# Pullback Attractors for Nonautonomous 2D-Navier-Stokes Models with Variable Delays

Authors: Xiaoli Liu; Yejuan Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Using a method based on the concept of the Kuratowski measure of the noncompactness of a bounded set as well as some new estimates of the equicontinuity of the solutions, we prove the existence of a unique pullback attractor in higher regularity space for the multivalued process associated with the nonautonomous 2D-Navier-Stokes model with delays and without the uniqueness of solutions.


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## Body

## 1. Introduction

It is well known that the Navier-Stokes equations are very important in the understanding of fluids motion and turbulence. These equations have been studied extensively over the last decades (see [1–3], and the references cited therein). Recently, Caraballo and Real [4] considered global attractors for functional Navier-Stokes models with the uniqueness of solutions and for the delay, so that a wide range of hereditary characteristics (constant or variable delay, distributed delay, etc.) can be treated in a unified way. Very recently, Marín-Rubio and Real [5] used the theory of multivalued dynamical system to establish the existence of attractors for the 2D-Navier-Stokes model with delays, when the forcing term containing the delay is sublinear and only continuous.For the study of asymptotic behavior for functional partial differential equations without the uniqueness of solutions, as far as we know, not many papers have been published. However, some results in the finite dimensional context can be found in [6, 7] (see also [8–10] for some preliminary and interesting results on the structure of the attractors for ordinary differential delay systems).The pullback attractor is a possible approach to define an “attractor” for the nonautonomous dynamical systems, the long time behavior of nonautonomous dynamical systems is an interesting and challenging problem; see, for example, [11–19], and so forth. The purpose of our current paper is to study existence of pullback attractors for the following functional Navier-Stokes problem:
(1)∂u∂t-νΔu+∑i=12‍ui∂u∂xi=f(t,u(t-ρ(t)))-∇p+g(t)in(τ,+∞)×Ω,divu=0in(τ,+∞)×Ω,u=0on(τ,+∞)×Γ,u(τ+t,x)=ϕ(t,x),t∈[-h,0],x∈Ω,
where Ω⊂ℝ2 is an open bounded set with regular boundary Γ, ν>0 is the kinematic viscosity, u is the velocity field of the fluid, p is the pressure, τ∈ℝ is the initial time, g is a nondelayed external force field, f is another external force term and contains some memory effects during a fixed interval of time of length h>0, ρ is an adequate given delay function, and ϕ the initial datum on the interval [-h,0].Using the technique of measure of noncompactness, noting that all norms on finite dimensional spaces are equivalent, we apply the new method to check the pullbackω-limit compactness given in [20] and then get the existence of the pullback attractors in CV.We consider the following usual abstract spaces:(2)𝒱={u∈(C0∞(Ω))2:divu=0},
where H=  the closure of 𝒱 in (L2(Ω))2 with norm |·| and inner product (·,·), where for u,v∈(L2(Ω))2,
(3)(u,v)=∑j=12‍∫Ω‍uj(x)vj(x)dx,
where V=  the closure of 𝒱 in (H01(Ω))2 with norm ∥·∥ and associated scalar product ((·,·)), where for u,v∈(H01(Ω))2,
(4)((u,v))=∑i,j=12‍∫Ω‍∂uj∂xi∂vj∂xidx.
Note that V⊂H≡H′⊂V′, where the injections are dense and compact. We will use ∥·∥* for the norm in V′ and 〈·,·〉 for the duality pairing between V and V′.Define the trilinear formb on V×V×V by
(5)b(u,v,w)=∑i,j=12‍∫Ω‍ui∂vj∂xiwjdx,∀u,v,w∈V.Now, let us establish some assumptions for (1).We assume that the given delay function satisfiesρ∈C1(ℝ;[0,h]), and there exists a constant ρ* satisfying
(6)ρ′(t)⩽ρ*<1,∀t∈ℝ.
Furthermore, we suppose that f and g satisfy the following assumptions:(H1)
f(·,v):ℝ→H is measurable for all v∈H,(H2)
f(t,·):H→H is continuous for all t∈ℝ,(H3)
there exist positive constantsk1, k2 such that for any v∈H,
(7)|f(t,v)|2⩽k12+k22|v|2,∀t∈ℝ,(H4)
there exists a fixedδ0>0 such that for any δ∈(0,δ0), the external force g∈Lloc2(ℝ;H) satisfies
(8)∫-∞t‍|g(r)|2eδrdr<∞,∀t∈ℝ.SetA:V→V′ as 〈Au,v〉=((u,v)), B:V×V→V′ by 〈B(u,v),w〉=b(u,v,w), for all u,v,w∈V. Denote by P the corresponding orthogonal projection P:(L2(Ω))2→H. We further set A:=-PΔ. The Stokes operator A is self-adjoint and positive from D(A)=V∩(H2(Ω))2 to H. The inverse operator is compact. Excluding the pressure, the system (1) can be written in the form
(9)ddtu(t)+νAu(t)+B(u(t),u(t))=f(t,u(t-ρ(t)))+g(t)in𝒟′(τ,+∞;V′),u(τ+t)=ϕ(t),t∈[-h,0],x∈Ω.


