A NONLOCAL PARABOLIC PROBLEM
S. B. DE MENEZES
Received 7 May 2005; Revised 20 September 2005; Accepted 28 November 2005 Dedicated to Professor L. A. Medeiros on the occasion of his 80th birthday
We prove a result on existence and uniqueness of weak solutions for a diffusion prob- lem associated with nonlinear diffusions of nonlocal type studied by Chipot and Lovat (1999) by an application of the fixed point result of Schauder. Moreover, making use of Faedo-Galerkin approximation, coupled with some technical ideas, we establish a result on existence of periodic solution.
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1. Introduction
In this work, we are going to study some questions concerning to the existence, unique- ness, and periodic solution for the parabolic problem
ut−al(u)Δu+f(u)=h inQ=Ω×(0,T), u(x,t)=0 onΣ=Γ×(0,T),
u(x, 0)=u0(x) inΩ,
(1.1)
whereΩis a smooth bounded open subset of RN with regular boundary Γ. In prob- lem (1.1)aand f are both continuous functions, whose properties will be introduced when necessary, l:L2(Ω)→Ris a nonlinear form, h∈L2(0,T;H−1(Ω)), and T >0 is some fixed time. System (1.1) is studied, for instance, in papers of Chipot and Lovat [4]
in case f = f(x) depends only on the variablex. In this work, we are going to present a simple extension of the results contained in [4], in which f = f(u) depends on the stateu, where we study, among other things, the case in whichuis periodic. This kind of problems, besides its mathematical motivation because of presence of the nonlocal term a=a(l(u)), arises from physical situations related to migration of a population of bac- terias in a container in which the velocity of migration−→v =a∇udepends on the global population in a subdomainΩ⊂Ωgiven bya=a(Ωudx). For more details see [4] and the references cited in this paper. Many books have dealt with parabolic equations and as- ymptotic analysis. See, for example, Amann [1], Haraux [5], Pao [6], and Zeidler [7] and
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 82654, Pages1–10
DOI10.1155/IJMMS/2006/82654
the references therein. However, because of the nonlocal termawe cannot, in the present problem, make a direct adaptation of the classical techniques. In view of this we have to appeal to another device in order to obtain results similar to those in which appear only local terms. In particular, our approach rests heavily on the ideas developed by Chipot and Lovat [4]. This work is organized as follows. InSection 2, we prove basic results on existence and uniqueness of weak solution for problem (1.1). The methodology of the proof of our results is based on the fixed point argument. Finally, inSection 3, we will prove the existence and uniqueness of periodic weak solution for problem (1.1). We use Faedo-Galerkin method and Brower fixed point theorem plus also some technical ideas.
2. Results on existence and uniqueness
The first object of this work is to prove existence and uniqueness for the problem (1.1).
In fact, concerning problem (1.1), we will suppose thata:R→Ris continuous and that for some constantsm,M,
0< m≤a(ξ)≤M, ∀ξ∈R, (2.1) l:L2(Ω)−→Ris a continuous nonlinear form, (2.2) that is, there isg∈L2(Ω) such thatl(u)=lg(u)=
Ωg(x)u(x)dx for allu∈L2(Ω) and f :R→Ris a Lipschitz continuous function, that is, there existsγ >0 such that
f(s)−f(t)≤γ|s−t|, ∀s,t∈R. (2.3) Moreover assume that f(0)=0, f(0) exists.
In this section, we present some notation that will be used throughout this work. By ·,·we will represent the duality pairing betweenX andX,Xbeing the topological dual of the spaceX, and byC(sometimesC1,C2,. . .) we denote various positive constants.
We represent byHm(Ω) the usual Sobolev space of orderm, byH0m(Ω) the closure of C∞0(Ω) inHm(Ω), and byL2(Ω) the class of square Lebesgue integrable real functions.
In particular,H01(Ω) has inner product ((·,·)) and norm · given by ((u,v))=
Ω∇u·
∇v dx;u2=
Ω|∇u|2dx. For the Hilbert spaceL2(Ω) we represent its inner and norm, respectively, by (·,·) and| · |, defined by (u,v)=
Ωuv dx;|u|2=
Ω|u|2dx. We have our first result.
