# Strong Uniform Attractors for Nonautonomous Suspension Bridge-Type Equations

Authors: Xuan Wang; Qiaozhen Ma
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We discuss long-term dynamical behavior of the solutions for the nonautonomous suspension bridge-type equation in the strong Hilbert spaceD(A)×H2(Ω)∩H01(Ω), where the
nonlinearity g(u,t) is translation compact and the time-dependent external forces h(x,t) only satisfy condition (C*) instead of translation compact. The existence of strong solutions and strong uniform attractors is investigated using a new process scheme. Since the solutions of the nonautonomous suspension bridge-type equation have no higher regularity and the process associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.


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## Body

## 1. Introduction

Consider the following equations:(1.1)utt+uxxxx+δut+ku+=l+ϵh(x,t),in(0,L)×ℝ,u(0,t)=u(L,t)=uxx(0,t)=uxx(L,t)=0,t∈ℝ.
Suspension bridge equations (1.1) have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and McKenna in 1990. This model has been derived as follows. In the suspension bridge system, suspension bridge can be considered as an elastic and unloaded beam with hinged ends. u(x,t) denotes the deflection in the downward direction; δut represents the viscous damping. The restoring force can be modeled owing to the cable with one-sided Hooke’s law so that it strongly resists expansion but does not resist compression. The simplest function to model the restoring force of the stays in the suspension bridge can be denoted by a constant k times u, the expansion, if u is positive, but zero, if u is negative, corresponding to compression; that is, ku+, where
(1.2)u+={u,ifu>0,0,ifu⩽0.
Besides, the right-hand side of (1.1) also contains two terms: the large positive term l corresponding to gravity, and a small oscillatory forcing term ϵh(x,t) possibly aerodynamic in origin, where ϵ is small.There are many results for (1.1) (cf. [1–9]), for instance, the existence, multiplicity, and properties of the traveling wave solutions, and so forth.In the study of equations of mathematical physics, attractor is a proper mathematical concept about the depiction of the behavior of the solutions of these equations when time is large or tends to infinity, which describes all the possible limits of solutions. In the past two decades, many authors have proved the existence of attractor and discussed its properties for various mathematical physics models (e.g., see [10–12] and the reference therein). About the long-time behavior of suspension bridge-type equations, for the autonomous case, in [13, 14] 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).It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding nonautonomous model is more natural and precise than the autonomous one, moreover, it always presents as a nonlinear equation, not just a linear one. Therefore, in this paper, we will discuss the following nonautonomous suspension bridge-type equation: letΩ be an open bounded subset of ℝ2 with smooth boundary, ℝτ=[τ,+∞], and we add the nonlinear forcing term g(u,t) (which is dependent on deflection u and time t) to (1.1) and neglect gravity, then we can obtain the following initial-boundary value problem:
(1.3)utt+Δ2u+αut+ku++g(u,t)=h(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 u(x,t) is an unknown function, which could represent the deflection of the road bed in the vertical plane; h(x,t) and g(u,t) are time dependant external forces; ku+ represents the restoring force, k denotes the spring constant; αut represents the viscous damping, α is a given positive constant.To our knowledge, this is the first time to consider the nonautonomous dynamics of (1.3) with the time dependant external forces h(x,t) and g(u,t) in the strong topological space D(A)×H2(Ω)∩H01(Ω). At the same time, in mathematics, we only assume that the force term h(x,t) satisfies the so-called condition (C*) (introduced in [15]), which is weaker than translation compact assumption (see [10] or Section 2 below).This paper is organized as follows. At first, in Section2, we give (recall) some preliminaries, including the notation we will use, the assumption on nonlinearity g(·,t), and some general abstract results about nonautonomous dynamical system. Then, in Section 3 we prove our main result about the existence of strong attractor for the nonautonomous dynamical system generated by the solution of (1.3).