## 2. Preliminaries

LetX be a complete metric space with metric dX(·,·), and denote by 𝒫(X) the class of nonempty subsets of X. As usual, let us denote by HX*(·,·) the Hausdorff semidistance between A and B, which are defined by
(10)HX*(A,B)=supa∈AdistX(a,B),
where distX(a,B)=infb∈BdX(a,b). Finally, denote by 𝒩(A,r) the open neighborhood {y∈X∣distX(y,A)<r} of radius r>0 of a subset A of a Banach space X.Definition 1.
A family of mappingsU(t,τ):X→𝒫(X),t⩾τ,τ∈ℝ is called to be a multivalued process (MVP in short) if it satisfies(1)
U(τ,τ)x={x},for all τ∈ℝ,x∈X;(2)
U(t,s)U(s,τ)x=U(t,τ)x,for all t⩾s⩾τ,τ∈ℝ,x∈X.Let𝒟 be a nonempty class of parameterized sets 𝒟={D(t)}t∈ℝ⊂𝒫(X).Definition 2.
Let{U(t,τ)} be a multivalued process on X. One says that {U(t,τ)} is(1)
pullback𝒟-dissipative, if there exists a family 𝒬={Q(t)}t∈ℝ∈𝒟, so that for any ℬ={B(t)}t∈ℝ∈𝒟 and each t∈ℝ, there exists a t0=t0(ℬ,t)∈ℝ+ such that
(11)U(t,t-s)B(t-s)⊂Q(t),∀s⩾t0;(2)
pullback𝒟-limit-set compact with respect to each t∈ℝ, if for any ℬ={B(t)}t∈ℝ∈𝒟 and ε>0, there exists a t1=t1(ℬ,t,ε)∈ℝ+ such that
(12)k(⋃s⩾t1‍U(t,t-s)B(t-s))⩽ε,where k is the Kuratowski measure of noncompactness.Definition 3.
A family of nonempty compact subsets𝒜={A(t)}t∈ℝ⊂𝒫(X) is called to be a pullback 𝒟-attractor for the multivalued process {U(t,τ)}, if it satisfies(1)
𝒜={A(t)}t∈ℝ is invariant; that is,
(13)U(t,τ)A(τ)=A(t),∀t⩾τ,τ∈ℝ,(2)
𝒜 is pullback 𝒟-attracting; that is, for every ℬ∈𝒟 and any fixed t∈ℝ,
(14)lims→+∞HX*(U(t,t-s)B(t-s),A(t))=0.LetX, Y be two Banach spaces, and let X*, Y* be their dual spaces, respectively. We also assume that X is a dense subspace of Y, the injection i:X↪Y is continuous, and its adjoint i*:Y*↪X* is densely injective.Theorem 4 (see [21, 22]).
LetX,Y be two Banach spaces satisfy the previous assumptions, and let {U(t,τ)} be a multivalued process on X and Y, respectively. Assume that {U(t,τ)} is upper semicontinuous or weak upper semicontinuous on Y. If for fixed t⩾τ,τ∈ℝ, U(t,τ) maps compact subsets of X into bounded subsets of 𝒫(X), then U(t,τ) is norm-to-weak upper semicontinuous on X.By slightly modifying the arguments of Theorem 3.4 and Remark 3.9 in [21], we have the following.Theorem 5.
LetX be a Banach space, and let {U(t,τ)} be a multivalued process on X. Also let U(t,τ)x be norm-to-weak upper semicontinuous in x for fixed t⩾τ, τ∈ℝ; that is, if xn→x, then for any yn∈U(t,τ)xn, there exist a subsequence ynk∈U(t,τ)xnk and a y∈U(t,τ)x such that ynk⇀y (weak convergence). Then the multivalued process {U(t,τ)} possesses a pullback 𝒟-attractor 𝒜={A(t)}t∈ℝ in X given by
(15)A(t)=ωt(Q)=⋂T∈ℝ+‍⋃s⩾T‍U(t,t-s)Q(t-s)¯⊂Q(t)
if and only if {U(t,τ)} is pullback 𝒟-dissipative and pullback 𝒟-limit-set compact with respect to each t∈ℝ, where 𝒬={Q(t)}t∈ℝ∈𝒟 is pullback 𝒟-absorbing for the multivalued process {U(t,τ)}.A multivalued process{U(t,τ)} is said to be pullback 𝒟-asymptotically upper-semicompact in X if for each fixed t∈ℝ, any ℬ={B(t)}t∈ℝ∈𝒟, any sequence {Tn} with Tn→+∞, {xn} with xn∈B(t-Tn), and any {yn} with yn∈U(t,t-Tn)xn; this last sequence {yn} is relatively compact in X.Remark 6.