Theorem 2.1. If (2.1)–(2.3) hold, then for
u0∈L2(Ω), h∈L20,T,H−1(Ω) (2.4) there exists a functionusuch that
u∈L20,T,H01(Ω)∩C[0,T],L2(Ω), ut∈L20,T,H−1(Ω), (2.5)
u(x, 0)=u0, (2.6)
d
dt(u,v) +al(u)
Ω∇u· ∇v dx+
Ωf(u)v dx= h,v, (2.7) for allv∈H01(Ω), where (2.7) must be understood as an equality inᏰ(0,T).
Proof. We argue using the Schauder fixed point theorem. For that purpose, considerw∈ L2(0,T;L2(Ω)) andu=N(w) the unique solution to “linearized” problem
u∈L20,T,H01(Ω)∩C[0,T],L2(Ω), ut∈L20,T,H−1(Ω), (2.8)
u(x, 0)=u0, (2.9)
d
dt(u,v) +al(w)
Ω∇u· ∇v dx+
Ωb(x,t,w)uv dx= h,v, (2.10) inᏰ(0,T) for allv∈H01(Ω).
Hereb:Ω×R+×R→Ris defined by
b(x,t,η)=
⎧⎪
⎪⎨
⎪⎪
⎩ f(η)
η forη=0, f(0) forη=0.
(2.11)
We note that the mapping
t−→l(w) (2.12)
is measurable, and due to (2.1)–(2.3), so is
t−→al(w). (2.13)
We also note thata(l(w))∈L∞(0,T), andb(x,t,η) is continuous, a.e. (x,t)∈Ω×R+, and measurable for allη∈Rand
b(x,t,η)≤C a.e. (x,t)∈Ω×R+,∀η∈R. (2.14) We know that such au=N(w) exists (see, e.g., [3]). Thus we would like to show that the mapping
w−→N(w) (2.15)
fromL2(0,T;L2(Ω)) into itself has a fixed point—this will be clearly a solution to our problem.
First let us remark that
ut−al(w)u+b(x,t,w)u=h inL20,T;H−1(Ω), (2.16) and thus for everyv∈L2(0,T;H01(Ω)),
ut,v+al(w)(∇u,∇v) +b(x,t,w)(u,v)= h,v a.e.t∈(0,T). (2.17) Takingv=uin (2.17), and in view of (2.1) and (2.14), one gets
1 2
d
dtu(t)2+mu(t)2≤Cu(t)2+u(t)|h|H−1(Ω). (2.18)
Applying Young’s inequality, we obtain 1
2 d
dtu(t)2+mu(t)2≤Cu(t)2+m
2u(t)2+ 1
2m|h|2H−1(Ω), (2.19) that is,
1 2
d
dtu(t)2+m
2u(t)2≤Cu(t)2+ 1
2m|h|2H−1(Ω). (2.20) Integrating it over (0,t) and employing Gronwall’s lemma, we obtain
|u|L2(0,T;H01(Ω))≤C, |u|L2(0,T;L2(Ω))≤C. (2.21) Setting
B=
v∈L20,T;L2(Ω)| |v|L2(0,T;L2(Ω))≤C, (2.22) it follows thatw→u=N(w) is a mapping fromBinto itself. The arguments above show that whenwlies in a bounded setBofL2(0,T;L2(Ω)),u=N(w) also lies in a bounded set ofL2(0,T;L2(Ω)). We have to show thatN(B) is relatively compact inL2(0,T;L2(Ω)).
