# Strong Pullback Attractors for Nonautonomous Suspension Bridge Equations

Authors: Wenchao Ju; Xuan Wang
Journal: ISRN Applied Mathematics
 (2014)

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## Abstract

We prove the existence of a pullback𝒟-attractor in DA×V for the nonautonomous suspension bridge equations.


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## Body

## 1. Introduction

In this paper, we consider the following nonautonomous suspension bridge equation:(1)utt+Δu+μut+ku++g(u)=f(x,t),inΩ×ℝτ,u(x,t)=Δu(x,t)=0,on⁡∂Ω×ℝτ,u(x,τ)=u1(x),ut(x,τ)=u2(x),x∈Ω,
where Ω is a bounded domain of ℝ2 with a smooth boundary ∂Ω, u(x,t) is an unknown function, which could represent the deflection of the road bed in the vertical plane, ku+ represents the restoring force, k denotes the spring constant, μut represents the viscous damping, and μ is a given positive constant.Suspension bridge equations have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and Mckenna in 1990. There are many results for the problem (1) (cf. [1–8]), for instance, the existence, multiplicity, and properties of the travelling wave solutions, and so forth. About the long-time behavior of suspension bridge equations, for the autonomous case, in [9, 10], the authors have discussed long-time behavior of the solutions of the problem on ℝ2 and obtained the existence of global attractors in the space H02(Ω)×L2(Ω) and D(A)×H02(Ω).Caraballo et al. advanced the concept of the pullback𝒟-attractor in [11], and the existence of the pullback attractors was proved under the assumptions of asymptotic compactness and existence of a family of absorbing sets. Recently, Park and Kang [12] studied the pullback 𝒟-attractor for suspension bridge equations in the weak space H02(Ω)×L2(Ω). Motivated by the ideas of [11, 13], we study the existence of a strong pullback 𝒟-attractor for the nonautonomous suspension bridge equations in the strong topological space D(A)×H02(Ω).The nonlinear functionsg∈C2(ℝ,ℝ) satisfy the following assumptions:
(2)lim⁡⁡inf⁡|u|→∞G(u)u2⩾0,G(u)=∫0u‍g(ζ)dζ,(3)|g(s)|⩽C(1+|s|p),∀p⩾1,(4)lim⁡⁡inf⁡|u|→∞ug(u)-C0G(u)u2⩾0,
where constant C,C0>0.With the usual notation, we introduce the spacesH=L2(Ω),V=H02(Ω), D(A)=u∈{H02(Ω)∣Au∈L2(Ω)}, where A=Δ2. We equip these spaces with inner product and norm  〈·,·〉,|·|, 〈·,·〉1,∥·∥1, and 〈·,·〉2,∥·∥2, respectively:
(5)〈u,v〉=∫Ω‍u(x)v(x)dx,|u|2=∫Ω‍|u(x)|2dx,∀u,v∈H;〈u,v〉1=∫Ω‍Δu(x)Δv(x)dx,∥u∥12=∫Ω‍|Δu(x)|2dx,∀u,v∈V;〈u,v〉2=∫Ω‍Δ2u(x)Δ2v(x)dx,∥u∥22=∫Ω‍|Δ2u(x)|2dx,∀u,v∈D(A).
Obviously, we have
(6)D(A)⊂V⊂H=H*⊂V*,
where H*,V* is dual space of H,V, respectively; the injections are continuous and each space is dense in the following one.Choosingλ=min⁡{λ1,λ12}, by the Poincaré inequality, we have
(7)λ|u|2⩽∥u∥12,∀u∈V;λ∥u∥12⩽∥u∥22,∀u∈D(A).We introduce the Hilbert spaces(8)ℰ0=V×H,ℰ1=D(A)×V
and endow this space with norm
(9)∥z∥ℰ0=∥(u,ut)∥ℰ0={∥u∥12+|ut|2}1/2,∥z∥ℰ1=∥(u,ut)∥ℰ1={∥u∥22+∥ut∥12}1/2.This paper is organized as follows. At first, in Section2, we recall some preliminaries and results concerning the pullback attractor. Then, in Section 3, we prove our main result about the existence of pullback 𝒟-attractor for the nonautonomous dynamical system generated by the solution of (1).