Let{U(t,τ)} be a multivalued process on X. Then {U(t,τ)} is pullback 𝒟-asymptotically upper-semicompact if and only if {U(t,τ)} is pullback 𝒟-limit-set compact; see [21].LetX be a Banach space, and let h>0 be a given positive number (the delay time). Denote by CX the Banach space C([-h,0];X) endowed with the norm
(16)∥ϕ∥CX=supθ∈[-h,0]∥ϕ(θ)∥X.
Let us consider 𝒟CX a class of sets parameterized in time, 𝒟={D(t)}t∈ℝ⊂𝒫(CX). To study the pullback 𝒟-limit-set compactness of the multivalued process on CX, we need the following result from [20].Theorem 7.
Let{U(t,τ)} be a multivalued process on CX. Suppose that for each t∈ℝ, any ℬ∈𝒟CX and ε>0, there exist τ0=τ0(t,ℬ,ε)>0, a finite dimensional subspace X1 of X, and a δ>0 such that(1)
for each fixedθ∈[-h,0],
(17)∥⋃s⩾τ0‍⋃ut(·)∈U(t,t-s)B(t-s)‍Pu(t+θ)∥Xisbounded;(2)
for alls⩾τ0,  ut(·)∈U(t,t-s)B(t-s),  θ1,θ2∈[-h,0] with |θ2-θ1|<δ,
(18)∥P(u(t+θ1)-u(t+θ2))∥X<ε;(3)
for alls⩾τ0,  ut(·)∈U(t,t-s)B(t-s),
(19)supθ∈[-h,0]∥(I-P)u(t+θ)∥X<ε,where P:X→X1 is the canonical projector. Then {U(t,τ)} is pullback 𝒟-limit-set compact in CX with respect to each t∈ℝ.


## 3. Existence of an Absorbing Family of Sets inCV

By the classical Faedo-Galerkin scheme and compactness method, analogous to the arguments in [5], we have the following.Theorem 8.
Let one considerϕ∈CH, g∈Lloc2(ℝ;H), and assume that f:ℝ×H→H satisfies the hypotheses (H1)–(H3). Then, for each τ∈ℝ,(a)
there exists a weak solutionu to problem (9) satisfying
(20)u∈C([τ-h,T];H)∩L∞(τ,T;H)∩L2(τ,T;V)∀T⩾τ;(b)
ifϕ∈CV, then there exists a strong solution u to problem (9); that is,
(21)u∈C([τ-h,T];V)∩L∞(τ,T;V)∩L2(τ,T;D(A)),∀T⩾τ.GivenT>τ and u:[τ-h,T)→H, for each t∈[τ,T), we denote by ut the function defined on [-h,0] by the relation ut(s)=u(t+s), s∈[-h,0]. We also denote CH=C([-h,0];H) and CV=C([-h,0];V). Let C be the arbitrary positive constants, which may be different from line to line and even in the same line.Thanks to Theorem8, we can define a multivalued process (CV,{U(·,·)}) as
(22)U(t,τ)(ϕ)={ut(·;τ,ϕ)∣u(·)isastrongsolutionof(9)withinitialdatumϕ∈CV}.We first need a priori estimates for the solutionu of (9) in the space CH and a necessary bound on the term ∫t-1t‍eαr∥u(r)∥2dr, which will be very useful in our analysis; it relates the absorption property for the multivalued process {U(t,τ)} on CV.Lemma 9.
In addition to the assumptions(H1)–(H4), assume that
(23)k22<(νλ12)2(1-ρ*)
holds true. Then
(24)∥ut∥CH2⩽(1+2k22eαhνλ1(1-ρ*)α)eα(τ-t+h)∥ϕ∥CH2+2k12νλ1α+e-α(t-h)2ε2∫-∞t‍eαs|g(s)|2ds,∀t⩾τ+h,(25)ν∫t-1t‍eαr∥u(r)∥2dr⩽Ceατ∥ϕ∥CH2+Ceαt+C∫-∞t‍eαr|g(r)|2dr,∀t⩾τ+h+1,
provided that α>0 is small enough.Proof.