Indeed, going back to (2.10), we have ut
L2(0,T;H−1(Ω))≤C. (2.23) Thenubelongs toW(0,T,H01(Ω),H−1(Ω)). We recall that
W(0,T,X,Y)=
v∈L2(0,T;X)|vt∈L2(0,T;Y). (2.24) Thus, the compactness ofN is a consequence of the compactness of the embedding of W(0,T,H01(Ω),H−1(Ω)) intoL2(0,T;L2(Ω)) (Aubin-Lions compactness result). In order to be able to apply the Schauder fixed point theorem, we now just need to prove thatNis continuous fromBinto itself. For that let (wn) be a sequence inBsuch that
wn−→w inL20,T;L2(Ω). (2.25) Setun=N(wn). From (2.25) we derive that
lwn
−→l(w) inL2(0,T). (2.26)
By the estimates above and Aubin-Lions compactness result we can findu∞∈W(0,T, H01(Ω),H−1(Ω)) and a subsequence fromnthat we will label alsonsuch that
un u∞ weakly inW0,T,H01(Ω),H−1(Ω), (2.27) un−→u∞ strongly inL20,T;L2(Ω), (2.28) lwn−→l(w) a.e.t∈(0,T). (2.29)
By continuity ofawe obtain alwn
−→al(w) a.e.t∈(0,T). (2.30) From (2.10) we have that
− T
0
Ωunvϕ(t)dx dt+ T
0
Ωalwn
∇un· ∇vϕ(t)dx dt +
T
0
Ωbx,t,wn
unvϕ(t)dx dt= T
0 h,vϕ(t)dt,
(2.31)
for anyϕ∈Ᏸ(0,T),v∈H01(Ω).
By the Lebesgue’s dominated convergence theorem we have ϕ(t)bx,t,wn
v−→ϕ(t)b(x,t,w)v, ϕ(t)alwn∂v
∂xi−→ϕ(t)al(w)∂v
∂xi, (2.32)
inL2(0,T;L2(Ω)). Taking limit in both sides of (2.31), we obtain thatu∞satisfies (2.8) and (2.10). By (2.28) one can assume without loss of generality that
un(t)−→u∞(t) inL2(Ω), a.e.t∈(0,T). (2.33) Thus
u∞(0)=u0. (2.34)
Then by the uniqueness of the solution to (2.8)–(2.10) we have thatu∞=u. Since any subsequence ofunhas the same limit,
un=Nwn−→u=N(w) inL20,T;L2(Ω), (2.35)
which concludes the proof ofTheorem 2.1.
Next, we will prove the following result.
Theorem 2.2. Assume thatais Lipschitz continuous in the sense that there exists a constant Asuch that
a(t)−a(t)≤A|t−t|, ∀t,t∈R. (2.36) Then under the assumptions ofTheorem 2.1, the problem (1.1) has a unique solution.
Proof. Let us denote byu1andu2two solutions of (1.1). Thus d
dt
u1−u2 ,v
+alu1
Ω∇u1· ∇v dx−alu2
×
Ω∇u2· ∇v dx+
Ω
fu1
−fu2
v dx=0.
(2.37)
From (2.3) we obtain d
dt
u1−u2
,v
+alu1
Ω∇ u1−u2
· ∇v dx
≤(alu2
−alu1
Ω∇u2· ∇v dx+γ
Ω
u1−u2v dx.
(2.38)
Takingv=(u1−u2)(t), for a.e.t, one gets 1
2 d
dtu1−u22+alu1u1−u22
≤alu2
−alu1
Ω
∇u2∇
u1−u2dx+γu1−u22.
(2.39)
So by the Cauchy-Schwarz inequality and by using (2.1)–(2.2) and (2.36), one gets 1
2 d
dtu1−u22+mu1−u22
≤Alu2
−lu1u2u1−u2+γu1−u22
≤Cu1−u2u2u1−u2+γu1−u22.
(2.40)
Then, applying Young’s inequality, we obtain 1
2 d
dtu1−u22+mu1−u22
≤m
2u1−u22+ 1
2mC2u22u1−u22+γu1−u22
=m
2u1−u22+ C2
2mu22+γu1−u22,
(2.41)
which gives
d
dtu1−u2≤ζ(t)u1−u22, (2.42) where the functionζ(t) belongs toL1(0,T). So this reads after multiplication by
exp
− t
0ζ(s)ds
, d
dt
exp
− t
0ζ(s)dsu1−u22
≤0.
(2.43)
Since this last function is nonincreasing and vanishes at 0 (u1(0)=u2(0)), the result fol-
lows.
