A NONLOCAL PARABOLIC PROBLEM

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.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

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 R^N 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 byH^m(Ω) the usual Sobolev space of orderm, byH₀^m(Ω) the closure of C^∞0(Ω) inH^m(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), u_t∈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), u_t∈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)),

u_t,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|²_H⁻¹_(Ω), (2.19) that is,

1 2

d

dtu(t)²+m

2u(t)²≤Cu(t)²+ 1

2m^|h|²_H−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)|v_t∈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

w_n−→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

u_n u_∞ weakly inW0,T,H₀¹(Ω),H⁻¹(Ω), (2.27) u_n−→u_∞ strongly inL²0,T;L²(Ω), (2.28) lw_n−→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

u_n(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,

u_n=Nw_n−→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)

u_j,v+alu_j∇u_j,∇v+fu_j,v= h,v, ∀v∈Vj, (3.5) u_j(0)=u⁰_j−→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=u_j 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

dtu_j(t)²+m

2u_j(t)²_≤γu_j(t)²+ 1

2m^|h_|²_H−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 solutionu_j(t) to [0,T].

Next, let us consider

u_j(0)=u⁰_j∈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

dtu_j(t)²+mλ1−2γu_j(t)²≤ 1

2m^|h|²_H−1(Ω). (3.13) We note that, by hypothesis (3.1),mλ1−2γ >0 and thus

d

dtu_j(t)²+Cu_j(t)²≤C|h|²_H−1(Ω). (3.14) We multiply both sides of (3.14) bye^Ctand we integrate on [0,t) to obtain

uj(t)²≤e^−Ctuj(0)²+C _T

0 e^−C(s−T)h(s)²_H−1(Ω)ds. (3.15) Sinceh∈L²(0,T,H⁻¹(Ω)), one has

u_j(t)²≤θ(t)u_j(0)²+C, (3.16) for all 0≤t≤T, withθ(t)=e^−Ct and C=_T

0 e^−C(s−T)|h(s)|²_H−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 eachu⁰_j∈B0(R)∩Vjthere exists a solutionu_j(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,

u⁰_j−→τu⁰_j=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 consideru⁰_1j,u⁰_2j 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

u1_j(T)−u2_j(T)≤Cu⁰_1j−u⁰_2j. (3.21) Returning to (3.20) and using the equivalence between the norms inVj, we obtain

u1j(T)−u2j(T)_Vj≤Cu⁰_1j−u⁰_2jVj (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 existsu⁰_1j∈Vjsuch that

τu⁰_1j=u⁰_1j. (3.23)

Thus,

u1_j(0)=u1_j(T), (3.24)

whereu1j is the solution of the approximate problem (3.4)–(3.6) with initial datumu⁰_1j. Since the estimates (3.9)–(3.11) were uniform inj, we can see that exist a subsequence of (u1_j), again called (u1_j), and a functionusuch that

u1_j ^∗ u weak star inL^∞0,T,L²(Ω), (3.25) u1_j 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

u1_j(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

u1_j(t),vψ(t)dt_−→

_T

0

d dt

u(t),vψ(t)dt, (3.32)

that is,

u1_j(0),v−→

u(0),v, ∀v∈H₀¹(Ω). (3.33) We use the preceding argument withψ∈H₀¹(Ω),ψ(0)=0, to get

u1_j(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]