By the energy inequality and the Poincaré inequality, we have(26)ddt|u(t)|2+νλ1|u(t)|2+ν∥u(t)∥2⩽2(f(t,u(t-ρ(t))),u(t))+2(g(t),u(t)).
We fixed two positive parameters ε1 and ε2 to be chosen later on. Then by (H3) and Young’s inequality, we can deduce that
(27)|(f(t,u(t-ρ(t))),u(t))|⩽|f(t,u(t-ρ(t)))||u(t)|⩽ε1|u(t)|2+k12+k22|u(t-ρ(t))|24ε1,|g(t),u(t)|⩽ε2|u(t)|2+14ε2|g(t)|2.
Therefore,
(28)ddt|u(t)|2+ν∥u(t)∥2⩽(2ε1+2ε2-νλ1)|u(t)|2+k12+k22|u(t-ρ(t))|22ε1+|g(t)|22ε2.
Let α>0 to be determined later on. Then it follows that
(29)ddt(eαt|u(t)|2)=αeαt|u(t)|2+eαtddt|u(t)|2⩽(α+2ε1+2ε2-νλ1)eαt|u(t)|2+eαtk122ε1+eαtk22|u(t-ρ(t))|22ε1+eαt|g(t)|22ε2.
Integrating betweenτ and t(⩾τ), we have
(30)eαt|u(t)|2⩽eατ|u(τ)|2+(α+2ε1+2ε2-νλ1)∫τt‍eαs|u(s)|2ds+k122ε1∫τt‍eαsds+k222ε1∫τt‍eαs|u(s-ρ(s))|2ds+12ε2∫τt‍eαs|g(s)|2ds.
Letr=s-ρ(s); note that ρ(s)∈[0,h] and 1/(1-ρ′(s))⩽1/(1-ρ*) for all s∈ℝ. Hence,
(31)k222ε1∫τt‍eαs|u(s-ρ(s))|2ds⩽k222ε111-ρ*∫τ-ht‍eα(r+h)|u(r)|2dr⩽k22eαh2ε1(1-ρ*)×(∫τ-hτ‍eαr|u(r)|2dr+∫τt‍eαr|u(r)|2dr)⩽k22eα(h+τ)∥ϕ∥CH22ε1(1-ρ*)α+k22eαh2ε1(1-ρ*)∫τt‍eαr|u(r)|2dr.
Combining (30) and (31) together, we get
(32)eαt|u(t)|2⩽(1+k22eαh2ε1(1-ρ*)α)eατ∥ϕ∥CH2+k12eαt2ε1α+12ε2∫-∞t‍eαs|g(s)|2ds+(α+2ε1+2ε2-νλ1+k22eαh2ε1(1-ρ*))×∫τt‍eαs|u(s)|2ds.
Let ε1=νλ1/4 and using (23), so we can choose positive constants α and ε2 small enough such that α+2ε1+2ε2-νλ1+(k22eαh/2ε1(1-ρ*))<0 and α<δ0 (where δ0 is given in the assumption (H4)). Then, it follows that
(33)eαt|u(t)|2⩽(1+2k22eαhνλ1(1-ρ*)α)eατ∥ϕ∥CH2+2k12eαtνλ1α+12ε2∫-∞t‍eαs|g(s)|2ds.
Setting now t+θ instead of t (where θ∈[-h,0]), multiplying by e-α(t+θ), it holds
(34)|u(t+θ)|2⩽(1+2k22eαhνλ1(1-ρ*)α)eα(τ-t-θ)∥ϕ∥CH2+2k12νλ1α+e-α(t+θ)2ε2∫-∞t+θ‍eαs|g(s)|2ds⩽(1+2k22eαhνλ1(1-ρ*)α)eα(τ-t+h)∥ϕ∥CH2+2k12νλ1α+e-α(t-h)2ε2∫-∞t‍eαs|g(s)|2ds.
Note that ∥ut∥CH2=supθ∈[-h,0]|u(t+θ)|2, thus the conclusion (24) follows immediately from (34).
Finally, we will obtain the bound on the termν∫t-1t‍eαr∥u(r)∥2dr. It follows from (28) that
(35)νeαt∥u(t)∥2⩽(2ε1+2ε2-νλ1)eαt|u(t)|2+k12eαt2ε1+k22eαt|u(t-ρ(t))|22ε1+eαt|g(t)|22ε2+αeαt|u(t)|2-ddt(eαt|u(t)|2).
Integrating from t-1 to t, we have
(36)ν∫t-1t‍eαr∥u(r)∥2dr⩽(α+2ε1+2ε2-νλ1)∫t-1t‍eαr|u(r)|2dr+k122ε1∫t-1t‍eαrdr+k222ε1∫t-1t‍eαr|u(r-ρ(r))|2dr+12ε2∫t-1t‍eαr|g(r)|2dr+eα(t-1)|u(t-1)|2.