## 2. Notation and Preliminaries

Let(E,d) be a complete metric space, (Q,ρ) be a metric space which will be called the parameter space. We define a nonautonomous dynamical system by a cocycle mapping ϕ:ℝ+×Q×E which is driven by an autonomous dynamical system θ acting on a parameter space Q. Specifically, θ={θt}t∈ℝ is a dynamical system on Q; that is, is a group of homeomorphisms under composition on Q with the properties that(i)
θ0(q)=q for all q∈Q;(ii)
θt+τ(q)=θt(θτ(q)) for all q∈Q,t,τ∈ℝ;(iii)
the mapping(t,q)→θt(q) is continuous.Definition 1.
A mappingϕ is said to be a cocycle on E with respect to group θ, if (i)
ϕ(0,q,x)=x for all (q,x)∈Q×E;(ii)
ϕ(t+s,q,x)=ϕ(s,θt(q),ϕ(t,q,x)) for all s,t∈ℝ+ and all (q,x)∈Q×E.
Let𝒫(E) denote the family of all nonempty subsets of E, let ℬ(E) be the set of all bounded subsets of E, and let 𝒦 be the class of all families D^={Dq}q∈Q⊂𝒫(E). We consider a nonempty subclass 𝒟∈𝒦.Definition 2 (see [11]).
Let(θ,ϕ) be a nonautonomous dynamical system on Q×E. (θ,ϕ) is said to be pullback 𝒟-asymptotically compact if, for any q∈Q, any D^∈𝒟, and any sequences tn→+∞, xn∈Dθ-tn(q), the sequence ϕ(tn,θ-tn(q),xn) possesses a convergent subsequence.Definition 3 (see [11]).
A familyB^={Bq}q∈Q∈𝒦 is said to be pullback 𝒟-absorbing if, for each q∈Q and D^∈𝒟, there exists t0(q,D^)⩾0 such that
(10)ϕ(t,θ-t(q),Dθ-t(q))⊂Bq∀t⩾t0(q,D^).Definition 4 (see [11]).
A familyC^={Cq}q∈Q∈𝒦 is said to be pullback 𝒟-attracting if
(11)lim⁡t→∞dist⁡(ϕ(t,θ-t(q),Dθ-t(q)),Cq)=0vvvvvvvvvvvvvvwv∀q∈Q,D^∈𝒟,
where dist⁡(X,Y)=sup⁡x∈Xinf⁡y∈Yd(x,y) is the Hausdorff semidistance between X and Y.Definition 5 (see [11]).
A familyA^={Aq}q∈Q∈𝒦 is called a global pullback 𝒟-attractor if it satisfies:(i)
Aq is compact for any q∈Q;(ii)
A^ is pullback 𝒟-attracting;(iii)
A^ is invariant; that is, ϕ(t,q,Aq)=Aθt(q) for all (t,q)∈ℝ+×Q.Definition 6 (see [14]).
Let(θ,ϕ) be a nonautonomous dynamical system on Q×E. (θ,ϕ) is said to be satisfying pullback 𝒟-Condition (C) if, for any q∈Q, D^∈𝒟, and any ϵ>0, there exist a t0=t0(q,D^,ϵ)⩾0 and a finite dimensional subspace E1 of E such that (i)
P(∪t⩾t0ϕ(t,θ-t(q),Dθ-t(q))) is bounded;(ii)
∥(I-P)(∪t⩾t0ϕ(t,θ-t(q),Dθ-t(q)))∥E⩽ϵ, where P:E→E1 is a bounded projector.Lemma 7 (see [14]).
Let(θ,ϕ) be a nonautonomous dynamical system on Q×E. (θ,ϕ) possesses a global pullback 𝒟-attractor A^={Aq}q∈Q satisfying Aq=Λ(D^,q) if it (i)
has a pullback𝒟-absorbing set B^={Bq}q∈Q∈𝒟;(ii)
satisfies pullback𝒟-Condition (C).Theorem 8 (see [12]).
Suppose thatk>0 and the assumption (2)–(4) hold. f(x,t)∈Lloc⁡2(ℝ,H) satisfies (17). Then there exists a unique global pullback 𝒟δ,ℰ0-attractor in ℰ0 for the nonautonomous dynamical system (θ,ϕ) defined by (15).We need the following lemmas in order to prove the main result.Lemma 9 (see [14]).
LetH be an infinite dimensional Hilbert space and let the family {ωi}i∈ℕ be an orthonormal of H. Suppose f(x,t)∈Lloc⁡2(ℝ;H) and, for any t∈ℝ, ∫-∞t‍eσs|f(x,s)|2ds<∞ for some σ⩾0. Then
(12)lim⁡n→∞∫-∞t‍eσs|(I-Pn)f(x,s)|H2ds=0,∀t∈ℝ,
where Pn:H→span⁡{ω1,ω2,…,ωn} is the orthogonal projector.Lemma 10 (see [10]).
Suppose thatg∈C2(ℝ,ℝ) and satisfying (3). Then g:D(A)→V are continuous compact.Lemma 11 (see [10]).
Leth(u,ut)=g′(u)ut, g∈C2(ℝ,ℝ) satisfying (3), and g(0)=0. Then h:D(A)×V→H is continuous compact.


## 3. Pullback𝒟-Attractors for Nonautonomous Suspension Bridge Equations

First, we give the following result.Theorem 12.
Suppose thatk>0, satisfying (2)–(4), if f∈Lloc⁡2(ℝτ;H) and (u1,u2)∈ℰ0. Then system (1) has a unique solution:
(13)(u,ut)∈C(ℝτ;ℰ0),
where ℝτ=[τ,∞). If, in addition, f′∈L2(τ,T;L2(Ω)) and (u1,u2)∈ℰ1, then
(14)(u,ut)∈C(ℝτ;ℰ1).
Moreover, the mapping (u1,u2)→(u(t),ut(t)) is continuous in ℰ1.We can construct the nonautonomous dynamical system generated by problem (1) in ℰ=ℰ0(or ℰ1). We consider Q=ℝ, θt(τ)=τ+t and define
(15)ϕ(t,τ,y0)=y(t+τ;τ,y0)=(u(t+τ),ut(t+τ)),ut(t+τ)0jvweτ∈ℝ,t⩾0,y0∈ℰ.
The uniqueness of solution to problem (1) implies that
(16)ϕ(t+s,τ,y0)=ϕ(t,s+τ,ϕ(s,τ,y0)),τ∈ℝ,s⩾0,y0∈ℰ.
Also, for all τ∈ℝ, t⩾0 the mapping ϕ(t,τ,·):ℰ→ℰ defined by (15) is continuous. Consequently, the mapping ϕ defined by (15) is a continuous cocycle on ℰ.Now, we assume thatf,f′∈Lloc⁡2(ℝ;H) and for any t∈ℝ,
(17)∫-∞t‍eδs|f(s)|2ds<∞,
where 0<δ<ϱ. Let ℜδ be the set of all functions r:ℝ→(0,+∞) such that
(18)lim⁡t→-∞eδtr2(t)=0,
and 𝒟δ,ℰ0 denotes the class of all families D^={D(t);t∈ℝ}⊂𝒫(ℰ0) such that D(t)=B-(0,rD^(t)) for some rD^∈ℜδ, where B-(0,rD^(t)) is the closed ball in ℰ0 centered at 0 with radius rD^(t).


