Attractors for a nonautonomous reaction-diffusion equation with delay

Open access

Abstract

In this paper, we discuss the existence and uniqueness of solutions for a non-autonomous reaction-diffusion equation with delay, after we prove the existence of a pullback π’Ÿ-asymptotically compact process. By a priori estimates, we show that it has a pullback π’Ÿ-absorbing set that allow us to prove the existence of a pullback π’Ÿ-attractor for the associated process to the problem.

Abstract

In this paper, we discuss the existence and uniqueness of solutions for a non-autonomous reaction-diffusion equation with delay, after we prove the existence of a pullback π’Ÿ-asymptotically compact process. By a priori estimates, we show that it has a pullback π’Ÿ-absorbing set that allow us to prove the existence of a pullback π’Ÿ-attractor for the associated process to the problem.

1 Introduction and statement of the problem

We consider the following nonautonomous functional reaction-diffusion equation

βˆ‚βˆ‚tu(t,x)βˆ’Ξ”u(t,x)=f(u(t,x))+b(t,ut)(x)+g(t,x)in(Ο„,∞)Γ—Ξ©,u=0on(Ο„,∞)Γ—βˆ‚Ξ©,u(Ο„,x)=u0(x),Ο„βˆˆRandx∈Ω,u(Ο„+ΞΈ,x)=Ο†(ΞΈ,x),θ∈[βˆ’r,0]andx∈Ω,

where Ξ© βŠ‚ ℝℕ is a bounded domain with smooth boundary βˆ‚ Ξ©, Ο„ ∈ ℝ, u0 ∈ L2(Ξ©) is the initial condition in Ο„ and Ο† ∈ L2([–r,0];L2(Ξ©)) is also the initial condition in [τ–r,Ο„],r > 0 is the length of the delay effect. For the rest we assume following assumptions conditions:

H1) Concerning the nonlinearity, we assume that f ∈ C1(ℝ, ℝ) there exist positive constants c, ΞΌ0, ΞΌ1, k and p > 2 N ≀ 2ppβˆ’2 such that

βˆ’cβˆ’ΞΌ0|u|p≀f(u)u≀cβˆ’ΞΌ1|u|pβˆ€u∈R,

f(u)βˆ’f(v)(uβˆ’v)≀k(uβˆ’v)2βˆ€u,v∈R.

Let us denote by

F(u):=∫0uf(s)ds.

From (1.2), there exist positive constants l,cβ€² ΞΌ0β€²,ΞΌ1β€² such that

|f(u)|≀l|u|pβˆ’1+1βˆ€u∈R,

βˆ’cβ€²βˆ’ΞΌ0β€²|u|p≀F(u)≀cβ€²βˆ’ΞΌ1β€²|u|pβˆ€u∈R.

H2) The operator b : ℝ Γ— L2([–r,0];L2(Ξ©)) β†’ L2(Ξ©) is a time-dependent external force with delay, such that

  • For all Ο• ∈ L2([–r,0]; L2(Ξ©)) the function β„βˆ‹ t ↦ b(t,Ο•) ∈ L2(Ξ©) is measurable;

  • b(t,0) = 0 for all t ∈ ℝ;

  • βˆƒ Lb > 0 s.t βˆ€ t ∈ ℝ and βˆ€ Ο•1, Ο•2 ∈ L2([–r,0];L2(Ξ©));

    βˆ₯b(t,Ο•1)βˆ’b(t,Ο•2)βˆ₯≀Lbβˆ₯Ο•1βˆ’Ο•2βˆ₯L2([βˆ’r,0];L2(Ξ©));

  • βˆƒ Cb > 0 s.t βˆ€ t β‰₯ Ο„, and βˆ€ u, v ∈ L2([Ο„-r, t]; L2(Ξ©));

    βˆ«Ο„tβˆ₯b(s,us)βˆ’b(s,vs)βˆ₯2ds≀Cbβˆ«Ο„βˆ’rtβˆ₯u(s)βˆ’v(s)βˆ₯2ds.

Remark 1

From (I)-(III), for T > Ο„ and u ∈ L2([τ–r,T];L2(Ξ©)) the function β„βˆ‹ t ↦ b(t,Ο•) ∈ L2(Ξ©) is measurable and belongs to L∞((Ο„,T);L2(Ξ©)).

H3) The function g ∈ Lloc2 (ℝ; L2(Ξ©)) is an another nondelayed time-dependent external force.

For more details on differential equations with delay, we refer the reader to J. Wu [9] and J.K. Hale [5]. The purpose of this paper is to discuss the existence of pullback π’Ÿ-attractor in L2(Ξ©)Γ— L2([–r,0];L2(Ξ©)) by using a priori estimates of solutions to the problem (1.1).

This work is motivated by the work of T. Caraballo and J. Real. [1], where they proved the existence of pullback attractors for the following 2D-Navier Stokes model with delays:

βˆ‚uβˆ‚tβˆ’Ξ½Ξ”u+βˆ‘i=12uiβˆ‚uβˆ‚xi=fβˆ’βˆ‡p+g(t,ut)in(Ο„,∞)Γ—Ξ©,divu=0in(Ο„,∞)Γ—Ξ©,u=0on(Ο„,∞)Γ—βˆ‚Ξ©,u(Ο„,x)=u0(x),x∈Ω,u(t,x)=Ο•(tβˆ’Ο„,x),t∈(Ο„βˆ’h,Ο„)andx∈Ω,

where Ξ½ > 0 is the kinematic viscosity, u is the velocity field of the fluid, p the pressure, Ο„ ∈ ℝ the initial time, u0 the initial velocity field, f a nondelayed external force field, g another external force with delay and Ο• the initial condition in (–h,0), where h is a fixed positive number.

On the other hand, the problem (1.1) without critical nonlinearity was treated by J. Li and J. Huang in [6], where they proved the existence of uniform attractor for the following non-autonomous parabolic equation with delays:

βˆ‚u(t,x)βˆ‚t+Au(t,x)+bu(t,x)=F(ut)(x)+g(t,x)xinΞ©,u(Ο„,x)=u0(x),u(Ο„+ΞΈ,x)=Ο•(ΞΈ,x),θ∈(βˆ’r,0).

Here Ξ© is a bounded domain in ℝn0 with smooth boundary, b β‰₯ 0, A is a densely-defined self-adjoint positive linear operator with domain D(A)βŠ‚ L2(Ξ©) and with compact resolvent, F is the nonlinear term which is locally Lipschitz continuous for the initial condition, g is an external force.

In [3], J.Garcia-Luengo and P.Marin-Rubio treated the following reaction-diffusion equation with non-autonomous force in H–1 and delays under measurability conditions on the driving delay term:

βˆ‚uβˆ‚tβˆ’Ξ”u=f(u)+g(t,ut)+k(t)in(Ο„,∞)Γ—Ξ©,u=0on(Ο„,∞)Γ—βˆ‚Ξ©,u(Ο„+s,x)=Ο•(s,x),s∈[βˆ’r,0]andx∈Ω,

where Ο„ ∈ ℝ, f ∈ C(ℝ) the nonlinear term with critical exponent, g is an external force with delay, k ∈ Lloc2 (ℝ;H–1(Ξ©)) a time-dependent force, Ο• the initial condition and h the lenght of the delay effect. In this work, the authors checked the existence of pullback π’Ÿ-attractor in C([–h,0]; L2(Ξ©)).

This paper is organized as follows. In section 2, we will prove the existence of weak solutions to the problem (1.1) by using the Faedo-Galerkin approximations, as well as the uniqueness and the continuous dependence of solution with respect to initial conditions. In section 3, we recall some definitions and abstract results on pullback π’Ÿ-attractor. Then we can prove the existence of pullback π’Ÿ-attractor for the nonautonomous problem with delay.

2 Existence and uniqueness of solution

First we give the concept of the solution.

Definition 1

A weak solution of (1.1) is a function u ∈ L2([τ–r,T];L2(Ξ©)) such that for all T > Ο„ we have

u∈L2((Ο„,T);H01(Ξ©))∩Lp((Ο„,T);Lp(Ξ©))∩C([Ο„,T];L2(Ξ©))

and

βˆ‚uβˆ‚t∈L2([Ο„,T];L2(Ξ©)),

with u(t) = Ο†(t–τ), for t ∈ [τ–r,Ο„], and it satisfies

βˆ«Ο„Tβˆ’γ€ˆu,v′〉+βˆ«Ο„Tβˆ«Ξ©βˆ‡uβˆ‡v=βˆ«Ο„T∫Ωf(u)v+βˆ«Ο„Tγ€ˆb(t,ut),v〉 +βˆ«Ο„T∫Ωgv+γ€ˆu0,v(Ο„)〉,

for all test functions v ∈ L2([Ο„, T]; H01 (Ξ©)) and vβ€²βˆˆ L2([Ο„, T]; H–1(Ξ©)) such that v(T) = 0.

Theorem 1

Assume that g ∈ Llog2 (ℝ;L2(Ξ©)), b and f satisfy (I)-(IV) and (1.2)-(1.5) respectively and if Ξ» 1 > 1 + Cb/2, Then for all T > Ο„ and all (u0,Ο†) in L2(Ξ©)Γ— L2([–r,0];L2(Ξ©)), there exists a unique weak solution u to the problem (1.1).

Proof

Let us consider {ek}k β‰₯ 1, the complete basis of H01 (Ξ©) which is given by the orthonormal eigenfunctions of Ξ” in L2(Ξ©). We consider

um(t)=βˆ‘k=1mΞ³k,m(t)ek,m=1,2,…

which is the approximate solutions of Faedo-Galerkin of order m, that is

γ€ˆdumdt,ek〉+γ€ˆΞ”um,ek〉=γ€ˆf(um),ek〉+γ€ˆb(t,utm),ek〉+γ€ˆg,ekγ€‰γ€ˆum(Ο„),ek〉=γ€ˆPmu0,ek〉=γ€ˆu0,ek〉i.e.Pmum(Ο„)β†’u0inL2(Ξ©)γ€ˆum(Ο„+ΞΈ),ek〉=γ€ˆPmΟ†(ΞΈ),ek〉=γ€ˆΟ†(ΞΈ),ekγ€‰βˆ€ΞΈβˆˆ(βˆ’r,0)

for all k = 1 … m. Where Ξ³k,m(t) = γ€ˆ um(t), ek 〉 denote the Fourier coefficients; such that Ξ³m,k ∈ C1((Ο„, T); ℝ) ∩ L2((τ–r, T), ℝ), Ξ³k,mβ€² (t) is absolutely continuous, and Pm u(t) = βˆ‘k=1m γ€ˆu,ek〉 ek is the orthogonal projection of L2(Ξ©) and H01 (Ξ©) in Vm = span{e1, …, em}.