Similar to the arguments of (31), we can deduce that
(37)k222ε1∫t-1t‍eαr|u(r-ρ(r))|2dr⩽k222ε111-ρ*∫t-1-ht‍eα(r+h)|u(r)|2dr⩽k22eαh2ε1(1-ρ*)×(∫t-1-ht-1‍eαr|u(r)|2dr+∫t-1t‍eαr|u(r)|2dr)⩽k22eα(h+t-1)∥ut-1∥CH22ε1(1-ρ*)α+k22eαh2ε1(1-ρ*)∫t-1t‍eαr|u(r)|2dr.
Recall that ε1=νλ1/4 and α+2ε1+2ε2-νλ1+(k22eαh/2ε1(1-ρ*))<0. By (24) and (36)-(37), we have (25) as desired, and thus the proof of this lemma is completed.By slightly modifying the proof of Lemma  1.1 in [23], we have the following result.Lemma 10.
Lett∈ℝ  be given arbitrarily. Let g, h, and y be three positive locally integrable functions on (-∞,t] such that y′ is locally integrable on (-∞,t], which satisfy that
(38)dyds⩽gy+hfors⩽t,∫t-1t‍g(s)ds⩽a1,∫t-1t‍h(s)ds⩽a2,∫t-1t‍y(s)ds⩽a3,
where a1, a2, and a3 are positive constants. Then
(39)y(t)⩽exp(a1)(a3+a2).Now we state and prove the main result in this section.Theorem 11.
Suppose in addition to the hypotheses in Lemma9, assume that
(40)limt→-∞∫-∞t‍e-γ(t-r)|g(r)|2dr<∞∀γ>0
holds true. Then the multivalued process {U(t,τ)} on CV is pullback 𝒟-dissipative.Proof.
We take the inner product of (9) with Au(t), we obtain
(41)12ddt∥u(t)∥2+ν|Au(t)|2+(B(u(t),u(t)),Au(t))=(f(t,u(t-ρ(t))),Au(t))+(g(t),Au(t)).
Now we evaluate the terms, using (H3) and Young’s inequality, and we arrive to
(42)|(f(t,u(t-ρ(t))),Au(t))|+|(g(t),Au(t))|⩽ν2|Au(t)|2+|f(t,u(t-ρ(t)))|2ν+|g(t)|2ν⩽ν2|Au(t)|2+k12+k22∥ut∥CH2ν+|g(t)|2ν.
Next,
(43)|(B(u(t),u(t)),Au(t))|⩽C1|u(t)|1/2∥u(t)∥|Au(t)|3/2⩽ν4|Au(t)|2+C2ν3|u(t)|2∥u(t)∥4.
Thanks to (41)–(43) and the fact that ∥φ∥2⩽λ1-1|Aφ|2 for φ∈D(A), we can deduce that
(44)ddt∥u(t)∥2+νλ12∥u(t)∥2⩽2k12+2k22∥ut∥CH2ν+2|g(t)|2ν+2C2ν3|u(t)|2∥u(t)∥4,
and consequently,
(45)ddt(eαt∥u(t)∥2)+(νλ12-α)eαt∥u(t)∥2⩽2k12+2k22∥ut∥CH2νeαt+2|g(t)|2νeαt+2C2eαtν3|u(t)|2∥u(t)∥4.
Since ε1=νλ1/4 and α+2ε1+2ε2-νλ1+(k22eαh/2ε1(1-ρ*))<0, it is easy to see that (νλ1/2)-α>0. Then
(46)ddt(eαt∥u(t)∥2)⩽2k12+2k22∥ut∥CH2νeαt+2|g(t)|2νeαt+2C2ν3|u(t)|2∥u(t)∥2∥u(t)∥2eαt.
Lett∈ℝ  be given arbitrarily and taking τ such that t⩾τ+h+1. In order to apply Lemma 10, in view of (24), now we firstly obtain
(47)∫t-1t‍(2k12+2k22∥ur∥CH2νeαr+2|g(r)|2νeαr)dr⩽∫t-1t‍2k12+2k22∥ur∥CH2νeαrdr+∫t-1t‍2|g(r)|2νeαrdr⩽Ceαt+Ceατ∥ϕ∥CH2+C∫-∞t‍eαr|g(r)|2dr.