### 3.1. Pullback𝒟-Attractors in ℰ1

In this subsection, we assume thatf,f′∈Lloc⁡2(ℝ;H) and for any t∈ℝ,
(19)∫-∞t‍eδs(|f(s)|2+|f′(s)|2)ds<∞.
Let ℛδ be the set of all functions r:ℝ→(0,+∞), which satisfies (18) with δ∈(0,ϱ), and 𝒟δ,ℰ1 denotes the class of all families D^={D(t);t∈ℝ}⊂𝒫(ℰ1) such that D(t)⊂B-(0,rD^(t)) for some rD^∈ℛδ, where B-(0,rD^(t)) is the closed ball in ℰ1 centered at 0 with rD^(t).Theorem 13.
Suppose thatf,f′∈Lloc⁡2(ℝ;H) satisfy (19). Then, there exists a unique global pullback 𝒟δ,ℰ1-attractor in ℰ1 for the nonautonomous dynamical system (θ,ϕ) defined by (15).Proof.
By Lemma7, we need to prove the existence of a pullback 𝒟δ,ℰ1-absorbing set belonging to 𝒟δ,ℰ1 and then show that the cocycle ϕ defined by (15) satisfies pullback 𝒟δ,ℰ1-Condition (C).
Multiplying (1) by Av(t)=Aut(t)+ϱAu(t) and integrating over Ω, we have
(20)12ddt(∥v∥12+∥u∥22)+ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ(μ-ϱ)〈u,v〉1+k〈u+,Av〉+〈g(u),Av〉=〈f(x,t),Av〉.
Using the Hölder and Young inequalities, we obtain(21)ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ(μ-ϱ)〈u,v〉1⩾ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ(μ-ϱ)λ∥u∥2·∥v∥1⩾ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ4∥u∥22-ϱμ2λ∥v∥12⩾3ϱ4∥u∥22+(μ-ϱ-ϱμ2λ)∥v∥12.
We can easily see that(22)〈ku+,Av〉=〈ku+,Aut〉+〈ku+,ϱAu〉=ddtk〈u+,Au〉+kϱ〈u+,Au〉-k〈(u+)t,Au〉⩾ddtk〈u+,Au〉+kϱ〈u+,Au〉-ϱ16∥u∥22-4k2ϱ|ut|2.
According to (3), Theorem 8, and the Sobolev embedding theorem, we know that g(u),g′(u) are uniformly bounded in L∞. That is, there exists a constant k0>0, such that
(23)|g(u)|⩽k0,|g′(u)|⩽k0.
In view of the Hölder inequality and (23), we can know
(24)〈g(u),Av〉=〈g(u),Aut〉+〈g(u),ϱAu〉=ddt〈g(u),Au〉-〈g′(u)ut,Au〉+ϱ〈g(u),Au〉⩾ddt〈g(u),Au〉+ϱ〈g(u),Au〉-∫Ω‍|g′(u)|·|ut|·|Au|dx⩾ddt〈g(u),Au〉+ϱ〈g(u),Au〉-ϱ16∥u∥22-4k02ϱ|ut|2,〈f(x,t),Av〉=〈f(x,t),Aut〉+〈f(x,t),ϱAu〉=ddt〈f(x,t),Au〉-〈f′(x,t),Au〉+ϱ〈f(x,t),Au〉⩽ddt〈f(x,t),Au〉+ϱ〈f(x,t),Au〉+ϱ8∥u∥22+2ϱ|f′(x,t)|2.
We chooseϱ small enough, such that μ-ϱ-(ϱμ2/λ)⩾ϱ/2; we get
(25)ddt(∥v∥12+∥u∥22+2k〈u+,Au〉+2〈g(u),Au〉-2〈f(x,t),Au〉∥v∥12)+ϱ(∥v∥12+∥u∥22vvvvvvvvvvvvvvvvvvvvv+2k〈u+,Au〉vvvvvvvvvvvvvvvvvvvvv+2〈g(u),Au〉nnnnnnnnnnnnnnnnnmn-2〈f(x,t),Au〉∥v∥12)⩽(8k2ϱ+8k02ϱ)|ut|2+4ϱ|f′(x,t)|2.
On the other hand, by the Hölder and Young inequalities, (7) and (23), it follows that
(26)ddt(12∥u∥22+2k〈u+,Au〉)=ddt|12Au+2ku+|2ddt(12∥u∥22+2k〈u+,Au〉)=-4k2∫Ω‍|u+|·|(u+)t|dxddt(12∥u∥22+2k〈u+,Au〉)⩾ddt|12Au+2ku+|2ddt(12∥u∥22+2k〈u+,Au〉)-2k2λ∥u∥12-2k2|ut|2,(27)ddt(12∥u∥22+2〈g(u),Au〉-2〈f(x,t),Au〉)⩾ddt|12Au+2g(u)-2f(x,t)|2-4∫Ω‍|g(u)|·|g′(u)ut|dx-4∫Ω‍|f(x,t)|·|f′(x,t)|dx+4∫Ω‍|g′(u)ut|·|f(x,t)|dx+4∫Ω‍|g(u)|·|f′(x,t)|dx⩾ddt|12Au+2g(u)-2f(x,t)|2-4k02-4k02|ut|2-4|f(x,t)|2-4|f′(x,t)|2,(28)ϱ(12∥u∥22+2k〈u+,Au〉)⩾ϱ|12Au+2ku+|2-2ϱk2|u+|2⩾ϱ|12Au+2ku+|2-2ϱk2λ∥u∥12,(29)ϱ(12∥u∥22+2〈g(u),Au〉-2〈f(x,t),Au〉)⩾ϱ|12Au+2g(u)-2f(x,t)|2-2ϱ|g(u)|2-2ϱ|f(x,t)|2+4ϱ∫Ω‍|g(u)|·|f(x,t)|dx⩾ϱ|12Au+2g(u)-2f(x,t)|2-4ϱk02-4ϱ|f(x,t)|2.