It is well-known that the above finite-dimensional delayed system is well-posed (e.g. cf. [2]), at least locally. We will provide a priori estimates for the Faedo-Galerkin approximate solutions.

  • For all m ∈ β„•βˆ— and all T > Ο„, the sequence {um} is bounded in

    L∞((Ο„,T);L2(Ξ©))∩L2((Ο„,T);H01(Ξ©))∩Lp((Ο„,T);Lp(Ξ©)).

    Multiplying (1.1) by um and integrating over Ξ©, we obtain

    12ddtβˆ₯um(t)βˆ₯2+βˆ₯βˆ‡um(t)βˆ₯2=∫Ωf(um)um+∫Ωb(t,utm)um+∫Ωgum.

    Using the hypothesis (1.2) and the Young inequality, we get

    12ddtβˆ₯um(t)βˆ₯2+βˆ₯βˆ‡um(t)βˆ₯2≀c|Ξ©|βˆ’ΞΌ1βˆ₯um(t)βˆ₯p+12βˆ₯b(t,utm)βˆ₯2+12βˆ₯um(t)βˆ₯2+12βˆ₯g(t)βˆ₯2+12βˆ₯um(t)βˆ₯2.

    So, one has

    ddtβˆ₯um(t)βˆ₯2+2βˆ₯βˆ‡um(t)βˆ₯2+2ΞΌ1βˆ₯um(t)βˆ₯p≀2c|Ξ©|+βˆ₯b(t,utm)βˆ₯2+βˆ₯g(t)βˆ₯2+βˆ₯um(t)βˆ₯2.

    After integrating this last estimate over [Ο„,t], Ο„ ≀ t ≀ T, we use (II) and (IV), so we get

    βˆ₯um(t)βˆ₯2+2βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds+2ΞΌ1βˆ«Ο„tβˆ₯um(s)βˆ₯pds≀2c|Ξ©|(tβˆ’Ο„)+βˆ₯um(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rtβˆ₯um(s)βˆ₯2ds+βˆ«Ο„tβˆ₯g(s)βˆ₯2ds+βˆ«Ο„tβˆ₯um(s)βˆ₯2ds,≀2c|Ξ©|(tβˆ’Ο„)+βˆ₯um(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rΟ„βˆ₯um(s)βˆ₯2ds+Cbβˆ«Ο„tβˆ₯um(s)βˆ₯2ds+βˆ«Ο„tβˆ₯g(s)βˆ₯2ds+βˆ«Ο„tβˆ₯um(s)βˆ₯2ds.

    By the fact that Ξ»1 β€– uβ€– 2 ≀ β€– βˆ‡ u β€– 2, one has

    βˆ₯um(t)βˆ₯2+2βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds+2ΞΌ1βˆ«Ο„tβˆ₯um(s)βˆ₯pds≀2c|Ξ©|(tβˆ’Ο„)+βˆ₯um(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rΟ„βˆ₯um(s)βˆ₯2ds+CbΞ»1βˆ’1βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds+βˆ«Ο„tβˆ₯g(s)βˆ₯2ds+Ξ»1βˆ’1βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds.

    Then, we find

    βˆ₯um(t)βˆ₯2+(2βˆ’CbΞ»1βˆ’1βˆ’Ξ»1βˆ’1)βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds+2ΞΌ1βˆ«Ο„tβˆ₯um(s)βˆ₯pds≀2c|Ξ©|(tβˆ’Ο„)+βˆ₯um(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rΟ„βˆ₯um(s)βˆ₯2ds+βˆ«Ο„tβˆ₯g(s)βˆ₯2ds,≀2c|Ξ©|(Tβˆ’Ο„)+βˆ₯um(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rΟ„βˆ₯um(s)βˆ₯2ds+βˆ«Ο„tβˆ₯g(s)βˆ₯2ds.

    Since g ∈ Llog2 (ℝ,L2(Ξ©)) and for Ξ»1 > 1+ Cb/2, we deduce by this last estimate that for all T > Ο„, the sequence

    {um} is bounded inL∞((Ο„,T);L2(Ξ©))∩L2((Ο„,T);H01(Ξ©))∩Lp((Ο„,T);Lp(Ξ©)).

    Also, the estimate (2.1) implies that the local solution can extended to the interval [Ο„, T].

  • {f(um)}isboundedinLq((Ο„,T);Lq(Ξ©)).

    Using (1.4), we have

    βˆ₯f(um(t)βˆ₯Lq(Ξ©)q=∫Ω|f(um(t,x))|qdx, ≀lq∫Ω|um(t,x)|pβˆ’1+1qdx.

    By the convexity of the power and the fact that p = q(p–1), one has

    βˆ₯f(um(t))βˆ₯Lq(Ξ©)q≀2qβˆ’1lq∫Ω|um(t,x)|q(pβˆ’1)dx+2qβˆ’1lq|Ξ©|,≀2qβˆ’1lqβˆ₯um(t)βˆ₯Lp(Ξ©)q(pβˆ’1)+2qβˆ’1lq|Ξ©|,≀2qβˆ’1lqβˆ₯um(t)βˆ₯Lp(Ξ©)p+2qβˆ’1lq|Ξ©|.

    Integrating this last estimate over [Ο„,t], Ο„ ≀ t ≀ T, one obtains

    βˆ«Ο„tβˆ₯f(um(s))βˆ₯Lq(Ξ©)qds≀2qβˆ’1lqβˆ«Ο„tβˆ₯um(s)βˆ₯Lp(Ξ©)pds+2qβˆ’1lq|Ξ©|(tβˆ’Ο„)

    From (2.1) we deduce that the term βˆ«Ο„tβˆ₯um(s)βˆ₯Lp(Ξ©)p ds is bounded, so by this last estimate we conclude that {f(um)} is bounded in Lq((Ο„, T);Lq(Ξ©)), for all T > Ο„.

  • βˆ‚βˆ‚tum is bounded in L2((Ο„,T);L2(Ξ©)).

Now, multiplying (1.1) by βˆ‚umβˆ‚t and integrating over Ξ©, one has

βˆ’ddtum(t)2+12ddtβˆ₯βˆ‡um(t)βˆ₯2=∫Ωf(um)βˆ‚umβˆ‚t+∫Ωb(t,utm)βˆ‚umβˆ‚t+∫Ωgβˆ‚umβˆ‚t.

On the other hand, we have

ddtF(u)=dFduβˆ‚uβˆ‚t,=f(u)βˆ‚uβˆ‚t.

So

ddt∫ΩF(u)=∫Ωf(u)βˆ‚uβˆ‚t.

Using this last equality in (2.4), we find

ddtum(t)2+12ddtβˆ₯βˆ‡um(t)βˆ₯2=ddt∫ΩF(um)+∫Ωb(t,utm)βˆ‚umβˆ‚t+∫Ωgβˆ‚umβˆ‚t.

From (1.5) and Cauchy inequality, we have

ddtum(t)2+12ddtβˆ₯βˆ‡um(t)βˆ₯2,≀ddt∫Ω(cβ€²βˆ’ΞΌ1β€²|u(t,x)|p)dx+Ξ΅12βˆ₯b(t,utm)βˆ₯2+12Ξ΅1ddtum(t)2+Ξ΅22βˆ₯g(t)βˆ₯2+12Ξ΅2ddtum(t)2.

After simplification, one obtains

2βˆ’1Ξ΅1βˆ’1Ξ΅2ddtum(t)2+ddtβˆ₯βˆ‡um(t)βˆ₯2+2ΞΌ1β€²βˆ₯um(t)βˆ₯Lp(Ξ©)p≀Ρ1βˆ₯b(t,utm)βˆ₯2+Ξ΅2βˆ₯g(t)βˆ₯2.

We can choose Ξ΅1 = Ξ΅2 = 2 to get

ddtum(t)2+ddtβˆ₯βˆ‡um(t)βˆ₯2+2ΞΌ1β€²βˆ₯um(t)βˆ₯Lp(Ξ©)p≀2βˆ₯b(t,utm)βˆ₯2+2βˆ₯g(t)βˆ₯2.

Integrating this last estimate over [Ο„,t] and using (II) and (IV), one has

βˆ«Ο„tddsum(s)2ds+βˆ₯βˆ‡um(t)βˆ₯2+2ΞΌ1β€²βˆ₯um(t)βˆ₯Lp(Ξ©)p≀βˆ₯βˆ‡um(Ο„)βˆ₯2+2ΞΌ1β€²βˆ₯um(Ο„)βˆ₯Lp(Ξ©)p+2Cbβˆ«Ο„βˆ’rtβˆ₯um(s)βˆ₯2ds+2βˆ«Ο„tβˆ₯g(s)βˆ₯2ds,≀βˆ₯βˆ‡um(Ο„)βˆ₯2+2ΞΌ1β€²βˆ₯um(Ο„)βˆ₯Lp(Ξ©)p+2Cbβˆ«Ο„βˆ’rΟ„βˆ₯um(s)βˆ₯2ds+2Cbβˆ«Ο„tβˆ₯um(s)βˆ₯2ds+2βˆ«Ο„tβˆ₯g(s)βˆ₯2ds

Since Ξ»1 β€– u β€–2 ≀ β€– βˆ‡uβ€–2, one has

βˆ«Ο„tddsum(s)2ds+βˆ₯βˆ‡um(t)βˆ₯2+2ΞΌ1β€²βˆ₯um(t)βˆ₯Lp(Ξ©)p≀βˆ₯βˆ‡um(Ο„)βˆ₯2+2ΞΌ1β€²βˆ₯um(Ο„)βˆ₯Lp(Ξ©)p+2Cbβˆ«Ο„βˆ’rΟ„βˆ₯um(s)βˆ₯2ds+2CbΞ»1βˆ’1βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds+2βˆ«Ο„tβˆ₯g(s)βˆ₯2ds

From (2.1), we have βˆ«Ο„tβˆ₯βˆ‡um(s)βˆ₯2ds

is bounded and since g ∈ Lloc2 (ℝ;L2(Ξ©)), this last estimate gives that

βˆ‚βˆ‚tumis bounded inL2((Ο„,T);L2(Ξ©)),

for all T > Ο„.

From the claims (1), (2) and (3), the hypothesis (IV) and the remark (1)1, we can extract a subsequence (relabelled the same) such that

um⇀uweakly* inL∞((Ο„,T);L2(Ξ©)),um⇀uweakly inL2((Ο„,T);H01(Ξ©)),um⇀uweakly inLp((Ο„,T);Lp(Ξ©)),βˆ‚umβˆ‚tβ†’βˆ‚uβˆ‚tstrongly inL2((Ο„,T);L2(Ξ©)),f(um)⇀σ′weakly inLq((Ο„,T);Lq(Ξ©)),b(.,u.m)β†’b(.,u.)strongly inL2((Ο„,T);L2(Ξ©)).

