Volume 2011, Article ID 965759,10pages doi:10.1155/2011/965759
Research Article
Discontinuous Parabolic Problems with
a Nonlocal Initial Condition
Abdelkader Boucherif
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 5046, Dhahran 31261, Saudi Arabia
Correspondence should be addressed to Abdelkader Boucherif,[email protected]
Received 28 February 2010; Revised 31 May 2010; Accepted 13 June 2010
Academic Editor: Feliz Manuel Minh ´os
Copyrightq2011 Abdelkader Boucherif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
LetΩbe a an open bounded domain inRN,N ≥ 2,with a smooth boundary∂Ω.LetQT
Ω×0, TandΓT ∂Ω×0, T whereT is a positive real number.ThenΓT is smooth and any
point onΓTsatisfies the insideand outsidestrong sphere propertysee1 . Foru:QT → R
we denote its partial derivatives in the distributional sensewhen they existbyDtu∂u/∂t,
Diu∂u/∂xi, DiDju∂2u/∂xi∂xj, i, j1, . . . , N.
In this paper, we study the following parabolic differential equation with a nonlocal initial condition
DtuLufx, t, u, x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0
T
0
kx, t, ux, tdt, x∈Ω,
wheref :QT×R → Ris not necessarily continuous, but is such that for every fixedu∈Rthe
functionx, t → fx, t, uis measurable andu → fx, t, uis of bounded variations over compact interval inRand nondecreasing, andk:QT×R → Ris continuous;Lis a strongly
elliptic operator given by
Lu−
N
i,j1
Di
aijx, tDju
cx, tu. 1.2
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 conduction6–10 and in thermoelasticity11 .
In order to investigate problem1.1, we introduce some notations, function spaces, and notions from set-valued analysis.
LetCQTdenote the Banach space of all continuous functionsu:QT → R, equipped
with the norm|u|0 maxx,t∈QT|ux, t|.LetC2,1QT {u : QT → R; u·, t ∈ C2Ωfor
eacht∈0, Tandux,·∈C10, Tfor eachx∈Ω}.For 1< p <∞,we say thatu:Q
T → R
is inLpQ
Tifuis measurable and
QT|ux, t|
pdx dt <∞,in which case we define its norm
by
|u|Lp
QT
|ux, t|pdx dt
1/p
. 1.3
Let J 0, T and let H1Ω denote the Sobolev space of functions z ∈ L2Ω
having first generalized derivatives in L2Ω and let H1Ω∗ be its corresponding dual
space. Then H1Ω ⊂ L2Ω ⊂ H1Ω∗ and they form an evolution triple with all
embeddings being continuous, dense, and compact see2,12 . The Bochner space W W2,2J, H1Ω,H1Ω∗ see13 is the set of functionsu∈L2J;H1
0Ωwith generalized derivativedu/dt∈L2J;H1Ω∗.Forz∈W,we define its norm by
zW zL2J;H1 0Ω
dzdt
L2J;H1Ω∗
. 1.4
Then W, · W is a separable reflexive Banach space. The embedding of W0
W2,2J, H1
0Ω,H1Ω
∗
into CJ;L2Ω is continuous and the embeddingW
0 ⊂ L2QT
is compact.
Now, we introduce some facts from set-valued analysis. For complete details, we refer the reader to the following books.14–16 . LetX, · X and Y, · Ybe Banach spaces. We will denote the set of all subsets, ofX having property byPX.For instance,PnX
denotes the set of all nonempty subsets ofX;V ∈PclXmeansV closed inX; when b we have the bounded subsets ofX, cv for convex subsets, cp for compact subsets and
cp,cv for compact and convex subsets. The domain of a multivalued mapR:X → PnY
is the set domR{z∈X; Rz/∅}. Ris convexclosedvalued ifRzis convexclosed for each z ∈ X. Ris bounded on bounded sets ifRA z∈ARz is bounded inY for allA ∈ PbX i.e., supz∈A{sup{yY;y ∈ Rz}} < ∞. Ris called upper semicontinuous
ofYcontainingRz, there exists an open neighborhoodΠofzsuch thatRΠ⊂Λ.In terms of sequences,Ris usc if for each sequencezn⊂X,zn → z0, andBis a closed subset ofY such thatRzn∩B /∅thenRz0∩B /∅.
The set-valued mapRis called completely continuous ifRAis relatively compact in
Y for everyA∈PbX.IfRis completely continuous with nonempty compact values, thenR
is usc if and only ifRhas a closed graphi.e.,zn → z, wn → w,wn ∈Rzn⇒w ∈Rz.