Therefore, combining (26)–(29), we get
(30)ddt(∥v∥12+|12Au+2ku+|2+|12Au+2g(u)-2f(x,t)|2)+ϱ(∥v∥12+|12Au+2ku+|2vvvv+|12Au+2g(u)-2f(x,t)|2)⩽(8k2ϱ+8k02ϱ+2k2+4k02)|ut|2+2k2λ(1+ϱ)∥u∥12+4(1+ϱ)|f(x,t)|2+4(1ϱ+1)|f′(x,t)|2+4(1+ϱ)k02.
Set(31)C1=max⁡{8k2ϱ+8k02ϱ+2k2+4k02,2k2λ(1+ϱ)},C2=max⁡{4(1+ϱ),4(1ϱ+1)}.
Thus, denote(32)y(t)=∥v∥12+|12Au+2ku+|2+|12Au+2g(u)-2f(x,t)|2,fort⩾τ.
We have(33)ddty(t)+ϱy(t)⩽C1(∥u∥12+|ut|2)+C2(|f(t)|2+|f′(t)|2)+4(1+ϱ)k02.
By the Gronwall lemma, we have(34)y(t)⩽e-ϱτy(t-τ)+4(1+ϱ)k02∫t-τt‍e-ϱ(t-s)dsn+C1∫t-τt‍e-ϱ(t-s)(∥u∥12+|ut|2)dsn+C2∫t-τt‍e-ϱ(t-s)(|f(t)|2+|f′(t)|2)ds.
Set(35)C3=max⁡{1+2ϱ2λ-1,2},ϱ0=1+(ϱ2+4k2λ)1λ.
Then(36)∥u∥22+∥ut∥12⩽4C3e-ϱτ(ϱ0∥u1∥22+∥u2∥12)+8C3e-ϱτ(2|g(u1)|2+2|f(t-τ)|2)+2C1C3∫t-τt‍e-ϱ(t-s)(∥u∥12+|ut|2)ds+2C2C3∫t-τt‍e-ϱ(t-s)(|f(t)|2+|f′(t)|2)ds+8C3(1+ϱ)k02∫t-τt‍e-ϱ(t-s)ds+4C3k2λ∥u∥12+8C3|g(u)|2+8C3|f(t)|2.
From Theorem8, we have
(37)∥u∥12+|ut|2⩽c1c2e-δτ(∥u1∥12+|u2|2)n+c1e-δtα1∫s-ττ‍eδξ|f(ξ)|2dξ+2c1αMδ,
where
(38)c1=max⁡{2,1+2α2λ-1},c2=max⁡{2,1+λ-1(2α2+k)},
and then
(39)∫t-τt‍e-ϱ(t-s)(∥u∥12+|ut|2)ds⩽c1c2e-δτ(∥u1∥12+|u2|2)∫-∞t‍e-ϱ(t-s)ds+2c1αMδϱn+c1α1(∫-∞t‍eδs|f(s)|2ds)(∫-∞t‍e-δse-ϱ(t-s)ds)⩽c1c2ϱλe-δτ(∥u1∥22+∥u2∥12)+2c1αMδϱ+c1α1(ϱ-δ)×∫-∞t‍e-δ(t-s)|f(s)|2ds.
Set(40)C4=max⁡{4C3,2c1c2C1C3ϱλ},C5=max⁡{2C2C3,2c1C1C3α1(ϱ-δ)},C6=8C3(1+1α)k02+4c1α0MC1C3δα1+8C3k02.
LetD={D(t)}t∈ℝ∈Dδ,ℰ1. Combining (36) and (39), we have
(41)∥ϕ(τ,t-τ,y0)∥ℰ12=∥u∥22+∥ut∥12⩽C4(e-ϱτ+e-δτ)(∥u1∥22+∥u2∥12)+C5(∫-∞t‍e-ϱ(t-s)(|f(s)|2+|f′(s)|2)dsvvvw+∫-∞t‍e-δ(t-s)|f(s)|2ds)+16C3e-ϱτ|f(t-τ)|2+16C3k02e-ϱτ+4C3k2λ∥u∥12+8C3∥f(t)∥2+C6,
for all y0∈D(t-τ), t∈ℝ, and τ⩾0.
Set(42)(Rδ,ℰ1(t))2=2C5(∫-∞t‍e-ϱ(t-s)(|f(s)|2+|f′(s)|2)dsvvvv+∫-∞t‍e-δ(t-s)|f(s)|2ds)+32C3k02e-ϱτ+8C3k2λ(Rδ,ℰ0(t))2+16C3∥f(t)∥2+2C6
and consider the family B^δ,ℰ1 of closed balls in ℰ1 defined by
(43)Bδ,ℰ1(t)={z∈ℰ1;∥z∥ℰ1⩽Rδ,ℰ1(t)}.
From (18) and (41), B^δ,ℰ1 is a pullback 𝒟δ,ℰ1-absorbing for the cocycle ϕ in ℰ1.
Next, we show that the cocycleϕ satisfies the pullback 𝒟δ,ℰ1-Condition (C).
We assume thatλ~i,i=1,2,…, are eigenvalue of operator A in D(A), satisfying
(44)0<λ~1<λ~2⩽⋯⩽λ~j⩽⋯,λ~j⟶∞,asj⟶∞,ω~i denotes eigenvector corresponding to eigenvalue λ~i,i=1,2,3,…, which forms an orthogonal basis in D(A), and at the same time they are also a group of canonical bases in D(A) or V and satisfy
(45)Aω~i=λ~iω~i,∀i∈ℕ.
LetVm=span⁡{ω~1,ω~2,…,ω~m} and Pm:V→Vm is an orthogonal projector. For any (u,ut)∈ℰ1, we write
(46)(u,ut)=(u1,u1t)+(u2,u2t),
where (u1,u1t)=(Pmu,Pmut).
Taking the scalar product withAv2(t)=Au2t(t)+σAu2(t) for (1) in H, we have
(47)12ddt(∥v2∥12+∥u2∥22)+σ∥u2∥22+(μ-σ)∥v2∥12-σ(μ-σ)〈u2,v2〉1+k〈u+,Av2〉+〈g(u),Av2〉=〈f(x,t),Av2〉.