By the Aubin-Lions lemma of compactness, we conclude that um β†’ u strongly in L2((Ο„,T);L2(Ξ©)). Thus um β†’ u a.e [Ο„,T]Γ— Ξ©.

Since f is continuous, we deduce that f(um) β†’ f(u) a.e [Ο„,T]Γ— Ξ©. So from (2.3) and (lemma 1.3 in [7], p.12) we can identify Οƒβ€² with f(u).

To prove that u(Ο„) = u0, we put v ∈ C1((Ο„,T); H01 (Ξ©)) such that v(T) = 0 and we note from (1.1) that

βˆ«Ο„Tβˆ’γ€ˆu,v′〉+βˆ«Ο„Tβˆ«Ξ©βˆ‡uβˆ‡v=βˆ«Ο„T∫Ωf(u)v+βˆ«Ο„Tγ€ˆb(t,ut),v〉 +βˆ«Ο„T∫Ωgv+γ€ˆu(Ο„),v(Ο„)〉.

In a similar way, from the Faedo-Galerkin approximations, we have

βˆ«Ο„Tβˆ’γ€ˆum,vβ€²γ€‰βˆ«Ο„Tβˆ«Ξ©βˆ‡umβˆ‡v=βˆ«Ο„T∫Ωf(um)v+βˆ«Ο„Tγ€ˆb(t,utm),v〉+βˆ«Ο„T∫Ωgv+γ€ˆum(Ο„),v(Ο„)〉.

Using the fact that um(Ο„) β†’ u0 in L2(Ξ©) and (2.6) to find

βˆ«Ο„Tβˆ’γ€ˆu,v′〉+βˆ«Ο„Tβˆ«Ξ©βˆ‡uβˆ‡v=βˆ«Ο„T∫Ωf(u)v+βˆ«Ο„Tγ€ˆb(t,ut),v〉 +βˆ«Ο„T∫Ωgv+γ€ˆu0,v(Ο„)〉.

Since v(Ο„) is given arbitrarily, comparing (2.7) and (2.9) we deduce that u(Ο„) = u0.

To prove that u ∈ C([Ο„,T];L2(Ξ©)), we put wm = um–u then we have

βˆ‚βˆ‚twmβˆ’Ξ”wm=f(um)βˆ’f(u)+b(t,utm)βˆ’b(t,ut).

Multiplying this equation by wm and integrating over Ξ©, we obtain

ddtβˆ₯wm(t)βˆ₯2+2βˆ₯βˆ‡wm(t)βˆ₯2=2∫Ωf(um)βˆ’f(u)wm+2∫Ω(b(t,utm)βˆ’b(t,ut))(umβˆ’u).

By (1.3), (I) and (1.6), we get

ddtβˆ₯wm(t)βˆ₯2+2βˆ₯βˆ‡wm(t)βˆ₯2≀2kβˆ₯wm(t)βˆ₯2+2Lbβˆ₯wtmβˆ₯L2([βˆ’r,0];L2(Ξ©))2.

Integrating over [Ο„,t], we get

βˆ₯wm(t)βˆ₯2βˆ’βˆ₯wm(Ο„)βˆ₯2+2βˆ«Ο„tβˆ₯βˆ‡wm(s)βˆ₯2ds≀2kβˆ«Ο„tβˆ₯wm(s)βˆ₯2+2Lbβˆ«Ο„tβˆ«βˆ’r0βˆ₯wm(s+ΞΈ)βˆ₯2dΞΈds,≀2kβˆ«Ο„tβˆ₯wm(s)βˆ₯2+2Lbβˆ«βˆ’r0βˆ«Ο„βˆ’rtβˆ₯wm(s)βˆ₯2dsdΞΈ,≀2kβˆ«Ο„tβˆ₯wm(s)βˆ₯2+2Lbrβˆ«Ο„βˆ’rΟ„βˆ₯wm(s)βˆ₯2ds+2Lbrβˆ«Ο„tβˆ₯wm(s)βˆ₯2ds.

Therefore by by this last estimate, we can deduce that

βˆ₯wm(t)βˆ₯2≀βˆ₯wm(Ο„)βˆ₯2+2Lbrβˆ«Ο„βˆ’rΟ„βˆ₯wm(s)βˆ₯2ds+(2k+2Lbr)βˆ«Ο„tβˆ₯wm(s)βˆ₯2ds.

Applying the Gronwall lemma to this estimate, we obtain

βˆ₯wm(t)βˆ₯2≀βˆ₯wm(Ο„)βˆ₯2+2Lbrβˆ«Ο„βˆ’rΟ„βˆ₯wm(s)βˆ₯2dse(2k+2Lbr)(tβˆ’Ο„).

Since um(Ο„)β†’ u0 and um(Ο„ + ΞΈ)β†’ Ο†(ΞΈ), the estimate (2.10) shows that um β†’ u uniformly in C([Ο„,T];L2(Ξ©)).

Finally, we prove the uniqueness and continuous dependence of the solution. Let u1; u2 be two solutions of problem (1.1) with the initial conditions u0,1, u0,2 and Ο†1, Ο†2. Denoting that w = u1 – u2 and repeating the argument as in the proof of (2.10), we find

βˆ₯w(t)βˆ₯2≀βˆ₯w(Ο„)βˆ₯2+2Lbrβˆ«Ο„βˆ’rΟ„βˆ₯w(s)βˆ₯2dse(2k+2Lbr)(tβˆ’Ο„).

and this completes the proof of the theorem. β—Ό

3 Existence of pullback D-attractors

3.1 Preliminaries of pullback D-attractors

First, we give some basic definitions and an abstract result on the existence of pullback attractors, which we need to obtain our results (we refer the reader to [2,3,4,8]). Let (X,d) be a complete metric space, 𝒫(X) be the class of nonempty subsets of X, and suppose π’Ÿ is a nonempty class of parameterized sets DΜ‚ = {D(t) : t ∈ ℝ}βŠ‚ 𝒫(X).

Definition 2

A two parameter family of mappings U(t,Ο„) : X β†’ X t β‰₯ Ο„, Ο„ ∈ ℝ, is called to be a process if

  1. S(Ο„,Ο„)x = {x},βˆ€ Ο„ ∈ ℝ, x ∈ Y;

  2. S(t,s)S(s,Ο„)x = S(t,Ο„)x, βˆ€ t β‰₯ s β‰₯ Ο„, Ο„ ∈ ℝ, x ∈ X.

Definition 3

A family of bounded sets BΜ‚ = {B(t) : t ∈ ℝ}∈ π’Ÿ is called pullback π’Ÿ-absorbing for the process S(t,Ο„)} if for any t ∈ ℝ and for any DΜ‚ ∈ π’Ÿ, there exists Ο„0(t,DΜ‚) ≀ t such that

S(t,Ο„)D(Ο„)βŠ‚B(t)forallτ≀τ0(t,D^).

Definition 4

The process S(t,Ο„) is said to be pullback π’Ÿ-asymptotically compact if for all t ∈ ℝ, all DΜ‚ ∈ π’Ÿ, any sequence Ο„ n β†’ -∞, and any sequence xn ∈ D(Ο„n), the sequence {S(t,Ο„n)xn} is relatively compact in X.

Definition 5

A family AΜ‚ = {A(t) : t ∈ ℝ}βŠ‚ 𝒫(X) is said to be a pullback 7 π’Ÿ-attractor for S(t,Ο„)} if

  1. A(t) is compact for all t ∈ ℝ;

  2. AΜ‚ is invariant; i.e., S(t,Ο„)A(Ο„) = A(t), for all t β‰₯ Ο„;

  3. AΜ‚ is pullback π’Ÿ-attracting; i.e.,

    limΟ„β†’βˆ’βˆždist(S(t,Ο„)D(Ο„),A(t))=0,

    for all DΜ‚ ∈ π’Ÿ and all t ∈ ℝ;

  4. If {C(t) : t ∈ ℝ} is another family of closed attracting sets then A(t) βŠ‚ C(t), for all t ∈ ℝ.

Theorem 2

Let us suppose that the process {S(t,Ο„)} is pullback π’Ÿ-asymptotically compact, and BΜ‚ = {B(t) : t ∈ ℝ}∈ π’Ÿ is a family of pullback π’Ÿ-absorbing sets for {S(t,Ο„)}. Then there exists a pullback π’Ÿ-attractor {A(t) : t ∈ ℝ} such that

A(t)=β‹‚s≀t⋃τ≀sS(t,Ο„)B(Ο„)Β―.

3.2 Construction of the associated process

Now, we will apply the above results in the phase space H: = L2(Ξ©) Γ— L2([–r,0]; L2(Ξ©)), which is a Hilbert space with the norm

βˆ₯(u0,Ο†)βˆ₯H2=βˆ₯βˆ‡u0βˆ₯2+βˆ«βˆ’r0βˆ₯Ο†(ΞΈ)βˆ₯2dΞΈ,

with (u0,Ο†)∈ H. To this aim, We consider g ∈ Lloc2 (ℝ;L2(Ξ©)), b: ℝ Γ— L2([–r,0];L2(Ξ©)) β†’ L2(Ξ©) with the hypotheses (I)-(IV) and f ∈ C1(ℝ;ℝ) verifying (1.2)-(1.5). Then the family of mappings

S(t,Ο„):Hβ†’H  (u0,Ο†)⟼S(t,Ο„)(u0,Ο†)=(u(t),ut),

with t β‰₯ Ο„, Ο„ ∈ ℝ and u is the weak solution to (1.1), defines a process.

On the other hand, we construct the family of mappings

U(t,Ο„):Hβ†’C([βˆ’r,0];L2(Ξ©))(u0,Ο†)⟼U(t,Ο„)(u0,Ο†)=ut,βˆ€tβ‰₯Ο„+r,

which we will use in our analysis. Of course, it is sensible to expect that the both operators should be related. Let us consider the linear mapping

j:C([βˆ’r,0];L2(Ξ©))β†’l2(Ξ©)Γ—C([βˆ’r,0];L2(Ξ©))β€‰β€‰β€‰β€‰β€‰Ο†βŸΌj(Ο†)=(Ο†(0),Ο†).

This map is obviously continuous from C([–r,0]; L2(Ξ©)) into H. We note that for all (u0,Ο†) ∈ H provided that t β‰₯ Ο„ + r, so we write

S(t,Ο„)(u0,Ο†)=j(U(t,Ο„)(u0,Ο†)),βˆ€(u0,Ο†)∈H,βˆ€tβ‰₯Ο„+r.

