any possible solution of (2.1) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ20_HTML.gif)
Let . Then
is nonempty, closed, and bounded subset of
.
Since the multifunction has nonempty, closed and convex values, it follows that
has nonempty, closed, and convex values. Since
is a continuous single valued operator, it is clear that
has nonempty, closed, and convex values. Next, we can easily show that
is a closed map (i.e., has a closed graph) and
is a bounded subset of
.
Finally, we show that for any bounded subset
. So, let
. Then, since
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ21_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ22_HTML.gif)
It follows from the assumption that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F109570/MediaObjects/13662_2010_Article_34_Equ23_HTML.gif)
This shows that is a condensing multivalued map.
By Theorem 3.1 in [26], has a fixed point in
, which is a solution of problem (1.10), (7), (8). This completes the proof of the main result.
References
Friedman A: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ, USA; 1964:xiv+347.
Ladyzhenskaya OA, Solonnikov VA, Uraltseva NN: Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow, Russia; 1967. English translation: American Mathematical Society, Providence, RI, USA, 1968
Lieberman GM: Second Order Parabolic Di¤erential Equations. World Scientific, River Edge, NJ, USA; 1996:xii+439.
Pao CV: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York, NY, USA; 1992:xvi+777.
Feireisl E, Norbury J: Some existence, uniqueness and nonuniqueness theorems for solutions of parabolic equations with discontinuous nonlinearities. Proceedings of the Royal Society of Edinburgh A 1991,119(1-2):1-17. 10.1017/S0308210500028262
Fleishman BA, Mahar TJ: A step-function model in chemical reactor theory: multiplicity and stability of solutions. Nonlinear Analysis 1981,5(6):645-654. 10.1016/0362-546X(81)90080-8
Deguchi H: On weak solutions of parabolic initial value problems with discontinuous nonlinearities. Nonlinear Analysis, Theory, Methods and Applications 2005,63(5–7):e1107-e1117.
Hofbauer J, Simon PL:An existence theorem for parabolic equations on
with discontinuous nonlinearity. Electronic Journal of Qualitative Theory of Differential Equations 2001, (8):-9.
Cardinali T, Fiacca A, Papageorgiou NS: Extremal solutions for nonlinear parabolic problems with discontinuities. Monatshefte für Mathematik 1997,124(2):119-131. 10.1007/BF01300615
Carl S, Grossmann Ch, Pao CV: Existence and monotone iterations for parabolic differential inclusions. Communications on Applied Nonlinear Analysis 1996,3(1):1-24.
Carl S, Heikkilä S: On a parabolic boundary value problem with discontinuous nonlinearity. Nonlinear Analysis: Theory, Methods & Applications 1990,15(11):1091-1095. 10.1016/0362-546X(90)90156-B
Pisani R: Problemi al contorno per operatori parabolici con non linearita discontinua. Rendiconti dell'Istituto di Matematica dell'Università di Trieste 1982, 14: 85-98.
Rauch J: Discontinuous semilinear differential equations and multiple valued maps. Proceedings of the American Mathematical Society 1977,64(2):277-282. 10.1090/S0002-9939-1977-0442453-6
Cannon JR: The solution of the heat equation subject to the specification of energy. Quarterly of Applied Mathematics 1963, 21: 155-160.
Ionkin NI: Solution of a boundary value problem in heat conduction theory with nonlocal boundary conditions. Differential Equations 1977, 13: 204-211.
Day WA: A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Quarterly of Applied Mathematics 1983,40(4):468-475.
Balachandran K, Uchiyama K: Existence of solutions of nonlinear integrodifferential equations of Sobolev type with nonlocal condition in Banach spaces. Proceedings of the Indian Academy of Sciences 2000,110(2):225-232. 10.1007/BF02829493
Byszewski L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. Journal of Mathematical Analysis and Applications 1991,162(2):494-505. 10.1016/0022-247X(91)90164-U
Dai D-Q, Huang Y: Remarks on a semilinear heat equation with integral boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2007,67(2):468-475. 10.1016/j.na.2006.06.012
Olmstead WE, Roberts CA: The one-dimensional heat equation with a nonlocal initial condition. Applied Mathematics Letters 1997,10(3):89-94. 10.1016/S0893-9659(97)00041-4
Aubin J-P, Cellina A: Differential Inclusions. Springer, Berlin, Germany; 1984:xiii+342.
Aubin J-P, Frankowska H: Set-Valued Analysis, Systems & Control: Foundations & Applications. Volume 2. Birkhäuser, Boston, Mass, USA; 1990:xx+461.
Deimling K: Multivalued Differential Equations, de Gruyter Series in Nonlinear Analysis and Applications. Volume 1. Walter de Gruyter, Berlin, Germany; 1992:xii+260.
Hu S, Papageorgiou NS: Handbook of Multivalued Analysis, Vol. I: Theory, Mathematics and Its Applications. Volume 419. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000.
Kamenskii M, Obukhovskii V, Zecca P: Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter Series in Nonlinear Analysis and Applications. Volume 7. Walter de Gruyter, Berlin, Germany; 2001:xii+231.
Agarwal RP, O'Regan D: Existence criteria for operator inclusions in abstract spaces. Journal of Computational and Applied Mathematics 2000,113(1-2):183-193. 10.1016/S0377-0427(99)00252-6
Byszewski L, Papageorgiou NS: An application of a noncompactness technique to an investigation of the existence of solutions to a nonlocal multivalued Darboux problem. Journal of Applied Mathematics and Stochastic Analysis 1999,12(2):179-190. 10.1155/S1048953399000180
Acknowledgments
This work is part of an ongoing research project FT090001. The author is grateful to KFUPM for its constant support. The author would like to thank an anonymous referee for his/her comments.
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Boucherif, A. Nonlocal Conditions for Lower Semicontinuous Parabolic Inclusions. Adv Differ Equ 2011, 109570 (2011). https://doi.org/10.1155/2011/109570
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DOI: https://doi.org/10.1155/2011/109570