3. Periodic solution
The second object of this work is to show the existence of periodic solution of problem (1.1). For this purpose, in addition to (2.1)–(2.3), we assume that
mλ1−2γ >0, (3.1)
withλ1the first eigenvalue of (−,H01(Ω)). We make use of Faedo-Galerkin approxima- tion and Brower fixed point theorem. The result on the periodic solution is given by the following theorem.
Theorem 3.1. Under the assumptions ofTheorem 2.2, assume that (3.1) holds. Ifh∈L2 (0,T,H−1(Ω)), then there exists a unique solutionuof the periodic problem
u∈L20,T,H01(Ω)∩C[0,T],L2(Ω), ut∈L20,T,H−1(Ω), d
dt(u,v) +al(u)
Ω∇u· ∇v dx+
Ωf(u)v dx= h,v,
(3.2)
for allv∈H01(Ω), in the sense ofᏰ(0,T),
u(0)=u(T) inL2(Ω). (3.3)
Proof. We employ the Faedo-Galerkin method. Let (ωj)j∈N be a Hilbertian basis of H01(Ω) (cf. Brezis [2]). Represent byVjthe subspace ofH01(Ω) generated by{ω1,. . .,ωj} and let us consider the approximate problem given by
uj∈Vj, (3.4)
uj,v+aluj∇uj,∇v+fuj,v= h,v, ∀v∈Vj, (3.5) uj(0)=u0j−→u0 strongly inL2(Ω). (3.6) The system of ordinary differential equations (3.4)–(3.6) has a local solution on an in- terval [0,tm[, 0< tm< T. We now have to establish an estimate that permits to extend the solution to the whole interval [0,T]. Takingv=uj in (3.5) and in view of (2.1)–(2.3), one gets
1 2
d
dtuj(t)2+muj(t)2≤γuj(t)2+uj|h|H−1(Ω). (3.7) Thanks to Young’s inequality, one gets
1 2
d
dtuj(t)2+m
2uj(t)2≤γuj(t)2+ 1
2m|h|2H−1(Ω). (3.8)
As in the proof ofTheorem 2.1, integrating it over (0,t) and employing Gronwall’s lemma, we obtain
uj
L2(0,T;H01(Ω))≤C, (3.9)
uj
L∞(0,T;L2(Ω))≤C, (3.10)
∂tuj
L2(0,T;H−1(Ω))≤C. (3.11)
In view of the estimates above, we can extend the approximate solutionuj(t) to [0,T].
Next, let us consider
uj(0)=u0j∈B0(R)∩Vj, (3.12) whereB0(R)= {u∈L2(Ω);|u|L2(Ω)< R}, withRa positive constant.
Letλ1 be the first eigenvalue of (−,H01(Ω)). Then, thanks to Poincare’s inequality and (3.8), we obtain
d
dtuj(t)2+mλ1−2γuj(t)2≤ 1
2m|h|2H−1(Ω). (3.13) We note that, by hypothesis (3.1),mλ1−2γ >0 and thus
d
dtuj(t)2+Cuj(t)2≤C|h|2H−1(Ω). (3.14) We multiply both sides of (3.14) byeCtand we integrate on [0,t) to obtain
uj(t)2≤e−Ctuj(0)2+C T
0 e−C(s−T)h(s)2H−1(Ω)ds. (3.15) Sinceh∈L2(0,T,H−1(Ω)), one has
uj(t)2≤θ(t)uj(0)2+C, (3.16) for all 0≤t≤T, withθ(t)=e−Ct and C=T
0 e−C(s−T)|h(s)|2H−1(Ω)ds. We note that 0<
θ(t)<1. Then,
uj(T)2≤θuj(0)2+C, (3.17) withθ=θ(T) constant, 0< θ <1. ChoosingR >0 such thatC/(1−θ)< R2, we have that
uj(T)2≤R2. (3.18)
Thus, for eachu0j∈B0(R)∩Vjthere exists a solutionuj(t) of the approximating problem (3.4)–(3.6) and, furthermore,uj(t) satisfies (3.18). So, we have defined a mapping
τ:B0(R)∩Vj−→B0(R)∩Vj,
u0j−→τu0j=uj(T). (3.19) We are ready to establish the following lemma.