To check the continuity of the process, we need the following lemma.

Lemma 1

Let (u0,Ο†), (v0,Ο•)∈ H be two couples of initial conditions for the problem (1.1) and u, v be the corresponding solutions to (1.1). Then there exists a positive constant Ξ½: = 2(12+k+Cb2βˆ’Ξ»1)>0, such that

βˆ₯u(t)βˆ’v(t)βˆ₯2≀βˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯2eΞ½(tβˆ’Ο„),βˆ€tβ‰₯Ο„.

It also holds

βˆ₯utβˆ’vtβˆ₯C([βˆ’r,0];L2(Ξ©))2≀βˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯2eΞ½(tβˆ’rβˆ’Ο„),βˆ€tβ‰₯Ο„+r.

Proof

From (1.1), one has

βˆ‚βˆ‚t(uβˆ’v)βˆ’Ξ”(uβˆ’v)=f(u)βˆ’f(v)+b(t,ut)βˆ’b(t,vt).

We put w = u–v, we obtain

βˆ‚wβˆ‚tβˆ’Ξ”w=f(u)βˆ’f(v)+b(t,ut)βˆ’b(t,vt).

Multiplying this equation by w and integrating it over Ξ©, one gets

12ddtβˆ₯w(t)βˆ₯2+βˆ₯βˆ‡w(t)βˆ₯2=∫Ωf(u)βˆ’f(v)w+∫Ωb(t,ut)βˆ’b(t,vt)w.

Using (1.3) and Cauchy-Schwarz inequality, one has

12ddtβˆ₯w(t)βˆ₯2+βˆ₯βˆ‡w(t)βˆ₯2≀kβˆ₯w(t)βˆ₯2+βˆ₯b(t,ut)βˆ’b(t,vt)βˆ₯βˆ₯w(t)βˆ₯.

Since Ξ»1 β€– w(t) β€–2 ≀ β€– βˆ‡ w(t) β€–2 and by the Young inequality, one finds

ddtβˆ₯w(t)βˆ₯2+2Ξ»1βˆ₯w(t)βˆ₯2≀ddtβˆ₯w(t)βˆ₯2+2βˆ₯βˆ‡w(t)βˆ₯2,≀2kβˆ₯w(t)βˆ₯2+βˆ₯b(t,ut)βˆ’b(t,vt)βˆ₯2+βˆ₯w(t)βˆ₯2.

Therefore, one has

ddtβˆ₯w(t)βˆ₯2≀212+kβˆ’Ξ»1βˆ₯w(t)βˆ₯2+βˆ₯b(t,ut)βˆ’b(t,vt)βˆ₯2.

Integrating this last estimate from Ο„ to t and using (1.7), one obtains

βˆ₯w(t)βˆ₯2≀βˆ₯w(Ο„)βˆ₯2+212+kβˆ’Ξ»1βˆ«Ο„tβˆ₯w(s)βˆ₯2ds +βˆ«Ο„tβˆ₯b(s,us)βˆ’b(s,vs)βˆ₯2ds, ≀βˆ₯w(Ο„)βˆ₯2+212+kβˆ’Ξ»1βˆ«Ο„tβˆ₯w(s)βˆ₯2ds+Cbβˆ«Ο„βˆ’rtβˆ₯w(s)βˆ₯2ds, ≀βˆ₯w(Ο„)βˆ₯2+212+kβˆ’Ξ»1βˆ«Ο„tβˆ₯w(s)βˆ₯2ds+Cbβˆ«Ο„βˆ’rΟ„βˆ₯w(s)βˆ₯2ds +Cbβˆ«Ο„tβˆ₯w(s)βˆ₯2ds, ≀βˆ₯w(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rΟ„βˆ₯w(s)βˆ₯2ds+212+k+Cb2βˆ’Ξ»1βˆ«Ο„tβˆ₯w(s)βˆ₯2ds.

By the Gronwall lemma, for all t β‰₯ Ο„, one deduces

βˆ₯w(t)βˆ₯2≀βˆ₯w(Ο„)βˆ₯2+Cbβˆ«Ο„βˆ’rΟ„βˆ₯w(s)βˆ₯2dseΞ½(tβˆ’Ο„), ≀βˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯L2([βˆ’r,0];L2(Ξ©))2eΞ½(tβˆ’Ο„),

and by this last estimate, we proved (3.3). Now, assume that t β‰₯ Ο„ + r, so t + ΞΈ β‰₯ Ο„ for all ΞΈ ∈ [–r,0] and one has

βˆ₯w(t+ΞΈ)βˆ₯2≀βˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯L2([βˆ’r,0];L2(Ξ©))2eΞ½(t+ΞΈβˆ’Ο„),    ≀βˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯L2([βˆ’r,0];L2(Ξ©))2eΞ½(tβˆ’rβˆ’Ο„).

Hence, we conclude

βˆ₯wtβˆ₯C([βˆ’r,0];L2(Ξ©))≀βˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯L2([βˆ’r,0];L2(Ξ©))2eΞ½(tβˆ’rβˆ’Ο„).

By this last estimate we finished the proof of this lemma. β—Ό

Theorem 3

Under the previous assumptions, the mapping S(.,.) defined in (3.1), is a continuous process for all Ο„ ≀ t.

Proof

The proof of this theorem is as the proof of Theorem 9 in [1]. The uniqueness of the solutions implies that S(.,.) is a process. For the continuity of S(.,.), we use the previous lemma. We consider (u0,Ο†), (v0,Ο•)∈ H and u, v are their corresponding solutions. Firstly, if we take t β‰₯ Ο„ + r, it follows from (3.4)

βˆ₯utβˆ’vtβˆ₯L2([βˆ’r,0];L2(Ξ©))2=βˆ«βˆ’r0βˆ₯u(t+ΞΈ)βˆ’v(t+ΞΈ)βˆ₯2dΞΈ,β€‰β‰€βˆ«βˆ’r0sups∈[βˆ’r,0]βˆ₯u(t+s)βˆ’v(t+s)βˆ₯2dΞΈ, ≀rβˆ₯u0βˆ’v0βˆ₯2+Cbβˆ₯Ο†βˆ’Ο•βˆ₯2eΞ½(tβˆ’rβˆ’Ο„).

Now, for t ∈[Ο„,Ο„ + r], we deduce

βˆ₯utβˆ’vtβˆ₯L2([βˆ’r,0];L2(Ξ©))2=βˆ«βˆ’r0βˆ₯u(t+ΞΈ)βˆ’v(t+ΞΈ)βˆ₯2dΞΈ, ≀rβˆ₯u0βˆ’v0βˆ₯2+(Cbr+1)βˆ₯Ο†βˆ’Ο•βˆ₯2eΞ½(tβˆ’rβˆ’Ο„).

So, for all t β‰₯ Ο„, we have

βˆ₯utβˆ’vtβˆ₯L2([βˆ’r,0];L2(Ξ©))2≀rβˆ₯u0βˆ’v0βˆ₯2+(Cbr+1)βˆ₯Ο†βˆ’Ο•βˆ₯2eΞ½(tβˆ’rβˆ’Ο„).

Hence, by this last estimate and (3.3) we deduce the continuity of S(t,Ο„).β—Ό

3.3 Existence of pullback D-absorbing set in C([–r,0]; L2(Ξ©)) and H

Firstly, we need to the following lemma, it relates the absorption properties for the mappings with those of process S in the fact that, proving those for U yields to similar properties for S.

Lemma 2

Assume that the family of bounded sets {B(t): t ∈ ℝ} in the space C([–r,0]; L2(Ξ©)) is pullback π’Ÿ-absorbing for the mapping U(.,.). Then the family of bounded sets {j(B(t)): t ∈ ℝ} in L2(Ξ©) Γ— C([–r,0]; L2(Ξ©)) is pullback π’Ÿ-absorbing for the process S(.,.).

Proof

Let {D(t): t ∈ ℝ} be a family bounded sets in H, so there exists T > r such that

U(t,Ο„)D(Ο„)βŠ‚B(t),βˆ€tβˆ’Ο„β‰₯T.

On the other hand, we have

S(t,Ο„)(u0,Ο†)=j(U(t,Ο„)(u0,Ο†)),

it follows that

S(t,Ο„)(u0,Ο†)=j(U(t,Ο„)(u0,Ο†))βŠ‚j(B(t)),βˆ€tβˆ’Ο„β‰₯T.

β—Ό

Remark 2

Noticing that the word absorbing used in this papier should be interpreted in a generalized sense, since U is not a process.

Now, we need the following estimations.

Lemma 3

Assume that g∈Lloc2(R;L2(Ξ©)), there exists a small enough Ξ± < 2Ξ»1 – 2–Cb such that

βˆ«βˆ’βˆžteΞ±tβˆ₯g(s)βˆ₯2ds<∞,

the function f satisfies (1.2)-(1.5) and b fulfills conditions (I)-(IV) and

βˆ«Ο„teΟƒsβˆ₯b(s,us)βˆ’b(s,vs)βˆ₯2ds≀Cbβˆ«Ο„βˆ’rteΟƒsβˆ₯u(s)βˆ’v(s)βˆ₯2ds.

Then we have

βˆ₯u(t)βˆ₯2≀eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+Cbeβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,

and

Ξ·eβˆ’Ξ±tβˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2ds+2ΞΌ1eβˆ’Ξ±tβˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯Lp(Ξ©)pds≀eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+Cbeβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,

where Ξ· : = 2Ξ»1 – 2 – Ξ± – Cb.

Proof

Multiplying (1.1) by u and integrating over Ξ©, one has

12ddtβˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2=∫Ωf(u)u+∫Ωb(t,ut)u+∫Ωgu.

By (1.2), Cauchy-Shwarz and Young inequalities, we obtain

12ddtβˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2+ΞΌ1βˆ₯u(t)βˆ₯Lp(Ξ©)p≀c|Ξ©|+12βˆ₯b(t,ut)βˆ₯2+12βˆ₯g(t)βˆ₯2+βˆ₯u(t)βˆ₯2.

Since Ξ»1βˆ₯uβˆ₯2 ≀ βˆ₯βˆ‡uβˆ₯2 and after calculation, one has

ddtβˆ₯u(t)βˆ₯2+2(Ξ»1βˆ’1)βˆ₯u(t)βˆ₯2+2ΞΌ1βˆ₯u(t)βˆ₯Lp(Ξ©)p≀2c|Ξ©|+βˆ₯b(t,ut)βˆ₯2+βˆ₯g(t)βˆ₯2.