Similar to the estimate of (21) and (22), we have
(48)σ∥u2∥22+(μ-σ)∥v2∥12-σ(μ-σ)〈u2,v2〉1⩾3σ4∥u2∥22+σ2∥v2∥12,(49)〈k(u+)2,Av2〉⩾ddtk〈(u+)2,Au2〉+kσ〈(u+)2,Au2〉-ϱ16∥u2∥22-4k2σ∥(u+)2t∥2.
Moreover, we obtain(50)〈g(u),Av2〉=ddt〈(g(u))2,Au2〉-〈(g′(u)ut)2,Au2〉+σ〈(g(u))2,Au2〉⩾ddt〈(g(u))2,Au2〉+σ〈(g(u))2,Au2〉-σ16∥u2∥22-4σ|(g′(u)ut)2|2,(51)〈(I-Pm)f(x,t),Av2〉=ddt〈fm(x,t),Au2〉-〈fm′(x,t),Au〉+σ〈f(x,t),Au2〉⩽ddt〈fm(x,t),Au2〉+σ〈fm(x,t),Au2〉+σ8∥u2∥22+2σ|fm′(x,t)|2,
where fm(x,t)=(I-Pm)f(x,t).
Combining (48)–(51), we obtain from (47)
(52)ddt(∥v2∥12+∥u2∥22+2k〈(u+)2,Au2〉+2〈(g(u))2,Au2〉-2〈fm(x,t),Au2〉∥v2∥12)+σ(∥v2∥12+∥u2∥22vvvvvvvvvvvvvvvvvvvvvvvwi+2k〈(u+)2,Au2〉vvvvvvvvvvvvvvvvvvvvvvvwi+2〈(g(u))2,Au2〉vvvvvvvvvvvvvvvvvvvvvvvii-2〈fm(x,t),Au2〉∥v2∥12)⩽8k2σ|(u+)2t|2+8σ|(g′(u)ut)2|2+4σ|fm′(x,t)|2.
Like for (26)–(29), using the Hölder and Young inequalities, we get
(53)ddt[|Au2+k(u+)2+(g(u))2-fm(t)|2+∥v2∥12]+σ[|Au2+k(u+)2+(g(u))2-fm(t)|2+∥v2∥12]⩽8k2σ|(u+)2t|2+8σ|(g′(u)ut)2|2+4σ|fm′(t)|2+2k2|(u+)2|·|(u+)2t|+2|(g(u))2|·|(g′(u)ut)2|+2|fm(t)|·|fm′(t)|+2|(g′(u)ut)2|·|fm(t)|+2|(g(u))2|·|fm′(t)|+2k|(u+)2t|·|(g(u))2|+2k|(u+)2|·|(g′(u)ut)2|+2k|(u+)2t|·|fm(t)|+2k|(u+)2|·|fm′(t)|+σk2|(u+)2|2+σ|(g(u))2|2+σ|fm(t)|2+2σ|(g(u))2|·|fm(t)|+2kσ|(u+)2|·|(g(u))2|+2kσ|(u+)2|·|fm(t)|⩽5σk2|(u+)2|2+11k2σ|(u+)2t|2+5σ|(g(u))2|2+11σ|(g′(u)ut)2|2+9σ|fm(t)|2+3σ|fm′(t)|2⩽l0(|(u+)2|2+|(u+)2t|2)+l1(|(g(u))2|2+|(g′(u)ut)2|2)+l2(|fm(t)|2+|fm′(t)|2),
where l0=max⁡{5σk2,11k2/σ}, l1=max⁡{5σ,11/σ}, and l2=max⁡{9/σ,3/σ}.
By the Gronwall lemma, we have(54)|Au2(t)|2+|Δu2t(t)|2⩽4l3e-στ(|Au12|2+|Δu22|2)+8l3e-στ(2|(g(u))12|2+2|fm(t-τ)|2)+4l3(k2|(u+)2|2+2|(g(u))2|2+2|fm(t)|2)+2l0l3∫t-τte-σ(t-s)(|(u+)2|2+|(u+)2t|2)ds+2l1l3∫t-τte-σ(t-s)(|(g(u))2|2+|(g′(u)ut)2|2)ds+2l2l3∫-∞t‍e-σ(t-s)(|fm(s)|2+|fm′(t)|2)ds:=I1+I2+I3+I4+I5+I6,
where l3=max⁡{1+2σ2λ-1,2}.
Then, given anyD^∈𝒟δ,ℰ1, we have
(55)∥ϕ2(τ,t-τ,y0)∥ℰ12⩽I1+I2+I3+I4+I5+I6,
for any y0∈D(t-τ) and τ>0.
Now we estimateI1,I2,I3,I4,I5, and I6 one by one. Given any ϵ>0 and any t∈ℝ, first, by the definition of 𝒟δ,ℰ1, it is easy to see that there exists τ1⩾0 such that, for τ⩾τ1, I1,I2⩽ϵ/6.
Second, it is easy to see that(56)|fm(t)|2=|(I-Pm)f(x,t)|2⟶0asn⟶∞.
By Theorem 8, there exists τ2⩾0 such that τ⩾τ2, |Δu(t)|2⩽(Rδ,ℰ0(t))2⩽∞. Lemma 10, we can choose n1 large enough such that I3⩽ϵ/6, for n⩾n1, τ⩾τ2.
Third, by Lemmas10 and 11, we know that there exist τ3⩾0 and n2 such that I4,I5⩽ϵ/6, for n⩾n2, τ⩾τ3.
Finally, by Lemma9, we can choose n3 large enough so that
(57)2l2l3∫-∞t‍e-σ(t-s)(|fm(s)|2+|fm′(t)|2)ds⩽ϵ6
for n⩾n3.
By the above analysis and (55), we know that, for any ϵ>0, there exist τ0=max⁡{τ1,τ2,τ3} and n0=max⁡{n1,n2,n3}; then
(58)andanyy0∈D(t-τ),∥ϕ2(τ,t-τ,y0)∥ℰ12⩽ϵforanyτ>τ0,n>n0andanyy0∈D(t-τ),
which implies the pullback Dδ,ℰ1-Condition (C).
We complete the proof.