## 2. Notation and Preliminaries

With the usual notation, we introduce the spacesH=L2(Ω),  V=H2(Ω)∩H01(Ω), D(A)={u∈H2(Ω)∩H01(Ω)∣Au∈L2(Ω)}, where A=Δ2. We equip these spaces with inner product and norm 〈·,·〉,∥·∥, 〈·,·〉1,∥·∥1 and 〈·,·〉2,∥·∥2, respectively,
(2.1)〈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
(2.2)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.In the following, the assumption on the nonlinearityg is given. Let g be a C1 function from ℝ×ℝ to ℝ and satisfy
(2.3)liminf|u|→∞G(u,s)u2⩾0,
where G(u,s)=∫0u‍g(w,s)dw, and there exists C0>0, such that
(2.4)liminf|u|→∞〈u,g(u,s)〉-C0G(u,s)u2⩾0.Suppose thatγ is an arbitrary positive constant, and
(2.5)|gu(u,s)|⩽C1(1+|u|γ),|gs(u,s)|⩽C1(1+|u|γ+1),(2.6)Gs(u,s)⩽δ2G(u,s)+C2,∀(u,s)∈ℝ×ℝ,
where δ is a sufficiently small constant.As a consequence of (2.3)-(2.4), if we denote 𝒢(u,s)=∫Ω‍G(u,s)dx, then there exist two positive constants K1, K2 such that
(2.7)𝒢(φ,s)+m∥φ∥2+K1⩾0,(2.8)〈φ,g(φ,s)〉-C0𝒢(φ,s)+m∥φ∥2+K2⩾0,∀(φ,s)∈ℝ×ℝ,where m,C0>0, and m is sufficiently small.By virtue of (2.5), we can get
(2.9)|g(u,s)|⩽C3(1+|u|γ+1),|G(u,s)|⩽C3(1+|u|γ+2).WhenA=Δ2, problem (1.3) is equivalent to the following equations in H:
(2.10)utt+αut+Au+ku++g(u,t)=h(x,t),u(τ)=u1,ut(τ)=u2.From the Poincaré inequality, there exists a proper constantλ1>0, such that
(2.11)λ1∥u∥2⩽∥u∥12,∀u∈V.We introduce the Hilbert space(2.12)ℰ0=V×H,ℰ1=D(A)×V,
and endow this space with norm:
(2.13)∥z∥ℰ0=∥(u,ut)∥ℰ0=(12(∥u∥12+∥ut∥2))1/2,∥z∥ℰ1=∥(u,ut)∥ℰ1=(12(∥u∥22+∥ut∥12))1/2.To prove the existence of uniform attractors corresponding to (2.10), we also need the following abstract results (e.g., see [10]).LetE be a Banach space, and let a two-parameter family of mappings {U(t,τ)}={U(t,τ)∣t⩾τ,τ∈ℝ} on E:
(2.14)U(t,τ):E→E,t⩾τ,τ∈ℝ.Definition 2.1 (see [10]).
LetΣ be a parameter set. {Uσ(t,τ)∣t⩾τ,τ∈ℝ},  σ∈Σ is said to be a family of processes in Banach space E, if for each σ∈Σ,  {Uσ(t,τ)} is a process; that is, the two-parameter family of mappings {Uσ(t,τ)} from E to E satisfy
(2.15)Uσ(t,s)∘Uσ(s,τ)=Uσ(t,τ),∀t⩾s⩾τ,τ∈ℝ,Uσ(τ,τ)=Iistheidentityoperator,τ∈ℝ,
where Σ is called the symbol space and σ∈Σ is the symbol.Note that the following translation identity is valid for a general family of processes{Uσ(t,τ)}, σ∈Σ,  if a problem is the unique solvability and for the translation semigroup {T(l)∣l⩾0} satisfying T(l)Σ=Σ:
(2.16)Uσ(t+l,τ+l)=UT(l)σ(t,τ),∀σ∈Σ,t⩾τ,τ∈ℝ,l⩾0.A setB0⊂E is said to be a uniformly (w.r.t σ∈Σ) absorbing set for the family of processes {Uσ(t,τ)}, σ∈Σ if for any τ∈ℝ and B∈ℬ(E), there exists t0=t0(τ,B)⩾τ such that ∪σ∈ΣUσ(t,τ)⊆B0 for all t⩾t0. A set Y⊂E is said to be uniformly (w.r.t. σ∈Σ) attracting for the family of processes {Uσ(t,τ)}, σ∈Σ if for any fixed τ∈ℝ and every B∈ℬ(E),
(2.17)limt→∞(supσ∈ΣdistE(Uσ(t,τ)B,Y))=0.Definition 2.2 (see [10]).
A closed setAΣ⊂E is said to be the uniform (w.r.t. σ∈Σ) attractor of the family of processes {Uσ(t,τ)}, σ∈Σ if it is uniformly (w.r.t. σ∈Σ) attracting (attracting property) and contained in any closed uniformly (w.r.t. σ∈Σ) attracting set A′ of the family of processes {Uσ(t,τ)}, σ∈Σ: AΣ⊆A′ (minimality property).Now we recalled the results in [16].Definition 2.3 (see [16, 17]).