Now, we multiply this last estimate by eΞ±t such that 0 < Ξ± < 2Ξ»1 –2 – Cb, so one gets

eΞ±tddtβˆ₯u(t)βˆ₯2+2(Ξ»1βˆ’1)eΞ±tβˆ₯u(t)βˆ₯2+2ΞΌ1eΞ±tβˆ₯u(t)βˆ₯Lp(Ξ©)p≀2c|Ξ©|eΞ±t+eΞ±tβˆ₯b(t,ut)βˆ₯2+eΞ±tβˆ₯g(t)βˆ₯2.

On the other hand, we have

ddteΞ±tβˆ₯u(t)βˆ₯2=Ξ±eΞ±tβˆ₯u(t)βˆ₯2+eΞ±tddtβˆ₯u(t)βˆ₯2

We substitute (3.9) in this equality, we find

ddteΞ±tβˆ₯u(t)βˆ₯2≀αeΞ±tβˆ₯u(t)βˆ₯2βˆ’2(Ξ»1βˆ’1)eΞ±tβˆ₯u(t)βˆ₯2βˆ’2ΞΌ1eΞ±tβˆ₯u(t)βˆ₯Lp(Ξ©)p+2c|Ξ©|eΞ±t+eΞ±tβˆ₯b(t,ut)βˆ₯2+eΞ±tβˆ₯g(t)βˆ₯2.

Integrating this last estimate over [Ο„, t], one obtains

eΞ±tβˆ₯u(t)βˆ₯2≀eΞ±Ο„βˆ₯u(Ο„)βˆ₯2+2c|Ξ©|Ξ±βˆ’1(eΞ±tβˆ’eΞ±Ο„)+(Ξ±+2βˆ’2Ξ»1)βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2dsβˆ’2ΞΌ1βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯Lp(Ξ©)pds+βˆ«Ο„teΞ±sβˆ₯b(s,us)βˆ₯2ds+βˆ«Ο„teΞ±sβˆ₯g(s)βˆ₯2ds.

Using (3.6) and (II), one has

eΞ±tβˆ₯u(t)βˆ₯2≀eΞ±Ο„βˆ₯u(Ο„)βˆ₯2+2c|Ξ©|Ξ±βˆ’1(eΞ±tβˆ’eΞ±Ο„)+(Ξ±+2βˆ’2Ξ»1)βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2dsβˆ’2ΞΌ1βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯Lp(Ξ©)pds+Cbβˆ«Ο„βˆ’rteΞ±sβˆ₯u(s)βˆ₯2ds+βˆ«Ο„teΞ±sβˆ₯g(s)βˆ₯2ds.

On the other hand, we have

βˆ«Ο„βˆ’rteΞ±sβˆ₯u(s)βˆ₯2ds=βˆ«Ο„βˆ’rΟ„eΞ±sβˆ₯u(s)βˆ₯2ds+βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2ds,≀eΞ±Ο„βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2ds.

So by (3.10) and (3.11), one finds

eΞ±tβˆ₯u(t)βˆ₯2≀eΞ±Ο„βˆ₯u(Ο„)βˆ₯2+2c|Ξ©|Ξ±βˆ’1(eΞ±tβˆ’eΞ±Ο„)+(Ξ±+2βˆ’2Ξ»1+Cb)βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2dsβˆ’2ΞΌ1βˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯Lp(Ξ©)pds+CbeΞ±Ο„βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+βˆ«Ο„teΞ±sβˆ₯g(s)βˆ₯2ds.

Hence, by (3.5) we obtain

βˆ₯u(t)βˆ₯2+(2Ξ»1βˆ’Ξ±βˆ’2βˆ’Cb)eβˆ’Ξ±tβˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2ds+2ΞΌ1eβˆ’Ξ±tβˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯Lp(Ξ©)pds≀eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2c|Ξ©|Ξ±βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+Cbeβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds.

Thus, for Ξ· : = 2 Ξ»1– Ξ± – 2 – Cb > 0, by this last estimate we get

βˆ₯u(t)βˆ₯2≀eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+Cbeβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,

and

Ξ·eβˆ’Ξ±tβˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯2ds+2ΞΌ1eβˆ’Ξ±tβˆ«Ο„teΞ±sβˆ₯u(s)βˆ₯Lp(Ξ©)pds≀eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+Cbeβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,

for all t β‰₯ Ο„. So by these two estimations the proof of the lemma is finished. β—Ό

Proposition 1

Under the assumptions in lemma (3). Then the family {B1(t) : t ∈ ℝ} given by

B1(t)=BΒ―C([βˆ’r,0];L2(Ξ©))(0,R1(t)),

with

R12(t)=eΞ±r2c|Ξ©|Ξ±βˆ’1+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±tβˆ₯g(s)βˆ₯2ds,βˆ€t∈R;

is pullback 𝓓-absorbing for the mapping U(t, Ο„). Moreover, the family {B0(t) : t ∈ ℝ} given by

B0(t)=BΒ―L2(Ξ©))(0,R1(t))Γ—BΒ―L2([βˆ’r,0];L2(Ξ©))0,rR1(t)βŠ‚H,βˆ€t∈R,

is pullback 𝓓-absorbing for the process S defined by (3.1).

Proof

The first part may be proved as follows.

By definition, we have

βˆ₯U(t,Ο„)(u0,Ο†)βˆ₯C([βˆ’r,0];L2(Ξ©))2=sups∈[βˆ’r,0]βˆ₯u(t+s)βˆ₯2.

From (3.7), if we take t β‰₯ Ο„ + r, so t + ΞΈ β‰₯ Ο„. Then one has

βˆ₯u(t+ΞΈ)βˆ₯2≀eβˆ’Ξ±(t+ΞΈβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+Cbeβˆ’Ξ±(t+ΞΈβˆ’Ο„)βˆ₯Ο†βˆ₯L2([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ’11βˆ’eβˆ’Ξ±(t+ΞΈβˆ’Ο„)+eβˆ’Ξ±(t+ΞΈ)βˆ«βˆ’βˆžt+ΞΈeΞ±sβˆ₯g(s)βˆ₯2ds,

which implies that

sups∈[βˆ’r,0]βˆ₯u(t+s)βˆ₯2≀eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+Cbeβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†βˆ₯L2([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ’1eΞ±reβˆ’Ξ±rβˆ’eβˆ’Ξ±(tβˆ’Ο„)+eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,

On the one hand, we have

βˆ₯Ο†βˆ₯L2([βˆ’r,0];L2(Ξ©))2=βˆ«βˆ’r0βˆ₯Ο†(ΞΈ)βˆ₯2dΞΈ,β‰€βˆ«βˆ’r0sups∈[βˆ’r,0]βˆ₯Ο†(s)βˆ₯2dΞΈ,≀rβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2.

Therefore by (3.12), (3.13) and the fact that u(Ο„) = Ο† (0), we obtain

sups∈[βˆ’r,0]βˆ₯u(t+s)βˆ₯2≀eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†(0)βˆ₯2+Cbreβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ’1eΞ±reβˆ’Ξ±rβˆ’eβˆ’Ξ±(tβˆ’Ο„)+eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,≀(1+Cbr)eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ’1eΞ±r+eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds.

Then, we find

βˆ₯U(t,Ο„)(u0,Ο†)βˆ₯C([βˆ’r,0];L2(Ξ©))2=sups∈[βˆ’r,0]βˆ₯u(t+s)βˆ₯2,≀(1+Cbr)eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+eΞ±r2c|Ξ©|Ξ±βˆ’1+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,

for all (u0, Ο†) ∈ H and all t β‰₯ Ο„ + r.

Let 𝓑 be the set of all functions ρ : ℝ ⟢ (0, + ∞) such that

limtβ†’βˆ’βˆžeΞ±tρ2(t)=0.

By 𝓓 we denote the class of all families DΜ‚ = {D(t) : t ∈ ℝ} βŠ‚ π“Ÿ(C([–r, 0];L2(Ξ©))) such that D(t) βŠ‚ BC([–r, 0];L2(Ξ©))(0,ρ(t)), for some ρ ∈ 𝓑, where we denote by BC([–r, 0];L2(Ξ©))(0, ρ(t)) the closed ball in C([–r, 0];L2(Ξ©)) centered at 0 with radius ρ(t). Let

R12(t)=eΞ±r2c|Ξ©|Ξ±βˆ’1+eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds.

Thus, for all DΜ‚ ∈ 𝓓 and all t ∈ ℝ, by (3.14) there exists Ο„0 (DΜ‚, t) ≀ t such that

βˆ₯U(t,Ο„)(u0,Ο†)βˆ₯C([βˆ’r,0];L2(Ξ©))2≀R12(t),

for all Ο„ ≀ Ο„0(DΜ‚, t); i.e., B1(t) = BC([–r, 0]; L2(Ξ©))(0, R1(t)) is pullback 𝓓-absorbing for the mapping U(t, Ο„).

Concerning the second part, we observe that {j(B(t)), t ∈ ℝ} is a family of pullback 𝓓-absorbing sets for the process S. On the other hand, since

βˆ₯Ο†βˆ₯L2([βˆ’r,0];L2(Ξ©))2≀rβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2,

and

j(B(t))=(Ο†(0),Ο†):Ο†βˆˆBΒ―C([βˆ’r,0];L2(Ξ©))(0,R1(t)),

we deduce that

j(B(t))βŠ‚BΒ―L2(Ξ©))(0,R1(t))Γ—BΒ―L2([βˆ’r,0];L2(Ξ©))0,rR1(t)=B0(t),

which implies that the family {B0(t): t ∈ ℝ} is pullback 𝓓-absorbing sets for the process S. β—Ό

3.4 Existence of pullback D-absorbing set in C([βˆ’r,0];H01(Ξ©))

Proposition 2

Suppose that conditions of lemma (3) are satisfied, if there exists a sufficiently small Ξ±βˆ— such that

Ξ±<Ξ±βˆ—<min2Ξ»1βˆ’1Ξ»1,2ΞΌ1.

Then the family {B2(t) : t ∈ ℝ} given by

B2(t)=BΒ―C([βˆ’r,0];H01(Ξ©))(0,R2(t)),

where

R22(t)=2c|Ξ©|Ξ±βˆ—βˆ’1eΞ±βˆ—r+2CbΞ±βˆ’1Ξ·βˆ’1eΞ±r+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds+2eβˆ’Ξ±βˆ—(tβˆ’r)βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds,βˆ€t∈R,

is pullback 𝓓-absorbing for the mapping U(t, Ο„).