## 3.1. Pullback𝒟-Attractors in ℰ1

In this subsection, we assume thatf,f′∈Lloc⁡2(ℝ;H) and for any t∈ℝ,
(19)∫-∞t‍eδs(|f(s)|2+|f′(s)|2)ds<∞.
Let ℛδ be the set of all functions r:ℝ→(0,+∞), which satisfies (18) with δ∈(0,ϱ), and 𝒟δ,ℰ1 denotes the class of all families D^={D(t);t∈ℝ}⊂𝒫(ℰ1) such that D(t)⊂B-(0,rD^(t)) for some rD^∈ℛδ, where B-(0,rD^(t)) is the closed ball in ℰ1 centered at 0 with rD^(t).Theorem 13.
Suppose thatf,f′∈Lloc⁡2(ℝ;H) satisfy (19). Then, there exists a unique global pullback 𝒟δ,ℰ1-attractor in ℰ1 for the nonautonomous dynamical system (θ,ϕ) defined by (15).Proof.
By Lemma7, we need to prove the existence of a pullback 𝒟δ,ℰ1-absorbing set belonging to 𝒟δ,ℰ1 and then show that the cocycle ϕ defined by (15) satisfies pullback 𝒟δ,ℰ1-Condition (C).
Multiplying (1) by Av(t)=Aut(t)+ϱAu(t) and integrating over Ω, we have
(20)12ddt(∥v∥12+∥u∥22)+ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ(μ-ϱ)〈u,v〉1+k〈u+,Av〉+〈g(u),Av〉=〈f(x,t),Av〉.
Using the Hölder and Young inequalities, we obtain(21)ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ(μ-ϱ)〈u,v〉1⩾ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ(μ-ϱ)λ∥u∥2·∥v∥1⩾ϱ∥u∥22+(μ-ϱ)∥v∥12-ϱ4∥u∥22-ϱμ2λ∥v∥12⩾3ϱ4∥u∥22+(μ-ϱ-ϱμ2λ)∥v∥12.
We can easily see that(22)〈ku+,Av〉=〈ku+,Aut〉+〈ku+,ϱAu〉=ddtk〈u+,Au〉+kϱ〈u+,Au〉-k〈(u+)t,Au〉⩾ddtk〈u+,Au〉+kϱ〈u+,Au〉-ϱ16∥u∥22-4k2ϱ|ut|2.
According to (3), Theorem 8, and the Sobolev embedding theorem, we know that g(u),g′(u) are uniformly bounded in L∞. That is, there exists a constant k0>0, such that
(23)|g(u)|⩽k0,|g′(u)|⩽k0.
In view of the Hölder inequality and (23), we can know
(24)〈g(u),Av〉=〈g(u),Aut〉+〈g(u),ϱAu〉=ddt〈g(u),Au〉-〈g′(u)ut,Au〉+ϱ〈g(u),Au〉⩾ddt〈g(u),Au〉+ϱ〈g(u),Au〉-∫Ω‍|g′(u)|·|ut|·|Au|dx⩾ddt〈g(u),Au〉+ϱ〈g(u),Au〉-ϱ16∥u∥22-4k02ϱ|ut|2,〈f(x,t),Av〉=〈f(x,t),Aut〉+〈f(x,t),ϱAu〉=ddt〈f(x,t),Au〉-〈f′(x,t),Au〉+ϱ〈f(x,t),Au〉⩽ddt〈f(x,t),Au〉+ϱ〈f(x,t),Au〉+ϱ8∥u∥22+2ϱ|f′(x,t)|2.
We chooseϱ small enough, such that μ-ϱ-(ϱμ2/λ)⩾ϱ/2; we get
(25)ddt(∥v∥12+∥u∥22+2k〈u+,Au〉+2〈g(u),Au〉-2〈f(x,t),Au〉∥v∥12)+ϱ(∥v∥12+∥u∥22vvvvvvvvvvvvvvvvvvvvv+2k〈u+,Au〉vvvvvvvvvvvvvvvvvvvvv+2〈g(u),Au〉nnnnnnnnnnnnnnnnnmn-2〈f(x,t),Au〉∥v∥12)⩽(8k2ϱ+8k02ϱ)|ut|2+4ϱ|f′(x,t)|2.
On the other hand, by the Hölder and Young inequalities, (7) and (23), it follows that
(26)ddt(12∥u∥22+2k〈u+,Au〉)=ddt|12Au+2ku+|2ddt(12∥u∥22+2k〈u+,Au〉)=-4k2∫Ω‍|u+|·|(u+)t|dxddt(12∥u∥22+2k〈u+,Au〉)⩾ddt|12Au+2ku+|2ddt(12∥u∥22+2k〈u+,Au〉)-2k2λ∥u∥12-2k2|ut|2,(27)ddt(12∥u∥22+2〈g(u),Au〉-2〈f(x,t),Au〉)⩾ddt|12Au+2g(u)-2f(x,t)|2-4∫Ω‍|g(u)|·|g′(u)ut|dx-4∫Ω‍|f(x,t)|·|f′(x,t)|dx+4∫Ω‍|g′(u)ut|·|f(x,t)|dx+4∫Ω‍|g(u)|·|f′(x,t)|dx⩾ddt|12Au+2g(u)-2f(x,t)|2-4k02-4k02|ut|2-4|f(x,t)|2-4|f′(x,t)|2,(28)ϱ(12∥u∥22+2k〈u+,Au〉)⩾ϱ|12Au+2ku+|2-2ϱk2|u+|2⩾ϱ|12Au+2ku+|2-2ϱk2λ∥u∥12,(29)ϱ(12∥u∥22+2〈g(u),Au〉-2〈f(x,t),Au〉)⩾ϱ|12Au+2g(u)-2f(x,t)|2-2ϱ|g(u)|2-2ϱ|f(x,t)|2+4ϱ∫Ω‍|g(u)|·|f(x,t)|dx⩾ϱ|12Au+2g(u)-2f(x,t)|2-4ϱk02-4ϱ|f(x,t)|2.