Then, it follows from (24) and (25) that
(48)2C2ν3∫t-1t‍|u(r)|2∥u(r)∥2dr⩽2C2ν3∫t-1t‍∥ur∥CH2∥u(r)∥2dr⩽2C2ν3∫t-1t‍(1+2k22eαhνλ1(1-ρ*)α)eα(τ-r+h)×∥ϕ∥CH2∥u(r)∥2dr+4C2k12ν4λ1α∫t-1t‍∥u(r)∥2dr+2C2ν3∫t-1t‍(e-α(r-h)2ε2∥u(r)∥2∫-∞r‍eαs|g(s)|2ds)dr⩽Ce2ατe-2αt∥ϕ∥CH2+C+Ce-αt∫-∞t‍eαr|g(r)|2dr+Ceατe-αt∥ϕ∥CH2+Ceατe-2αt∥ϕ∥CH2∫-∞t‍eαr|g(r)|2dr+Ce-2αt(∫-∞t‍eαr|g(r)|2dr)2.
Combining (25) and (47)-(48) together, by Lemma 10, we can conclude that
(49)∥u(t)∥2⩽(a3+a2)ea1∀t⩾τ+h+1,
where
(50)a3=Ceατ∥ϕ∥CH2+Ceαt+C∫-∞t‍eαr|g(r)|2dr,a2=Ceαt+Ceατ∥ϕ∥CH2+C∫-∞t‍eαr|g(r)|2dr,a1=Ce2ατe-2αt∥ϕ∥CH2+C+Ce-αt∫-∞t‍eαr|g(r)|2dr+Ceατe-αt∥ϕ∥CH2+Ceατe-2αt∥ϕ∥CH2∫-∞t‍eαr|g(r)|2dr+Ce-2αt(∫-∞t‍eαr|g(r)|2dr)2.
Therefore, if we take τ such that t⩾τ+1+2h, then similar to the above mentioned, we get
(51)∥ut∥CV2=supθ∈[-h,0]∥u(t+θ)∥2⩽(a3+a2)ea1.
We denote byℛ the set of all functions r:ℝ→(0,+∞) such that
(52)limt→-∞r2(t)=0,
and denote by 𝒟CV the class of all families 𝒟={D(t)}t∈ℝ⊂𝒫(CV) such that D(t)⊂𝒩¯(0,r𝒟(t)), for some r𝒟∈ℛ, where 𝒫(CV) denotes the family of all nonempty subsets of CV and 𝒩¯(0,r𝒟(t)) denotes the closed ball in CV centered at zero with radius r𝒟(t).
Denote byR(t) the nonnegative number given for each t∈ℝ by
(53)(R(t))2=(Ceαt+C∫-∞t‍eαr|g(r)|2dr)×exp((∫-∞t‍eαr|g(r)|2dr)2Ce-αt∫-∞t‍eαr|g(r)|2dr+Ce-2αt(∫-∞t‍eαr|g(r)|2dr)2),
and consider the family of closed balls 𝒬={Q(t)}t∈ℝ in CV defined by
(54)Q(t)={ψ∈CV:∥ψ∥CV⩽R(t)}.
It is straightforward to check that 𝒬∈𝒟CV, and moreover, by (51) and (52), the family of 𝒬 is pullback 𝒟-absorbing for the multivalued process {U(t,τ)} on CV.
The proof of Theorem11 is completed.


## 4. Existence of the Pullback Attractors inCV

Theorem 12.
Suppose in addition to the hypotheses in Theorem11 that g∈C(ℝ;H). Then there exists a unique pullback 𝒟-attractor {ACV(t)}t∈R for the multivalued process {U(t,τ)} in CV.Proof.
SinceA-1 is a continuous compact operator in H, by the classical spectral theory, there exist a sequence {λj}j=1∞,
(55)0<λ1⩽λ2⩽⋯⩽λj⩽⋯,λj⟶+∞,asj⟶+∞,
and a family of elements {wj}j=1∞ of D(A) which are orthonormal in H such that
(56)Awj=λjwj∀j∈ℕ.
Let Vm=span{w1,…,wm} in V and Pm:V→Vm be an orthogonal projector.
Letu=u1+u2, where u1=Pmu and u2=(I-Pm)u. We decompose (9) as follows:
(57)∂u2(t)∂t+νAu2(t)+B(u(t),u(t))-PmB(u1(t),u1(t))=f(t,u(t-ρ(t)))-Pmf(t,u1(t-ρ(t)))+(I-Pm)g(t),u2(τ+t)=(I-Pm)ϕ(t),t∈[-h,0],(58)∂u1(t)∂t+νAu1(t)+PmB(u1(t),u1(t))=Pmf(t,u1(t-ρ(t)))+Pmg(t),u1(τ+t)=Pmϕ(t),t∈[-h,0].
We divide the proof into three steps.