Proof

Multipying (1.1) by u+βˆ‚uβˆ‚t and integrating over Ξ©, we obtain

ddtu(t)2+12ddtβˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2=∫Ωf(u)u+βˆ‚uβˆ‚t+∫Ωb(t,ut)u+βˆ‚uβˆ‚t+∫Ωgu+βˆ‚uβˆ‚t.

Using (1.2), (1.5), (2.5) and Young inequality, one finds

2ddtu(t)2+ddtβˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2+2ΞΌ1β€²βˆ₯u(t)βˆ₯Lp(Ξ©)p+2βˆ₯βˆ‡u(t)βˆ₯2+2ΞΌ1βˆ₯u(t)βˆ₯Lp(Ξ©)p≀2c|Ξ©|+2βˆ₯b(t,ut)βˆ₯2+2βˆ₯g(t)βˆ₯2+2ddtu(t)2+2βˆ₯u(t)βˆ₯2.

By the fact that Ξ»1βˆ₯uβˆ₯2 ≀ βˆ₯βˆ‡uβˆ₯2, after simplification one has

ddtβˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2+2ΞΌ1β€²βˆ₯u(t)βˆ₯Lp(Ξ©)p+2(1βˆ’Ξ»1βˆ’1)βˆ₯βˆ‡u(t)βˆ₯2+2ΞΌ1βˆ₯u(t)βˆ₯Lp(Ξ©)p≀2c|Ξ©|+2βˆ₯b(t,ut)βˆ₯2+2βˆ₯g(t)βˆ₯2.

Since Ξ± in lemma (3) is small enough, we can choose a positive constant Ξ±βˆ— sufficiently small with Ξ± < Ξ±* < min2Ξ»1βˆ’1Ξ»1,2ΞΌ1, such that

2(1βˆ’Ξ»1βˆ’1)βˆ₯βˆ‡u(t)βˆ₯2β‰₯Ξ±βˆ—(βˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2).

So, we can write

ddtΞ³1(t)+Ξ±βˆ—Ξ³1(t)≀2c|Ξ©|+2βˆ₯b(t,ut)βˆ₯2+2βˆ₯g(t)βˆ₯2,

where

Ξ³1(t)=βˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2+2ΞΌ1β€²βˆ₯u(t)βˆ₯Lp(Ξ©)p.

Multiplying (3.16) by eΞ±βˆ—t, one has

eΞ±βˆ—tddtΞ³1(t)+Ξ±βˆ—eΞ±βˆ—tΞ³1(t)≀2c|Ξ©|eΞ±βˆ—t+2eΞ±βˆ—tβˆ₯b(t,ut)βˆ₯2+2eΞ±βˆ—tβˆ₯g(t)βˆ₯2.

On the other hand, we have

ddteΞ±βˆ—tΞ³1(t)=Ξ±βˆ—eΞ±βˆ—tΞ³1(t)+eΞ±βˆ—tddtΞ³1(t)

Then, by (3.18) and (3.19), we obtain

ddteΞ±βˆ—tΞ³1(t)β‰€Ξ±βˆ—eΞ±βˆ—tΞ³1(t)βˆ’Ξ±βˆ—eΞ±βˆ—tΞ³1(t)+2c|Ξ©|eΞ±βˆ—t+2eΞ±βˆ—tβˆ₯b(t,ut)βˆ₯2+2eΞ±βˆ—tβˆ₯g(t)βˆ₯2,≀2c|Ξ©|eΞ±βˆ—t+2eΞ±βˆ—tβˆ₯b(t,ut)βˆ₯2+2eΞ±βˆ—tβˆ₯g(t)βˆ₯2.

Integrating this last one from Ο„ to t, one gets

eΞ±βˆ—tΞ³1(t)≀eΞ±βˆ—Ο„Ξ³1(Ο„)+2c|Ξ©|βˆ«Ο„teΞ±βˆ—sds+2βˆ«Ο„teΞ±βˆ—sβˆ₯b(s,us)βˆ₯2ds+2βˆ«Ο„teΞ±βˆ—sβˆ₯g(s)βˆ₯2ds,+≀eΞ±βˆ—Ο„Ξ³1(Ο„)+2c|Ξ©|Ξ±βˆ—βˆ’1eΞ±βˆ—tβˆ’eΞ±βˆ—Ο„+2βˆ«Ο„teΞ±βˆ—sβˆ₯b(s,us)βˆ₯2ds++2βˆ«Ο„teΞ±βˆ—sβˆ₯g(s)βˆ₯2ds.

From (3.5) and (3.6), one finds

eΞ±βˆ—tΞ³1(t)≀eΞ±βˆ—Ο„Ξ³1(Ο„)+2c|Ξ©|Ξ±βˆ—βˆ’1eΞ±βˆ—tβˆ’eΞ±βˆ—Ο„+2Cbβˆ«Ο„βˆ’rteΞ±βˆ—sβˆ₯u(s)βˆ₯2ds+2βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds,≀eΞ±βˆ—Ο„Ξ³1(Ο„)+2c|Ξ©|Ξ±βˆ—βˆ’1eΞ±βˆ—tβˆ’eΞ±βˆ—Ο„+2CbeΞ±βˆ—Ο„βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2Cbβˆ«Ο„teΞ±βˆ—sβˆ₯u(s)βˆ₯2ds+2βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds.

We multiply this estimate by eβ€“Ξ±βˆ—t, we obtain

Ξ³1(t)≀eβˆ’Ξ±βˆ—(tβˆ’Ο„)Ξ³1(Ο„)+2Cbeβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’Ο„)+2Cbeβˆ’Ξ±βˆ—tβˆ«Ο„teΞ±βˆ—sβˆ₯u(s)βˆ₯2ds+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds.

On the one hand, since H01(Ξ©)βŠ‚L2(Ξ©)andH01(Ξ©)βŠ‚Lp(Ξ©),we have

Ξ³1(Ο„)=βˆ₯u(Ο„)βˆ₯2+βˆ₯βˆ‡u(Ο„)βˆ₯2+2ΞΌ1β€²βˆ₯u(Ο„)βˆ₯Lp(Ξ©)p,≀(1+Ξ»1βˆ’1)βˆ₯βˆ‡u(Ο„)βˆ₯2+2ΞΌ1β€²βˆ₯u(Ο„)βˆ₯Lp(Ξ©)p,≀(1+Ξ»1βˆ’1)βˆ₯βˆ‡u(Ο„)βˆ₯2+k1βˆ₯βˆ‡u(Ο„)βˆ₯p,≀k2(1+Ξ»1βˆ’1)βˆ₯βˆ‡u(Ο„)βˆ₯p+k1βˆ₯βˆ‡u(Ο„)βˆ₯p,≀k3βˆ₯βˆ‡u(Ο„)βˆ₯p.

So, by (3.17), (3.20) and (3.21), one finds

βˆ₯u(t)βˆ₯2+βˆ₯βˆ‡u(t)βˆ₯2+2ΞΌ1β€²βˆ₯u(t)βˆ₯Lp(Ξ©)p≀k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2Cbeβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’Ο„)+2Cbeβˆ’Ξ±βˆ—tβˆ«Ο„teΞ±βˆ—sβˆ₯u(s)βˆ₯2ds+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds.

From this last estimate and (3.8), we have

βˆ₯βˆ‡u(t)βˆ₯2≀k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2Cbeβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’Ο„)+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2Cbeβˆ’Ξ±βˆ—tβˆ«Ο„teΞ±βˆ—sβˆ₯u(s)βˆ₯2ds,≀k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2Cbeβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’Ο„)+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cb2Ξ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+2CbΞ·βˆ’1eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds,≀k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbeβˆ’Ξ±βˆ—(tβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ«Ο„βˆ’rΟ„βˆ₯u(s)βˆ₯2ds+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’Ο„)+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds.

In the fact that

βˆ₯Ο†βˆ₯L2([βˆ’r,0];L2(Ξ©))2≀rβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2,

one has

βˆ₯βˆ‡u(t)βˆ₯2≀k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbreβˆ’Ξ±βˆ—(tβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’Ο„)+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’11βˆ’eβˆ’Ξ±(tβˆ’Ο„)+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds≀k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbreβˆ’Ξ±βˆ—(tβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ—βˆ’1+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’1+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds.

If we take t β‰₯ Ο„ + r i.e. t + ΞΈ β‰₯ Ο„, it follows

βˆ₯βˆ‡u(t+ΞΈ)βˆ₯2≀k3eβˆ’Ξ±βˆ—(t+ΞΈβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(t+ΞΈβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbreβˆ’Ξ±βˆ—(t+ΞΈβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(t+ΞΈβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(t+ΞΈβˆ’Ο„)+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’11βˆ’eβˆ’Ξ±(t+ΞΈβˆ’Ο„)+2eβˆ’Ξ±βˆ—(t+ΞΈ)βˆ«βˆ’βˆžt+ΞΈeΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±(t+ΞΈ)βˆ«βˆ’βˆžt+ΞΈeΞ±sβˆ₯g(s)βˆ₯2ds.

Hence,

βˆ₯U(t,Ο„)(u0,Ο†)βˆ₯C([βˆ’r,0];H01(Ξ©))2=supθ∈[βˆ’r,0]βˆ₯βˆ‡u(t+ΞΈ)βˆ₯2,≀k3eβˆ’Ξ±βˆ—(tβˆ’rβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbreβˆ’Ξ±βˆ—(tβˆ’rβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(tβˆ’rβˆ’Ο„)+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’11βˆ’eβˆ’Ξ±(tβˆ’rβˆ’Ο„)+2eβˆ’Ξ±βˆ—(tβˆ’r)βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds.

So, we obtain

βˆ₯U(t,Ο„)(u0,Ο†)βˆ₯C([βˆ’r,0];H01(Ξ©))2≀k3eβˆ’Ξ±βˆ—(tβˆ’rβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbreβˆ’Ξ±βˆ—(tβˆ’rβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’rβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ—βˆ’1eΞ±βˆ—r+2CbΞ±βˆ’1Ξ·βˆ’1eΞ±r+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds+2eβˆ’Ξ±βˆ—(tβˆ’r)βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds.

Similarly to the Lemma 3, let 𝓑 be the set of all functions ρ : ℝ ⟢ (0, + ∞) such that

limtβ†’βˆ’βˆžeΞ±βˆ—tρ2(t)=0,

by 𝓓 we denote the class of all families D^={D(t):t∈R}βŠ‚P(C([βˆ’r,0];H01(Ξ©))) such that D(t) βŠ‚ BΒ―C([βˆ’r,0];H01(Ξ©))(0,ρ(t)), for some ρ ∈ 𝓑, where we denote by BΒ―C([βˆ’r,0];H01(Ξ©))(0,ρ(t)) the closed ball in C([βˆ’r,0];H01(Ξ©)) centered at 0 with radius ρ(t). Let

R22(t)=2c|Ξ©|Ξ±βˆ—βˆ’1eΞ±βˆ—r+2CbΞ±βˆ’1Ξ·βˆ’1eΞ±r+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’r)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds+2eβˆ’Ξ±βˆ—(tβˆ’r)βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds.