A family of processes{Uσ(t,τ)}, σ∈Σ is said to be satisfying uniform (w.r.t. σ∈Σ). Condition (C) if for any fixed τ∈ℝ,  B∈ℬ(E) and ϵ>0, there exist a t0=t0(τ,B,ϵ)⩾τ and a finite dimensional subspace Em of E such that  (i)
Pm(∪σ∈Σ∪t⩾t0Uσ(t,τ)B) is bounded;(ii)
∥(I-Pm)(∪σ∈Σ∪t⩾t0Uσ(t,τ)x)∥E⩽ϵ,∀x∈B, where dimEm=m and Pm:E→Em is abounded projector.Theorem 2.4 (see [16]).
LetΣ be a complete metric space, and let {T(t)} be a continuous invariant T(t)Σ=Σ semigroup on Σ satisfying the translation identity. A family of processes {Uσ(t,τ)}, σ∈Σ possesses compact uniform (w.r.t. σ∈Σ) attractor AΣ in E satisfying
(2.18)AΣ=ω0,Σ(B0)=ωτ,Σ(B0),∀t∈ℝ,
if it  (i)
has a bounded uniformly (w.r.t.σ∈Σ) absorbing set B0;(ii)
satisfies uniform (w.r.t.σ∈Σ) condition (C), where ωτ,Σ(B0)=∩t⩾τ[∪σ∈Σ∪s⩾tUσ(s,t)B0]. Moreover, if E is a uniformly convex Banach space, then the converse is true.LetX be a Banach space. Consider the space Lloc2(ℝ;X) of functions ϕ(s),s∈ℝ with values in X that are 2-power integrable in the Bochner sense. Lc2(ℝ;X) is a set of all translation compact functions in Lloc2(ℝ;X), Lb2(ℝ;X) is a set of all translation bound functions in Lloc2(ℝ;X).In [15], the authors have introduced a new class of functions which are translation bounded but not translation compact. In the third section, let the forcing term h(x,t) satisfy condition (C*), we can prove the existence of compact uniform (w.r.t. σ∈ℋ(σ0), σ0(s)=(g0(u,s),h(x,s))) attractor for nonautonomous suspension bridge equation in ℰ1.Definition 2.5 (see [15]).
LetX be a Banach space. A function f∈Lb2(ℝ;X) is said to satisfy condition (C*) if for any ϵ>0, there exists a finite dimensional subspace X1 of X such that
(2.19)supt∈ℝ∫tt+1‍∥(I-Pm)f(s)∥X2ds<ϵ,
where Pm:X→X1 is the canonical projector.Denote byLc*2(ℝ;X) the set of all functions satisfying condition (C*). From [15], we can see that Lc2(ℝ;X)⊂Lc*2(ℝ;X)⊂Lb2(ℝ;X).Remark 2.6.
In fact, the function satisfying condition (C*) implies the dissipative property in some sense, and the condition (C*) is very natural in view of the compact condition, uniform condition (C).Lemma 2.7 (see [15]).
Iff∈Lc*2(ℝ;X), then for any ϵ>0 and τ∈ℝ we have
(2.20)supt⩾τ∫τt‍e-δ(t-s)∥(I-Pm)f(s)∥X2ds⩽ϵ,
where Pm:X→X1 is the canonical projector and δ is a positive constant.In order to define the family of processes of (2.10), we also need the following results.Proposition 2.8 (see [10]).
IfX is reflexive separable, then  (i)
for allh1∈ℋ(h0), ∥h1∥Lb2(ℝ;X)⩽∥h0∥Lb2(ℝ;X);(ii)
the translation group{T(t)}is weakly continuous onℋ(h0);(iii)
T(t)ℋ(h0)=ℋ(h0)for allt∈ℝ+.Proposition 2.9 (see [10]).
Letg0(s)∈Lc2(ℝ;X), then  (i)
for allg1∈ℋ(g0), g1∈Lc2(ℝ;X), and the setℋ(g0)is bound inLb2(ℝ;X);(ii)
the translation group{T(t)}is continuous onℋ(g0)with the topology ofLloc2(ℝ,X);(iii)
T(t)ℋ(g0)=ℋ(g0)for allt∈ℝ+.