Therefore, combining (26)–(29), we get
(30)ddt(∥v∥12+|12Au+2ku+|2+|12Au+2g(u)-2f(x,t)|2)+ϱ(∥v∥12+|12Au+2ku+|2vvvv+|12Au+2g(u)-2f(x,t)|2)⩽(8k2ϱ+8k02ϱ+2k2+4k02)|ut|2+2k2λ(1+ϱ)∥u∥12+4(1+ϱ)|f(x,t)|2+4(1ϱ+1)|f′(x,t)|2+4(1+ϱ)k02.
Set(31)C1=max⁡{8k2ϱ+8k02ϱ+2k2+4k02,2k2λ(1+ϱ)},C2=max⁡{4(1+ϱ),4(1ϱ+1)}.
Thus, denote(32)y(t)=∥v∥12+|12Au+2ku+|2+|12Au+2g(u)-2f(x,t)|2,fort⩾τ.
We have(33)ddty(t)+ϱy(t)⩽C1(∥u∥12+|ut|2)+C2(|f(t)|2+|f′(t)|2)+4(1+ϱ)k02.
By the Gronwall lemma, we have(34)y(t)⩽e-ϱτy(t-τ)+4(1+ϱ)k02∫t-τt‍e-ϱ(t-s)dsn+C1∫t-τt‍e-ϱ(t-s)(∥u∥12+|ut|2)dsn+C2∫t-τt‍e-ϱ(t-s)(|f(t)|2+|f′(t)|2)ds.
Set(35)C3=max⁡{1+2ϱ2λ-1,2},ϱ0=1+(ϱ2+4k2λ)1λ.
Then(36)∥u∥22+∥ut∥12⩽4C3e-ϱτ(ϱ0∥u1∥22+∥u2∥12)+8C3e-ϱτ(2|g(u1)|2+2|f(t-τ)|2)+2C1C3∫t-τt‍e-ϱ(t-s)(∥u∥12+|ut|2)ds+2C2C3∫t-τt‍e-ϱ(t-s)(|f(t)|2+|f′(t)|2)ds+8C3(1+ϱ)k02∫t-τt‍e-ϱ(t-s)ds+4C3k2λ∥u∥12+8C3|g(u)|2+8C3|f(t)|2.
From Theorem8, we have
(37)∥u∥12+|ut|2⩽c1c2e-δτ(∥u1∥12+|u2|2)n+c1e-δtα1∫s-ττ‍eδξ|f(ξ)|2dξ+2c1αMδ,
where
(38)c1=max⁡{2,1+2α2λ-1},c2=max⁡{2,1+λ-1(2α2+k)},
and then
(39)∫t-τt‍e-ϱ(t-s)(∥u∥12+|ut|2)ds⩽c1c2e-δτ(∥u1∥12+|u2|2)∫-∞t‍e-ϱ(t-s)ds+2c1αMδϱn+c1α1(∫-∞t‍eδs|f(s)|2ds)(∫-∞t‍e-δse-ϱ(t-s)ds)⩽c1c2ϱλe-δτ(∥u1∥22+∥u2∥12)+2c1αMδϱ+c1α1(ϱ-δ)×∫-∞t‍e-δ(t-s)|f(s)|2ds.
Set(40)C4=max⁡{4C3,2c1c2C1C3ϱλ},C5=max⁡{2C2C3,2c1C1C3α1(ϱ-δ)},C6=8C3(1+1α)k02+4c1α0MC1C3δα1+8C3k02.
LetD={D(t)}t∈ℝ∈Dδ,ℰ1. Combining (36) and (39), we have
(41)∥ϕ(τ,t-τ,y0)∥ℰ12=∥u∥22+∥ut∥12⩽C4(e-ϱτ+e-δτ)(∥u1∥22+∥u2∥12)+C5(∫-∞t‍e-ϱ(t-s)(|f(s)|2+|f′(s)|2)dsvvvw+∫-∞t‍e-δ(t-s)|f(s)|2ds)+16C3e-ϱτ|f(t-τ)|2+16C3k02e-ϱτ+4C3k2λ∥u∥12+8C3∥f(t)∥2+C6,
for all y0∈D(t-τ), t∈ℝ, and τ⩾0.
Set(42)(Rδ,ℰ1(t))2=2C5(∫-∞t‍e-ϱ(t-s)(|f(s)|2+|f′(s)|2)dsvvvv+∫-∞t‍e-δ(t-s)|f(s)|2ds)+32C3k02e-ϱτ+8C3k2λ(Rδ,ℰ0(t))2+16C3∥f(t)∥2+2C6
and consider the family B^δ,ℰ1 of closed balls in ℰ1 defined by
(43)Bδ,ℰ1(t)={z∈ℰ1;∥z∥ℰ1⩽Rδ,ℰ1(t)}.
From (18) and (41), B^δ,ℰ1 is a pullback 𝒟δ,ℰ1-absorbing for the cocycle ϕ in ℰ1.
Next, we show that the cocycleϕ satisfies the pullback 𝒟δ,ℰ1-Condition (C).
We assume thatλ~i,i=1,2,…, are eigenvalue of operator A in D(A), satisfying
(44)0<λ~1<λ~2⩽⋯⩽λ~j⩽⋯,λ~j⟶∞,asj⟶∞,ω~i denotes eigenvector corresponding to eigenvalue λ~i,i=1,2,3,…, which forms an orthogonal basis in D(A), and at the same time they are also a group of canonical bases in D(A) or V and satisfy
(45)Aω~i=λ~iω~i,∀i∈ℕ.
LetVm=span⁡{ω~1,ω~2,…,ω~m} and Pm:V→Vm is an orthogonal projector. For any (u,ut)∈ℰ1, we write
(46)(u,ut)=(u1,u1t)+(u2,u2t),
where (u1,u1t)=(Pmu,Pmut).
Taking the scalar product withAv2(t)=Au2t(t)+σAu2(t) for (1) in H, we have
(47)12ddt(∥v2∥12+∥u2∥22)+σ∥u2∥22+(μ-σ)∥v2∥12-σ(μ-σ)〈u2,v2〉1+k〈u+,Av2〉+〈g(u),Av2〉=〈f(x,t),Av2〉.
Similar to the estimate of (21) and (22), we have