Thus, for all DΜ‚ ∈ 𝓓 and all t ∈ ℝ, by (3.23) there exists Ο„0 (DΜ‚, t) ≀ t such that

βˆ₯U(t,Ο„)(u0,Ο†)βˆ₯C([βˆ’r,0];H01(Ξ©))2≀R22(t),

for all Ο„ ≀ Ο„0(DΜ‚, t), this means that B2(t)=BΒ―C([βˆ’r,0];H01(Ξ©))(0,R2(t)) is pullback 𝓓-absorbing for the mapping U(t, Ο„).

The proof of the proposition is completed. β—Ό

3.5 Existence of pullback D-attractor

To prove the existence of pullback 𝓓-attractor, we need to prove the following lemma.

Lemma 4

Assume that conditions of lemma (3) are satisfied. Then the process {S(t, Ο„)} corresponding to (1.1) is pullback 𝓓-asymptotically compact.

Proof

Let t ∈ ℝ, DΜ‚ ∈ 𝓓, a sequences Ο„n β†’nβ†’+∞ – ∞ and (u0, n, Ο†n ∈ D(Ο„n), be fixed. We have to check that the sequence

{S(t,Ο„n)(u0,n,Ο†n)}={(u(t,Ο„n,(u0,n,Ο†n)),ut(.,Ο„n,(u0,n,Ο†n)))},

is relatively compact in H. In order to show this, we need to prove that the sequence

{U(t,Ο„n)(u0,n,Ο†n)}={ut(.,Ο„n,(u0,n,Ο†n))}

is relatively compact in C([–r, 0];L2(Ξ©)). To this end, we use the Ascoli-Arzela theorem. In other words, we check

  • the equicontinuity property for the sequence {ut(.,Ο„n,(u0,n,Ο†n))}:={utn(.)},i.e.βˆ€Ξ΅>0,βˆƒΞ΄>0 such that if |ΞΈ1βˆ’ΞΈ2|≀δ,thenβˆ₯utn(ΞΈ1)βˆ’utn(ΞΈ2)βˆ₯≀Ρ,forallΞΈ1>ΞΈ2∈[βˆ’r,0];

  • the uniform boundedness of {utn(ΞΈ)}, for all ΞΈ ∈ [–r, 0].

In order to prove (b), we consider un, u the corresponding solutions to (1.1), so by Lemma 1 we can deduce that {utn} and {ut} are uniformly bounded in C([–r, 0]; L2(Ξ©)).

To prove (a), we proceed as follows:

βˆ₯utn(ΞΈ1)βˆ’utn(ΞΈ2)βˆ₯=βˆ₯u(t+ΞΈ1)βˆ’u(t+ΞΈ2)βˆ₯,=∫t+ΞΈ2t+ΞΈ1uβ€²(s)ds,β‰€βˆ«t+ΞΈ2t+ΞΈ1βˆ₯uβ€²(s)βˆ₯ds,β‰€βˆ«t+ΞΈ2t+ΞΈ1(βˆ₯Ξ”u(s)βˆ₯+βˆ₯f(u(s))βˆ₯+βˆ₯b(s,us)βˆ₯+βˆ₯g(s)βˆ₯)ds.

Now, we estimate the terms on the right hand side of this inequality

  1. From the Holder inequality, we have

    ∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯dsβ‰€βˆ«t+ΞΈ2t+ΞΈ1ds1/2∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯2ds1/2,≀|ΞΈ1βˆ’ΞΈ2|1/2∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯2ds1/2.

    On the one hand, we have

    βˆ₯Ξ”uβˆ₯2≀λmβˆ₯βˆ‡uβˆ₯2.

    So, using this inequality in (3.22) and integrating it over [t + ΞΈ2, t + ΞΈ1], one obtain

    ∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯2ds≀λm∫t+ΞΈ2t+ΞΈ1βˆ₯βˆ‡u(s)βˆ₯2ds≀k3Ξ»m∫t+ΞΈ2t+ΞΈ1eβˆ’Ξ±βˆ—(sβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯pds+2CbΞ·βˆ’1Ξ»m∫t+ΞΈ2t+ΞΈ1eβˆ’Ξ±(sβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2ds+2CbrΞ»m∫t+ΞΈ2t+ΞΈ1eβˆ’Ξ±βˆ—(sβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(sβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2ds+2c|Ξ©|Ξ»m∫t+ΞΈ2t+ΞΈ1Ξ±βˆ—βˆ’11βˆ’eβˆ’Ξ±βˆ—(sβˆ’Ο„)+CbΞ±βˆ’1Ξ·βˆ’11βˆ’eβˆ’Ξ±(sβˆ’Ο„)ds+2Ξ»m∫t+ΞΈ2t+ΞΈ1eβˆ’Ξ±βˆ—sβˆ«βˆ’βˆžseΞ±βˆ—sβ€²βˆ₯g(sβ€²)βˆ₯2dsβ€²ds+2CbΞ·βˆ’1Ξ»m∫t+ΞΈ2t+ΞΈ1eβˆ’Ξ±sβˆ«βˆ’βˆžseΞ±sβ€²βˆ₯g(sβ€²)βˆ₯2dsβ€²ds.

    Therefore, one obtains

    ∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯2ds≀k3Ξ»mβˆ₯βˆ‡u(Ο„)βˆ₯pΞ±βˆ—βˆ’1eβˆ’Ξ±βˆ—(tβˆ’Ο„)eβˆ’Ξ±βˆ—ΞΈ2βˆ’eβˆ’Ξ±βˆ—ΞΈ1+2CbΞ·βˆ’1Ξ»mβˆ₯u(Ο„)βˆ₯2Ξ±βˆ’1eβˆ’Ξ±(tβˆ’Ο„)eβˆ’Ξ±ΞΈ2βˆ’eβˆ’Ξ±ΞΈ1+2CbrΞ»mβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2Ξ±βˆ—βˆ’1eβˆ’Ξ±βˆ—(tβˆ’Ο„)eβˆ’Ξ±βˆ—ΞΈ2βˆ’eβˆ’Ξ±βˆ—ΞΈ1+2Cb2rΞ»mβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2Ξ·βˆ’1Ξ±βˆ’1eβˆ’Ξ±(tβˆ’Ο„)eβˆ’Ξ±ΞΈ2βˆ’eβˆ’Ξ±ΞΈ1+2c|Ξ©|Ξ»mΞ±βˆ—βˆ’11βˆ’Ξ±βˆ—βˆ’1eβˆ’Ξ±βˆ—(tβˆ’Ο„)eβˆ’Ξ±βˆ—ΞΈ2βˆ’eβˆ’Ξ±βˆ—ΞΈ1+2c|Ξ©|CbΞ·βˆ’1Ξ»mΞ±βˆ’11βˆ’Ξ±βˆ’1eβˆ’Ξ±(tβˆ’Ο„)eβˆ’Ξ±ΞΈ2βˆ’eβˆ’Ξ±ΞΈ1+2Ξ»mΞ±βˆ—βˆ’1eβˆ’Ξ±βˆ—ΞΈ2βˆ’eβˆ’Ξ±βˆ—ΞΈ1eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβ€²βˆ₯g(sβ€²)βˆ₯2dsβ€²+2CbΞ·βˆ’1Ξ»mΞ±βˆ’1eβˆ’Ξ±ΞΈ2βˆ’eβˆ’Ξ±ΞΈ1eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβ€²βˆ₯g(sβ€²)βˆ₯2dsβ€²β†’0whenΞΈ1β†’ΞΈ2.

    Hence, it follows that

    ∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯ds≀|ΞΈ1βˆ’ΞΈ2|1/2∫t+ΞΈ2t+ΞΈ1βˆ₯Ξ”u(s)βˆ₯2ds1/2β†’0whenΞΈ1β†’ΞΈ2.

  2. From the Holder inequality, we have

    ∫t+ΞΈ2t+ΞΈ1βˆ₯f(u(s))βˆ₯ds≀|ΞΈ1βˆ’ΞΈ2|1/2β‹…βˆ«t+ΞΈ2t+ΞΈ1βˆ₯f(u(s))βˆ₯2ds1/2.

    Using (1.4) and the convexity of the power, one gets

    βˆ₯f(u(t))βˆ₯2=∫Ω|f(u(t,x))|2dx≀2l2βˆ₯u(t)βˆ₯2(pβˆ’1)+2l2|Ξ©|.

    Integrating this estimate over [t + ΞΈ2, t + ΞΈ1], one finds

    ∫t+ΞΈ2t+ΞΈ1βˆ₯f(u(s))βˆ₯2ds≀2l2∫t+ΞΈ2t+ΞΈ1βˆ₯u(s)βˆ₯2(pβˆ’1)ds+2l2|Ξ©|β‹…|ΞΈ1βˆ’ΞΈ2|.

    Since Ξ»1βˆ₯uβˆ₯2 ≀ βˆ₯βˆ‡uβˆ₯2, we have

    ∫t+ΞΈ2t+ΞΈ1βˆ₯f(u(s))βˆ₯2ds≀2l2Ξ»1(pβˆ’1)∫t+ΞΈ2t+ΞΈ1βˆ₯βˆ‡u(s)βˆ₯2(pβˆ’1)ds+2l2|Ξ©|β‹…|ΞΈ1βˆ’ΞΈ2|.

    From (3.22), one has

    βˆ₯βˆ‡u(t)βˆ₯2(pβˆ’1)≀{k3eβˆ’Ξ±βˆ—(tβˆ’Ο„)βˆ₯βˆ‡u(Ο„)βˆ₯p+2CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯u(Ο„)βˆ₯2+2Cbreβˆ’Ξ±βˆ—(tβˆ’Ο„)+CbΞ·βˆ’1eβˆ’Ξ±(tβˆ’Ο„)βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2+2c|Ξ©|Ξ±βˆ—βˆ’1+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’1+2eβˆ’Ξ±βˆ—tβˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds+2CbΞ·βˆ’1eβˆ’Ξ±tβˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds}(pβˆ’1).