Lemma 3.2. Assume that hypotheses ofTheorem 2.2hold. Then the mappingτis continuous.
Proof. Let us consideru01j,u02j inB0(R)∩Vj andu1,u2the corresponding solutions of (3.4)–(3.6). From (2.41) we obtain
1 2
d
dtu1j−u2j2+m
2u1j−u2j2≤Cu1j−u2j2, (3.20) since, byTheorem 2.1,u2j< C. By Gronwall’s lemma, one gets
u1j(T)−u2j(T)≤Cu01j−u02j. (3.21) Returning to (3.20) and using the equivalence between the norms inVj, we obtain
u1j(T)−u2j(T)Vj≤Cu01j−u02jVj (3.22) and, therefore, Lemma 3.2follows. Returning to the map τ and usingLemma 3.2, we obtain, from the fixed Brower theorem, that there existsu01j∈Vjsuch that
τu01j=u01j. (3.23)
Thus,
u1j(0)=u1j(T), (3.24)
whereu1j is the solution of the approximate problem (3.4)–(3.6) with initial datumu01j. Since the estimates (3.9)–(3.11) were uniform inj, we can see that exist a subsequence of (u1j), again called (u1j), and a functionusuch that
u1j ∗ u weak star inL∞0,T,L2(Ω), (3.25) u1j u weakly inL20,T,H01(Ω), (3.26)
∂tu1j ∂tu weakly inL20,T,H−1(Ω), (3.27) d
dt(u,v) +al(u)
Ω∇u· ∇v dx+
Ωf(u)v dx= h,v, (3.28) for allv∈H01(Ω), in the sense ofᏰ(0,T).
Finally, we will now prove that
u(0)=u(T). (3.29)
Indeed, by (3.26) we have T
0
u1j(t),vψ(t)dt−→
T
0
u(t),vψ(t)dt, (3.30)
for allv∈H01(Ω) andψ∈H01(0,T), withψ(T)=0.
Also by (3.27) we obtain T
0
d dt
u1j(t),vψ(t)dt−→
T
0
d dt
u(t),vψ(t)dt, (3.31)
for allv∈H01(Ω) andψ∈H01(0,T), withψ(T)=0.
It follows from (3.30) and (3.31) that T
0
d dt
u1j(t),vψ(t)dt−→
T
0
d dt
u(t),vψ(t)dt, (3.32)
that is,
u1j(0),v−→
u(0),v, ∀v∈H01(Ω). (3.33) We use the preceding argument withψ∈H01(Ω),ψ(0)=0, to get
u1j(T),v−→
u(T),v, ∀v∈H01(Ω). (3.34) Sinceu1j(0)=u1j(T), it follows from (3.33) and (3.34) thatu(T)=u(0) inL2(Ω), which
concludes the proof of ourTheorem 3.1.
References
[1] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, Monographs in Mathematics, vol.
89, Birkh¨auser Boston, Massachusetts, 1995.
[2] H. Brezis, Analyse Fonctionnelle: Th´eorie et Applications [Functional Analysis: Theory and Appli- cations], Masson, Paris, 1983.
[3] M. Chipot, Elements of Nonlinear Analysis, Birkh¨auser Advanced Texts, Birkh¨auser, Basel, 2000.
[4] M. Chipot and B. Lovat, On the asymptotic behaviour of some nonlocal problems, Positivity 3 (1999), no. 1, 65–81.
[5] A. Haraux, Syst`emes Dynamiques Dissipatifs et Applications [Dissipative Dynamical Systems and Applications], Research in Applied Mathematics, vol. 17, Masson, Paris, 1991.
[6] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
[7] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operator, Springer, New York, 1990.
S. B. de Menezes: Departamento de Matem´atica, Universidade Federal do Par´a, 66075-110 Bel´em, Par´a, Brazil
E-mail address:[email protected]
Special Issue on
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