## 3. Uniform Attractors inℰ1

To describe the asymptotic behavior of the solutions of our system, we seth0∈Lc*2(ℝτ;V)⊂Lb2(ℝτ;V) and ℋ(h0)=[h0(x,s+h)∣h∈ℝ]Lloc2,w(ℝτ;V), where [] denotes the closure of a set in topological space Lloc2,w(ℝτ;V). If h∈ℋ(h0), then h∈Lb2(ℝτ;V), that is to be
(3.1)supt⩾τ∫tt+1‍∥h(x,s)∥1ds<∞,
where ∥·∥1 denotes the norm in V.


### 3.1. Existence and Uniqueness of Strong Solutions

At first, we give the concept of strong solutions for the initial-boundary value problem (2.10).Definition 3.1.
SetI=[τ,T], for T>τ⩾0. We suppose that k>0, h∈Lb2(ℝτ;V), g∈C1(ℝ×ℝ;ℝ) satisfying (2.3)–(2.6) and g(0,0)=0. The function z=(u,ut)∈L∞(I;ℰ1) is said to be a strong solution to problem (2.10) in the time interval I, with initial data z(τ)=zτ=(u1,u2)∈ℰ1, provided
(3.2)〈utt,v-〉+α〈ut,v-〉+∫Ω‍ΔuΔv-dx+∫Ω‍g(u,t)v-dx+k〈u+,v-〉=∫Ω‍h(x,t)v-dx,
for all v-∈V and a.e. t∈I.Then, by using the methods in [18] (Galerkin approximation method), we can get the following result about the existence and uniqueness of strong solutions.Theorem 3.2 (existence and uniqueness of strong solutions).
DefineI=[τ,T], for all T>τ. Let k>0, h∈Lb2(ℝτ;V), g∈C1(ℝ×ℝ;ℝ) satisfying (2.3)–(2.6). Then for any given zτ∈ℰ1, there is a unique solution z=(u,ut) for problem (2.10) in ℰ1. Furthermore, for i=1,2, let {zτi,hi}(zτi∈ℰ1  and  hi∈Lb2(ℝτ;V)) be two initial conditions, and denote by zi corresponding solutions to problem (2.10). Then the estimates hold as follows: for all τ⩽t⩽T+τ,
(3.3)∥z1(t)-z2(t)∥ℰ12⩽Q(∥zτi∥ℰ1,T)(∥zτ1-zτ2∥ℰ12+∥h1-h2∥Lb2(ℝτ;V)2).Thus, (2.10) will be written as an evolutionary system introduced z(t)=(u(t),ut(t)) and zτ=z(τ)=(u1,u2) for brevity, as ∥z∥ℰ12=(1/2)(∥u∥22+∥ut∥12), the system (2.10) can be written in the operator form
(3.4)∂tz=Aσ(t)(z),z|t=τ=zτ,
where σ(s)=(g(u,s),h(x,s)) is the symbol of (3.4). If zτ∈ℰ1, then problem (3.4) has a unique solution z(t)∈L∞(ℝτ,ℰ1). This implies that the process {Uσ(t,τ)} given by the formula Uσ(t,τ)zτ=z(t) is defined in ℰ1.Now we define the symbol space. A fixed symbolσ0(s)=(g0(u,s),h0(x,s)) can be given, where h0(x,s) is in Lc*2(ℝτ;V), the function g0(u,s)∈Lc2(ℝτ;ℳ) satisfying (2.3)–(2.6), and ℳ is a Banach space,
(3.5)ℳ={g∈C1(ℝ×ℝ,ℝ)∥g(u)∥1+∥gs(u)∥1∥u∥1γ+1+1+∥gu(u)∥1∥u∥1γ+1<∞},
endowed with the following norm:
(3.6)∥g∥ℳ=supu∈ℝ{∥g(u)∥1+∥gs(u)∥1∥u∥1γ+1+1+∥gu(u)∥1∥u∥1γ+1}.
Obviously, the function σ0(s)=(g0(u,s),h0(x,s)) is in Lc2(ℝτ;ℳ)×Lc*2(ℝτ;V). we define ℋ(σ0)=ℋ(g0)×ℋ(h0)=[g0(u,s+l)∣l∈ℝ]Lloc2,w(ℝτ;ℳ)×[h0(x,s+l)∣l∈ℝ]Lloc2,w(ℝτ;V), where [] denotes the closure of a set in topological space Lloc2,w(ℝτ;ℳ)  (or Lloc2,w(ℝτ;V)). So, if (g,h)∈ℋ(σ0), then g(u,t) and h(x,t) all satisfy condition (C*).Applying Propositions2.8 and 2.9 and Theorem 3.2, we can easily know that the family of processes {Uσ(t,τ)}:ℰ1→ℰ1,  σ∈ℋ(σ0), t⩾τ are defined. Furthermore, the translation semigroup {T(l)∣l∈ℝ+} satisfies that for all l∈ℝ+, T(l)ℋ(σ0)=ℋ(σ0), and the following translation identity:
(3.7)Uσ(t+l,τ+l)=UT(l)σ(t,τ),∀σ∈ℋ(σ0),fort⩾τ⩾0,l⩾0
holds.Then for anyσ∈ℋ(σ0), the problem (3.4) with σ instead of σ0 possesses a corresponding to process {Uσ(t,τ)} acting on ℰ1.Consequently, for eachσ∈ℋ(σ0), σ0(s)=(g0(u,s),h0(x,s)) (here h0(x,s)∈Lc*2(ℝτ;V), g0(u,s)∈Lc2(ℝτ;ℳ) satisfying (2.3)–(2.6)), we can define a process
(3.8)Uσ(t,τ):ℰ1→ℰ1,zτ=(u1,u2)→(u(t),ut(t))=Uσ(t,τ)zτ,
and {Uσ(t,τ)},  σ∈ℋ(σ0) is a family of processes on ℰ1.