    By applying the convexity of power three times, one gets

    βˆ₯βˆ‡u(t)βˆ₯2(pβˆ’1)≀22(pβˆ’2)k3βˆ₯βˆ‡u(Ο„)βˆ₯p+2Cbrβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2(pβˆ’1)eβˆ’(pβˆ’1)Ξ±βˆ—(tβˆ’Ο„)+22(pβˆ’2)2CbΞ·βˆ’1βˆ₯u(Ο„)βˆ₯2+2Cb2rΞ·βˆ’1βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2(pβˆ’1)eβˆ’(pβˆ’1)Ξ±(tβˆ’Ο„)+22(pβˆ’2)2c|Ξ©|Ξ±βˆ—βˆ’1+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’1(pβˆ’1)+23(pβˆ’2)2(pβˆ’1)βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds(pβˆ’1)eβˆ’(pβˆ’1)Ξ±βˆ—t+23(pβˆ’2)(2CbΞ·βˆ’1)(pβˆ’1)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds(pβˆ’1)eβˆ’(pβˆ’1)Ξ±t.

    Integrating it over [t + ΞΈ2, t + ΞΈ1], one obtains

    ∫t+ΞΈ2t+ΞΈ1βˆ₯βˆ‡u(s)βˆ₯2(pβˆ’1)ds≀22(pβˆ’2)k3βˆ₯βˆ‡u(Ο„)βˆ₯p+2Cbrβˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2(pβˆ’1)∫t+ΞΈ2t+ΞΈ1eβˆ’(pβˆ’1)Ξ±βˆ—(sβˆ’Ο„)ds+22(pβˆ’2)2CbΞ·βˆ’1βˆ₯u(Ο„)βˆ₯2+2Cb2rΞ·βˆ’1βˆ₯Ο†βˆ₯C([βˆ’r,0];L2(Ξ©))2(pβˆ’1)∫t+ΞΈ2t+ΞΈ1eβˆ’(pβˆ’1)Ξ±(sβˆ’Ο„)ds+22(pβˆ’2)2c|Ξ©|Ξ±βˆ—βˆ’1+4Cbc|Ξ©|Ξ±βˆ’1Ξ·βˆ’1(pβˆ’1)|ΞΈ1βˆ’ΞΈ2|+23(pβˆ’2)2(pβˆ’1)βˆ«βˆ’βˆžteΞ±βˆ—sβˆ₯g(s)βˆ₯2ds(pβˆ’1)∫t+ΞΈ2t+ΞΈ1eβˆ’(pβˆ’1)Ξ±βˆ—sds+23(pβˆ’2)(2CbΞ·βˆ’1)(pβˆ’1)βˆ«βˆ’βˆžteΞ±sβˆ₯g(s)βˆ₯2ds(pβˆ’1)∫t+ΞΈ2t+ΞΈ1eβˆ’(pβˆ’1)Ξ±sds.

    Therefore, we get

    ∫t+ΞΈ2t+ΞΈ1βˆ₯βˆ‡u(s)βˆ₯2(pβˆ’1)ds≀C1β€²eβˆ’(pβˆ’1)Ξ±βˆ—(tβˆ’Ο„)eβˆ’(pβˆ’1)Ξ±βˆ—ΞΈ2βˆ’eβˆ’(pβˆ’1)Ξ±βˆ—ΞΈ1+C2β€²eβˆ’(pβˆ’1)Ξ±(tβˆ’Ο„)eβˆ’(pβˆ’1)Ξ±ΞΈ2βˆ’eβˆ’(pβˆ’1)Ξ±ΞΈ1+C3β€²|ΞΈ1βˆ’ΞΈ2|+C4β€²eβˆ’(pβˆ’1)Ξ±teβˆ’(pβˆ’1)Ξ±ΞΈ2βˆ’eβˆ’(pβˆ’1)Ξ±ΞΈ1+C5β€²eβˆ’(pβˆ’1)Ξ±βˆ—teβˆ’(pβˆ’1)Ξ±βˆ—ΞΈ2βˆ’eβˆ’(pβˆ’1)Ξ±βˆ—ΞΈ1β†’0asΞΈ1β†’ΞΈ2.

    Hence by (3.27), (3.28) and this last estimate we deduce that

    ∫t+ΞΈ2t+ΞΈ1βˆ₯f(u(s))βˆ₯dsβ†’0asΞΈ1β†’ΞΈ2.

  3. Similarly, by the Holder inequality, we have

    ∫t+ΞΈ2t+ΞΈ1βˆ₯b(s,us)βˆ₯ds≀|ΞΈ1βˆ’ΞΈ2|1/2β‹…βˆ«t+ΞΈ2t+ΞΈ1βˆ₯b(s,us)βˆ₯2ds1/2.

    On the other hand, by (II), (1.7) and since Ξ»1βˆ₯uβˆ₯2 ≀ βˆ₯βˆ‡uβˆ₯2, one has

    ∫t+ΞΈ2t+ΞΈ1βˆ₯b(s,us)βˆ₯2ds≀Cb∫t+ΞΈ2βˆ’rt+ΞΈ1βˆ₯u(s)βˆ₯2dsβ‰€βˆ«t+ΞΈ2βˆ’rt+ΞΈ2βˆ₯u(s)βˆ₯2ds+∫t+ΞΈ2t+ΞΈ1βˆ₯u(s)βˆ₯2ds≀βˆ₯Ο†βˆ₯L2([βˆ’r,0];L2(Ξ©))2+Ξ»1βˆ’1∫t+ΞΈ2t+ΞΈ1βˆ₯βˆ‡u(s)βˆ₯2ds.

    By (3.26), it follows that

    ∫t+ΞΈ2t+ΞΈ1βˆ₯βˆ‡u(s)βˆ₯2dsβ†’0asΞΈ1β†’ΞΈ2.

    Then, (3.29), (3.30) and this last estimate, we deduce that

    ∫t+ΞΈ2t+ΞΈ1βˆ₯b(s,us)βˆ₯dsβ†’0whenΞΈ1β†’ΞΈ2.

  4. Finally, we use the Holder inequality to obtain

    ∫t+ΞΈ2t+ΞΈ1βˆ₯g(s)βˆ₯ds≀|ΞΈ1βˆ’ΞΈ2|1/2β‹…βˆ«t+ΞΈ2t+ΞΈ1βˆ₯g(s)βˆ₯2ds1/2.

    Since g∈Lloc2(R;L2(Ω)), one gets

    ∫t+ΞΈ2t+ΞΈ1βˆ₯g(s)βˆ₯ds≀|ΞΈ1βˆ’ΞΈ2|1/2β‹…βˆ₯gβˆ₯L2([t+ΞΈ2,t+ΞΈ1];L2(Ξ©))β†’0whenΞΈ1β†’ΞΈ2.

    Consequently, by 1), 2), 3), 4) and (3.25), we deduce that

    βˆ₯u(t+ΞΈ1)βˆ’u(t+ΞΈ2)βˆ₯β†’0whenΞΈ1β†’ΞΈ2,

    and this ensures the equicontinuity property in C([–r, 0]; L2(Ξ©)); i.e. the sequence {U(t, Ο„n)(u0, n, Ο†n)} is relatively compact in C([–r, 0]; L2(Ξ©)).

    Since we have S(t, Ο„n)(u0, n, Ο†n) = j(U(t, Ο„n)(u0, n, Ο†n)), so {S(t, Ο„n)(u0, n, Ο†n)} is relatively compact in the space L2(Ξ©) Γ— C([–r, 0]; L2(Ξ©)) and by the continuous injection of L2(Ξ©) Γ— C([–r, 0]; L2(Ξ©)) in H, we deduce that {S(t, Ο„n)(u0, n, Ο†n)} is relatively compact in H. The proof of this lemma is completed. β—Ό

By Proposition 1 and Lemma 4, we proved that the process S(t, Ο„) has a pullback 𝓓-absorbing set and it is pullback 𝓓-asymptotically compact, then by Theorem 2 we can deduce the following result.

Theorem 4

The process {S(t, Ο„)} corresponding to (1.1) has a pullback 𝓓-attractor AΜ‚ = {A(t) : t ∈ ℝ} in H. Furetheremore, AΜ‚ βŠ‚ L2(Ξ©) Γ— C([–r, 0]; L2(Ξ©)).

Communicated by Juan L.G. Guirao

References

  • [1]↑

    T. Caraballo, J. Real. Attractors for 2D-Navier-Stokes models with delays. J. Differential Equations 205, 271-297 (2004).

  • [2]↑

    T. Caraballo, J. Real, G. Lukaszewicz. Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. 64, 484-498 (2006).

  • [3]↑

    J. Garcia-Luengo, P. Marin-Rubio. Reaction-diffusion equations with non-autonomous force in H–1 and delays under measurability conditions on the driving delay term. J. Math.Anal.Appl. 417, 80-95 (2014).

  • [4]↑

    J. K. Hale, Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, AMS, Providence, RI, 25 (1988).

  • [5]↑

    J. K. Hale, S. M. Verduyn-Lunel, Introduction to Functional Differential Equations. Springer-Verlag (1993).

  • [6]↑

    J. Li, J. Huang. Uniform attractors for non-autonomous parabolic equations with delays. Nonlinear Analysis 71, 2194-2209 (2009).

  • [7]↑

    J. L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris, (1969)

  • [8]↑

    P. Marin-Rubio, J. Real. Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains. J. Nonlinear Anal. 67, 2784-2799 (2007).

  • [9]↑

    J. Wu, Theory and Applications of Partial Functional Differential Equations. Springer-Verlag, New York (1996).

[1]

T. Caraballo, J. Real. Attractors for 2D-Navier-Stokes models with delays. J. Differential Equations 205, 271-297 (2004).

[2]

T. Caraballo, J. Real, G. Lukaszewicz. Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. 64, 484-498 (2006).

[3]

J. Garcia-Luengo, P. Marin-Rubio. Reaction-diffusion equations with non-autonomous force in H–1 and delays under measurability conditions on the driving delay term. J. Math.Anal.Appl. 417, 80-95 (2014).

[4]

J. K. Hale, Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monographs, AMS, Providence, RI, 25 (1988).

[5]

J. K. Hale, S. M. Verduyn-Lunel, Introduction to Functional Differential Equations. Springer-Verlag (1993).

[6]

J. Li, J. Huang. Uniform attractors for non-autonomous parabolic equations with delays. Nonlinear Analysis 71, 2194-2209 (2009).

[7]

J. L. Lions. Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris, (1969)

[8]

P. Marin-Rubio, J. Real. Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains. J. Nonlinear Anal. 67, 2784-2799 (2007).

[9]

J. Wu, Theory and Applications of Partial Functional Differential Equations. Springer-Verlag, New York (1996).

Journal Information

Metrics

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 28 28 17
PDF Downloads 16 16 7