WhenX ⊂Y thenRhas a fixed point if there existsz∈Xsuch thatz∈Rz.A multivalued mapR:J → PclXis called measurable if for everyx∈X, the functionθ:J → Rdefined byθt distx, Rt inf{|x−z|;z ∈Rt}is measurable.RzY denotes sup{yY;y ∈ Rz}.The Kuratowski measure of noncompactness see15, page 113 of A ∈ PbX is
defined by
αA inf
>0; A⊂
m
i1
Ai, diamAi≤
. 1.5
The Kuratowski measure of noncompactness satisfies the following properties.
iαA 0 if and only ifAis compact; iiαA αA;
iiiαAB≤αA αB; ivαcA |c|αA,c∈R;
vαconvA αA,where convAdenotes the convex hull ofA.
Definition 1.1see17 . A functionf:QT×R → Ris calledN-measurable onRif for every
measurable functionu:QT → Rthe functionx, t → fx, t, ux, tis measurable.
Examples of N-measurable functions are Carath´eodory functions, Baire measurable functions.
Letgx, t, u lim infz→ufx, t, uandhx, t, u lim supz→ufx, t, u.Thensee17,
Proposition 1 the functionu → gx, t, uis lower semicontinuous, that is, for everyx, t∈ QT the set{u:gx, t, u> r}is open for anyr ∈R, and the functionu → hx, t, uis upper
semicontinuous, that is, for everyx, t∈QT,the set{u:hx, t, u< r}is open for anyr∈R.
Moreover, the functionsu → gx, t, uandu → hx, t, uare nondecreasing.
Definition 1.2. The multivalued functionFdefined byFx, t, u gx, t, u, hx, t, u for all x, t∈QT is calledN-measurable onRif both functionsgandhareN-measurable onR.
Definition 1.3. The operatorNF :L2QT → L2QTdefined by
NFu
q∈L2QT; gx, t, u≤qx, t≤hx, t, u,x, t∈QT
1.6
is called the Nemitskii operator of the multifunctionF.
Since F is an N-measurable and upper semicontinuous multivalued function with compact and convex values, we have the following properties for the operatorNF see17,
Lemma 1.4. NF is N-measurable, compact and convex-valued, upper semicontinuous and maps
bounded sets into precompact sets.
We will consider solutions of problem1.1 as solutions of the following parabolic problem with multivalued right-hand side:
DtuLu∈Fx, t, u, x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0
T
0
kx, t, ux, tdt, x∈Ω,
1.7
whereFx, t, u gx, t, u, hx, t, u for allx, t∈QT.As pointed out in15, Example 1.3
page 5 , this is the most general upper semicontinuous set-valued map with compact and convex values inR.
Theorem 1.5see18 . LetEbe a Banach space andΥ:E → Pcp,cvEa condensing map. If the
setS:{z∈E;λz∈Υzfor someλ >1}is bounded, thenΥhas 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 aij, c : QT −→ R are H ¨older
continuous,aijajiand moreover, there exist positive numbersλ0, andλ1such that
λ0ξ2≤
N
i,j1
aijx, tξiξj ≤λ1ξ2, ∀ξ∈RN, ∀x, t∈QT. 2.1
Given a continuous functionu0:Ω → R,the linear parabolic problem
DtuLufx, t x, t∈QT,
ux, t 0 x, t∈ΓT,
ux,0 u0x, x∈Ω
2.2
is well known and completely solvedsee the books1,19,20 . The linear homogeneous problem
DtuLu0, x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0 0, x∈Ω
has only the trivial solution. There exists a unique function,Gx, t;y, s,called Green’s function corresponding to the linear homogeneous problem. This function satisfies the followingsee1,20 :
iDtGLGδt−sδx−y, s < t, x, y∈Ω;
iiGx, t;y, s 0, s > t, x, y∈Ω; iiiGx, t;y, s 0,x, t,y, s∈ΓT;
ivGx, t;y, s>0 forx, t∈QT;
vG, DtG, DiG,andDiDjGare continuous functions ofx, t,y, s∈QT, t−s >0;
vi|Gx, t;y, s| ≤ Ct−s−N/2exp−ax−yR2n/t−s,for some positive constants
C, asee19 ;
viifor any H ¨older continuous functionf:QT → R, the functionu :QT → R, given
forx, t∈QT byux, t
ΩGx, t;y,0 u0ydy
t
0
ΩGx, t;y, sfy, sdy ds,is
the unique classical solution, that is,u∈C2,1Q
T∩CQT,of the nonhomogeneous
problem2.2.