(1) For every fixedt∈ℝ, any ℬ={B(t)}t∈ℝ∈𝒟CV and ε>0, we observe that for any T⩾t-s with s⩾0,
(59)U(T,t-s)(ϕ)={ϕuT(·;t-s,ϕ)∣u(·)isastrongsolutionof(9)withϕ∈B(t-s)uT}.
Taking the inner product in H of (57) with Au2=A(I-Pm)u, we get
(60)12ddT∥u2(T)∥2+ν|Au2(T)|2⩽|(f(T,u(T-ρ(T))),Au2(T))|+|(Pmf(T,u1(T-ρ(T))),Au2(T))|+|(B(u(T),u(T)),Au2(T))|+|(PmB(u1(T),u1(T)),Au2(T))|+|((I-Pm)g(T),Au2(T))|.
By (H3) and Young’s inequality, we have
(61)|(f(T,u(T-ρ(T))),Au2(T))|+|(g(T),Au2(T))|⩽ν8|Au2(T)|2+C+C∥uT∥CH2+C|g(T)|2.
To estimate (B(u(T),u(T)),Au2(T)), we recall some inequalities [19]:
(62)|φ|(L∞(Ω))2⩽C3∥φ∥(1+log|Aφ|2λ1∥φ∥2)1/2∀φ∈D(A),
and thus
(63)|B(u,v)|⩽C4|(u·∇)v|⩽C4|u|L∞(Ω)∥v∥⩽C4C3∥u∥∥v∥(1+log|Au|2λ1∥u∥2)1/2.
Note that |Au1|2⩽λm∥u1∥2, and set L=1+log(λm+1/λ1). Then by Young’s inequality, we can deduce that
(64)|(B(u(T),u(T)),Au2(T))|⩽|(B(u2(T),u1(T)+u2(T)),Au2(T))|+|(B(u1(T),u1(T)+u2(T)),Au2(T))|⩽C1|u2(T)|1/2|Au2(T)|3/2×(∥u1(T)∥+∥u2(T)∥)+C3C4L1/2∥u1(T)∥|Au2(T)|×(∥u1(T)∥+∥u2(T)∥)⩽ν8|Au2(T)|2+C|u(T)|2∥u(T)∥4+C∥u(T)∥4.
By (60)–(64) and Poincaré inequality, we obtain
(65)ddT∥u2(T)∥2+νλm+1∥u2(T)∥2⩽C+C∥uT∥CH2+C|g(T)|2+C|u(T)|2∥u(T)∥4+C∥u(T)∥4.
Applying the Gronwall’s lemma in the interval [t-s,t+θ], it yields(66)∥u2(t+θ)∥2⩽∥u2(t-s)∥2e-νλm+1(θ+s)+C∫t-st+θ‍e-νλm+1(t+θ-r)×(1+∥ur∥CH2+|g(r)|2+|u(r)|2∥u(r)∥4+∥u(r)∥4)dr.Let ε>0 be given arbitrarily. Note that g∈C(ℝ;H), then we can take m+1 large enough such that for any fixed η>0,
(67)C∫t-h-ηt+θ‍e-νλm+1(t+θ-r)|g(r)|2dr⩽Cνλm+1<ε4,(68)supθ∈[-h,0]C∫-∞t-h-η‍e-νλm+1(t+θ-r)|g(r)|2dr⩽C∫-∞t-h-η‍e-νλm+1(t-h-r)|g(r)|2dr⩽Ce-νλm+1(t-h)×(∫t-h-η-1t-h-η‍eνλm+1r|g(r)|2dr+∫t-h-η-2t-h-η-1‍eνλm+1r|g(r)|2dr+⋯)⩽Ce-νλm+1(t-h)×(e(νλm+1-α)(t-h-η)+e(νλm+1-α)(t-h-η-1)+⋯)×∫-∞t‍eαr|g(r)|2dr⩽Ce-νλm+1ηe-α(t-h-η)1-e-(νλm+1-α)∫-∞t‍eαr|g(r)|2dr<ε4.
Combining (67) and (68) together, we can get for m+1 large enough,
(69)supθ∈[-h,0]C∫-∞t+θ‍e-νλm+1(t+θ-r)|g(r)|2dr<ε2.
On the other hand, thanks to Lemma 9 and Theorem 11, we can deduce that when m+1 and s are large enough,
(70)supθ∈[-h,0]∥u2(t-s)∥2e-νλm+1(θ+s)⩽∥u2(t-s)∥2e-νλm+1(s-h)<ε4,supθ∈[-h,0]C∫t-st+θ‍e-νλm+1(t+θ-r)×(C+∥ur∥CH2+|u(r)|2∥u(r)∥4+∥u(r)∥4)dr<ε4.