(48)σ∥u2∥22+(μ-σ)∥v2∥12-σ(μ-σ)〈u2,v2〉1⩾3σ4∥u2∥22+σ2∥v2∥12,(49)〈k(u+)2,Av2〉⩾ddtk〈(u+)2,Au2〉+kσ〈(u+)2,Au2〉-ϱ16∥u2∥22-4k2σ∥(u+)2t∥2.
Moreover, we obtain(50)〈g(u),Av2〉=ddt〈(g(u))2,Au2〉-〈(g′(u)ut)2,Au2〉+σ〈(g(u))2,Au2〉⩾ddt〈(g(u))2,Au2〉+σ〈(g(u))2,Au2〉-σ16∥u2∥22-4σ|(g′(u)ut)2|2,(51)〈(I-Pm)f(x,t),Av2〉=ddt〈fm(x,t),Au2〉-〈fm′(x,t),Au〉+σ〈f(x,t),Au2〉⩽ddt〈fm(x,t),Au2〉+σ〈fm(x,t),Au2〉+σ8∥u2∥22+2σ|fm′(x,t)|2,
where fm(x,t)=(I-Pm)f(x,t).
Combining (48)–(51), we obtain from (47)
(52)ddt(∥v2∥12+∥u2∥22+2k〈(u+)2,Au2〉+2〈(g(u))2,Au2〉-2〈fm(x,t),Au2〉∥v2∥12)+σ(∥v2∥12+∥u2∥22vvvvvvvvvvvvvvvvvvvvvvvwi+2k〈(u+)2,Au2〉vvvvvvvvvvvvvvvvvvvvvvvwi+2〈(g(u))2,Au2〉vvvvvvvvvvvvvvvvvvvvvvvii-2〈fm(x,t),Au2〉∥v2∥12)⩽8k2σ|(u+)2t|2+8σ|(g′(u)ut)2|2+4σ|fm′(x,t)|2.
Like for (26)–(29), using the Hölder and Young inequalities, we get
(53)ddt[|Au2+k(u+)2+(g(u))2-fm(t)|2+∥v2∥12]+σ[|Au2+k(u+)2+(g(u))2-fm(t)|2+∥v2∥12]⩽8k2σ|(u+)2t|2+8σ|(g′(u)ut)2|2+4σ|fm′(t)|2+2k2|(u+)2|·|(u+)2t|+2|(g(u))2|·|(g′(u)ut)2|+2|fm(t)|·|fm′(t)|+2|(g′(u)ut)2|·|fm(t)|+2|(g(u))2|·|fm′(t)|+2k|(u+)2t|·|(g(u))2|+2k|(u+)2|·|(g′(u)ut)2|+2k|(u+)2t|·|fm(t)|+2k|(u+)2|·|fm′(t)|+σk2|(u+)2|2+σ|(g(u))2|2+σ|fm(t)|2+2σ|(g(u))2|·|fm(t)|+2kσ|(u+)2|·|(g(u))2|+2kσ|(u+)2|·|fm(t)|⩽5σk2|(u+)2|2+11k2σ|(u+)2t|2+5σ|(g(u))2|2+11σ|(g′(u)ut)2|2+9σ|fm(t)|2+3σ|fm′(t)|2⩽l0(|(u+)2|2+|(u+)2t|2)+l1(|(g(u))2|2+|(g′(u)ut)2|2)+l2(|fm(t)|2+|fm′(t)|2),
where l0=max⁡{5σk2,11k2/σ}, l1=max⁡{5σ,11/σ}, and l2=max⁡{9/σ,3/σ}.
By the Gronwall lemma, we have(54)|Au2(t)|2+|Δu2t(t)|2⩽4l3e-στ(|Au12|2+|Δu22|2)+8l3e-στ(2|(g(u))12|2+2|fm(t-τ)|2)+4l3(k2|(u+)2|2+2|(g(u))2|2+2|fm(t)|2)+2l0l3∫t-τte-σ(t-s)(|(u+)2|2+|(u+)2t|2)ds+2l1l3∫t-τte-σ(t-s)(|(g(u))2|2+|(g′(u)ut)2|2)ds+2l2l3∫-∞t‍e-σ(t-s)(|fm(s)|2+|fm′(t)|2)ds:=I1+I2+I3+I4+I5+I6,
where l3=max⁡{1+2σ2λ-1,2}.
Then, given anyD^∈𝒟δ,ℰ1, we have
(55)∥ϕ2(τ,t-τ,y0)∥ℰ12⩽I1+I2+I3+I4+I5+I6,
for any y0∈D(t-τ) and τ>0.
Now we estimateI1,I2,I3,I4,I5, and I6 one by one. Given any ϵ>0 and any t∈ℝ, first, by the definition of 𝒟δ,ℰ1, it is easy to see that there exists τ1⩾0 such that, for τ⩾τ1, I1,I2⩽ϵ/6.
Second, it is easy to see that(56)|fm(t)|2=|(I-Pm)f(x,t)|2⟶0asn⟶∞.
By Theorem 8, there exists τ2⩾0 such that τ⩾τ2, |Δu(t)|2⩽(Rδ,ℰ0(t))2⩽∞. Lemma 10, we can choose n1 large enough such that I3⩽ϵ/6, for n⩾n1, τ⩾τ2.
Third, by Lemmas10 and 11, we know that there exist τ3⩾0 and n2 such that I4,I5⩽ϵ/6, for n⩾n2, τ⩾τ3.
Finally, by Lemma9, we can choose n3 large enough so that
(57)2l2l3∫-∞t‍e-σ(t-s)(|fm(s)|2+|fm′(t)|2)ds⩽ϵ6
for n⩾n3.
By the above analysis and (55), we know that, for any ϵ>0, there exist τ0=max⁡{τ1,τ2,τ3} and n0=max⁡{n1,n2,n3}; then
(58)andanyy0∈D(t-τ),∥ϕ2(τ,t-τ,y0)∥ℰ12⩽ϵforanyτ>τ0,n>n0andanyy0∈D(t-τ),
which implies the pullback Dδ,ℰ1-Condition (C).
We complete the proof.



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*Source: 253532-2014-03-09.xml*