It is clear from property vi above that G ∈ L2Q
T × QT. Also, the integral
representation inviiimplies that the functionx, t → ΩGx, t;y,0dyis continuous. Let
γ0maxx,t∈QT
ΩGx, t;y,0dy.
Lemma 2.1. Iff ∈ L2Q
T,then2.2has a unique weak solutionu ∈ L2QT.Moreover, there
exists a positive constantM, depending only onu0, γ0, T,andΩ,such that
|u|L2QT≤M|G|L2QT×QTfL2QT. 2.4
Proof. Consider the following representationsee propertyviiabove:
ux, t
ΩG
x, t;y,0u0
ydy
t
0
ΩG
x, t;y, sfy, sdy ds, x, t∈QT. 2.5
Define an operatorG:L2Q
T → L2QTby
Gfx, t
t
0
ΩG
x, t;y, sfy, sdy ds, x, t∈QT. 2.6
ThenGis a bounded linear operator with
GfL2QT≤ |G|L2QT×QTfL2QT. 2.7
Then for eachx, t∈QT,
ux, t
ΩG
x, t;y,0u0
This implies that for eachx, t∈QT
|ux, t| ≤γ0|u0|0Gfx, t. 2.9
Minkowski’s inequality leads to
|u|L2QT≤M|G|L2QT×QTfL2QT. 2.10
3. Problem with a Discontinuous Nonlinearity
In this section, we investigate the multivalued problem1.7. We define the notion of a weak solution.
Definition 3.1. A solution of1.7is a functionu∈W0such that
ithere existsw∈L2Q
Twithgx, t, u≤wx, t≤hx, t, u, x, t∈QT;
iiDtuLuwx, t, x, t∈QT;
iiiux,0 0Tkx, t, ux, tdt, x∈Ω.
We introduce the notion of lower and upper solutions of problem1.7.
Definition 3.2. U∈W0is a weak lower solution of1.7if
iDtULU≤gx, t, U, x, t∈QT;
iiUx, t≤0, x, t∈ΓT;
iiiUx,0≤0Tkx, t, Ux, tdt, x∈Ω.
Definition 3.3. U∈W0is a weak upper solution of1.7if
jDtULU≥hx, t, U, x, t∈QT;
jjUx, t≥0, x, t∈ΓT;
jjjUx,0≥0Tkx, t, Ux, tdt, x∈Ω.
We will assume that the functionf :QT×R → R, generating the multivalued function
F, isN-measurable on R, which implies thatF is anN-measurable, upper semicontinuous multivalued function with nonempty, compact, and convex values. In addition, we will need the following assumptions:
H1there existsp∈L2Q
Tsuch that|fx, t, u| ≤px, t, x, t∈QT;
H2there exist a lower solutionUand an upper solutionUof1.7such thatU≤U; H3k:QT×R → Ris continuous, andu→kx, t, uis nondecreasing withkx, t,0 0.
We state and prove our main result.
Theorem 3.4. Assume that (H1), (H2), and (H3) are satisfied. Then the multivalued problem1.7
Proof. First, it is clear that the operatorδ:L2Q
T → U, U defined by
δu maxU,minu, U 3.1
is continuous and uniformly bounded. Consider the modified problem
DtuLu∈Fx, t, δu, x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0
T
0
kx, t, δux, tdt, x∈Ω.
3.2
We show that possible solutions of3.2are a priori bounded. Letu∈L2Q
Tbe a solution of
3.2. It follows from the definition and the representation2.5that for eachx, t∈QT,
ux, t
T
0
ΩG
x, t;y,0ky, s, δuy, sdy ds
t
0
ΩG
x, t;y, swy, sdy ds, 3.3
where w ∈ L2Q
T with gx, t, δu ≤ wx, t ≤ hx, t, δu,x, t ∈ QT. Since k is
continuous andδis uniformly bounded there existsmk>0 such that|kx, t, δu| ≤mk.Also,
assumptionH1implies that|wx, t| ≤ px, t.The relation3.3together withLemma 2.1
yields
|u|L2QT≤C:M1|G|L2QT×QTpL2QT, 3.4
whereM1depends only onmk, T, γ0.LetV {u∈L2QT;|u|L2QT≤C}.