Thanks to (69) and (70), it follows from (66) that when m+1 and s are large enough,
(71)∥u2t∥CV2=supθ∈[-h,0]∥u2(t+θ)∥2⩽supθ∈[-h,0]∥u2(t-s)∥2e-νλm+1(θ+s)+supθ∈[-h,0]C∫t-st+θ‍e-νλm+1(t+θ-r)×(C+∥ur∥CH2+|g(r)|2+|u(r)|2∥u(r)∥4+∥u(r)∥4∥ur∥CH2)dr<ε.
(2) Now we consider the ordinary functional differential system (58) and check the condition (2) in Theorem 7. Note that |Au1|2⩽λm∥u1∥2⩽λm2|u1|2. Without generality, we assume that θ1,θ2∈[-h,0] with 0<θ1-θ2<1. Hence
(72)∥u1(t+θ1)-u1(t+θ2)∥⩽λm|u1(t+θ1)-u1(t+θ2)|⩽λm∫t+θ2t+θ1‍|du1(T)dT|dT⩽λm∫t+θ2t+θ1‍(ν|Au1(T)|+|B(u1(T),u1(T))|+|f(T,u1(T-ρ(T)))|+|Pmg(T)|)dT.
Notice that
(73)|B(u1,u1)|⩽C|Au1|∥u1∥⩽Cλm∥u1∥2⩽Cλm3/2|u1|2.
Then, it follows from (H3), (H4), and (24) that
(74)∫t+θ2t+θ1‍(|Au1(T)|+|B(u1(T),u1(T))|)dT⩽C∫t+θ2t+θ1‍|u1(T)|dT+C∫t+θ2t+θ1‍|u1(T)|2dT⩽C∫t+θ2t+θ1‍|u1(T)|2dT+C(θ1-θ2)⩽C(e-αθ2-e-αθ1)+C(θ1-θ2)+C(e-αθ2-e-αθ1)e-αt∫-∞t‍eαr|g(r)|2dr,∫t+θ2t+θ1‍|f(T,u1(T-ρ(T)))|dT⩽∫t+θ2t+θ1‍(|f(T,u1(T-ρ(T)))|2+C)dT⩽∫t+θ2t+θ1‍(k22∥u1T∥CH2+C)dT⩽C(e-αθ2-e-αθ1)+C(θ1-θ2)+C(e-αθ2-e-αθ1)e-αt∫-∞t‍eαr|g(r)|2dr.
Since g∈C(ℝ;H) and t is fixed,
(75)∫t+θ2t+θ1‍|Pmg(T)|dT⩽C(θ1-θ2).
Equations (74)–(75) imply that the condition (2) in Theorem 7 is proved.
(3) Invoking Theorem7, in view of the previous arguments and Theorem 11, we can see that the multivalued process {U(t,τ)} is pullback 𝒟-limit-set compact and pullback 𝒟-dissipative in CV.
In order to get the existence of pullback𝒟-attractors, by the proof of Theorem 3.2 in [21], now we only need to show the negative invariance of {ACV(t)}t∈ℝ, where
(76)ACV(t)=ωt(𝒬)=⋂T∈ℝ+‍⋃s⩾T‍U(t,t-s)Q(t-s)¯,∀t∈ℝ,
and 𝒬={Q(t)}t∈ℝ∈𝒟CV is a pullback 𝒟-absorbing set of {U(t,τ)} in CV.
Lety∈ACV(t). Then there exist sequences sn∈ℝ+,sn→+∞(n→∞), xn∈Q(t-sn), and yn∈U(t,t-sn)xn such that
(77)yn⟶yinCVasn⟶∞.
On the other hand, for n sufficiently large,
(78)yn∈U(t,t-sn)xn=U(t,τ)U(τ,t-sn)xn.
Then by the pullback 𝒟-limit-set compactness of the multivalued process {U(t,τ)}, there is a subsequence of x~n∈U(τ,t-sn)xn=U(τ,τ-(τ+sn-t))xn, which we still relabel as x~n such that yn∈U(t,τ)x~n and
(79)x~n⟶xinCVasn⟶∞.
Clearly, x∈ACV(τ).
We observe thatyn is bounded in CV for n sufficiently large. Then by slightly modifying the proof of the existence of solutions (see [16] for details), in view of Theorem 2.11 in [21], we can see that
(80)yn(·)⇀u(·+t,τ,x)inL2([-h,0];V).
This together with (77)–(79), we can deduce that y∈U(t,τ)x⊂U(t,τ)ACV(τ), and thus the proof of Theorem 12 is finished.



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*Source: 425031-2013-04-17.xml*