It is clear that solutions of 3.2 are fixed point of the multivalued operator :
L2Q
T → L2QT, defined by
uku GNFu. 3.5
Here,k:L2Q
T → L2QTis a single-valued operator defined by
kux, t
T
0
ΩG
x, t;y,0ky, s, δuy, sdy ds, 3.6
andGNF :L2QT → L2QTis a multivalued operator defined by
GNFux, t
t
0
ΩG
x, t;y, sNF
δuy, sdy ds. 3.7
Claim 1. kVis compact inL2Q
T. Since the functionk is continuous and the operatorδ
continuous and has no singularity fort >0. It follows that the operatorkis continuous and there existsρ, depending only onT and Ω,such thatkuW0 ≤ ρTγ0mk,so thatkVis
uniformly bounded inW0.Since the embeddingW0⊂L2QTis compact it follows thatkV
is compact inL2Q
T.
Claim 2. GNFVis also compact inL2QT. This follows from the continuity of the Green’s
function and the properties of the Nemitski operatorNF.SeeLemma 1.4.
Claim 3. αV 0,that is, it is a condensing multifunction.We haveαV αkV
GNFV≤αkV αGNFV 0.
Also Lemma 1.4implies thatNF has nonempty, compact, convex values. Sincek is
single-valued, the operator has nonempty compact and convex values. We show that has a closed graph. Letvn → v∗, hn ∈vn,andhn → h∗.We show thath∗∈v∗.Now,
hn ∈ vnimplies thathn−kvn ∈ GNFvn.It is clear thathn−kvn → h∗−kv∗in
L2Q
T.We can use the last part of Lemma 4.1 in13 to conclude thath∗−kv∗∈GNFv∗,
which, in turn, implies thath∗ ∈kv∗ GNFv∗ v∗.This will imply thatis upper
semicontinuous.
Therefore,:L2Q
T → Pcp,cvL2QTis condensing. ˙It remains to show that the set
{z∈L2Q
T; λz∈zfor someλ >1}is bounded; but this is a consequence of inequality
3.4.Theorem 1.5implies that the operatorhas a fixed pointz∈V ,which is a solution of 3.2.
We, now, show thatz∈U, U .We prove thatz≥U.It follows from the definition of a solution of3.2that there existsw∈L2Q
Twithgx, t, δz≤wx, t≤hx, t, δz,x, t∈
QT, such that
zx, t
T
0
ΩG
x, t;y,0ky, s, δzy, sdy ds
t
0
ΩG
x, t;y, swy, sdy ds. 3.8
On the other hand,Usatisfies
Ux, t≤
T
0
ΩG
x, t;y,0ky, s, Uy, sdy dt
t
0
ΩG
x, t;y, sgy, s, Uy, sdy ds.
3.9
Letφx, t zx, t−Ux, tfor eachx, t∈QT.Then
φx, t≥
T
0
ΩG
x, t;y,0ky, s, δzy, s−ky, s, Uy, sdy ds
t 0 ΩG
x, t;y, swy, s−gy, s, Uy, sdy ds.
3.10
Sinceδz ≥ Uand the functionsu → ky, s, uand u → gy, s, uare nondecreasing, it follows thatφx, t≥0,so thatzx, t≥Ux, tfor a.e.x, t∈QT.We can show in a similar
way thatzx, t≤Ux, tfor a.e.x, t∈QT.In this caseδz z, and3.2reduces to1.7.
4. Example
Consider the problem
DtuLu∈Fx, t, u −1,1 , x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0 μ
T
0
ux, tdt, x∈Ω.
4.1
Letξx, t 0tΩGx, t;y, sdy ds −ηx, t.It is clear thatξis a classical solution of the problem
DtuLu1, x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0 0, x∈Ω,
4.2
andηis a classical solution of the problem
DtuLu−1, x, t∈QT,
ux, t 0, x, t∈ΓT,
ux,0 0, x∈Ω.
4.3
LetUx, t ξx, t ax, t,whereais a solution of the problemDtuLu0, u0
onΓT,andux,0 1.Thenax, t
ΩGx, t;y,0dyandUis an upper solution of problem 4.1provided thatμsupx∈Ω0Tξx, t ax, tdt <1.
Similarly, letb be a solution ofDtuLu 0, u 0 onΓT, andux,0 −1.Then
bx, t −ax, tandUx, t ηx, t bx, tis a lower solution of problem4.1provided that 1μinfx∈Ω0Tηx, t bx, tdt≥0.
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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