### 3.2. A Priori Estimates

#### 3.2.1. A Priori Estimates inℰ0

Theorem 3.3.
Assume thatz(t) is a solution of (2.10) with initial data z0∈B. If the nonlinearity g(u,t) satisfies (2.3)–(2.6), h0∈Lc*2(ℝτ;H),  h∈ℋ(h0), k>0, then there is a positive constant μ0 such that for any bounded (in  ℰ0) subset B, there exists t0=t0(∥B∥ℰ0) such that
(3.9)∥z(t)∥ℰ02=12(∥u∥12+∥ut∥2)⩽μ02,t⩾t0=t0(∥B∥ℰ0).Proof.
Now we will prove thatz=(u,ut) are bounded in ℰ0=V×H.
We assume thatϱ is positive and satisfies
(3.10)0<ϱ(α-ϱ)<λ1.
Multiplying (2.10) by v(t)=ut(t)+ϱu(t) and integrating over Ω, we have
(3.11)12ddt(∥v∥2+∥u∥12)+ϱ∥u∥12+(α-ϱ)∥v∥2-ϱ(α-ϱ)〈u,v〉+k〈u+,v〉+〈g(u,t),v〉=〈h(t),v〉.
We can easily see that(3.12)ϱ(α-ϱ)〈u,v〉⩽(α-ϱ)∥v∥24+(α-ϱ)ϱ2∥u∥2,(3.13)〈h(t),v〉⩽(α-ϱ)∥v∥24+∥h(t)∥2α-ϱ.
Then, substituting (3.12)-(3.13) into (3.11), we can obtain that
(3.14)ddt(∥v∥2+∥u∥12)+2ϱ∥u∥12+(α-ϱ)∥v∥2-2ϱ2(α-ϱ)∥u∥2+2k〈u+,v〉+2〈g,v〉⩽2∥h(t)∥2α-ϱ.
In view of (2.6) and (2.8), we can know
(3.15)〈g,v〉=〈g,ut+ϱu〉=ddt∫Ω‍G(u(x,t),t)dx+ϱ〈g(u,t),u〉-∫Ω‍Gs(u(x,t),t)dx=ddt∫Ω‍G(u(x,t),t)dx+ϱ∫Ω‍g(u(x,t),t)u(x,t)dx-ϱC0∫Ω‍G(u(x,t),t)dx+ϱC0∫Ω‍G(u(x,t),t)dx-∫Ω‍Gs(u(x,t),t)dx⩾ddt𝒢(u(x,t),t)+ϱC0𝒢(u(x,t),t)-ϱ(m∥u∥2+K2)-δ2𝒢(u(x,t),t)-C2|Ω|,k〈u+,v〉=12ddtk∥u+∥2+ϱk∥u+∥2.
Consequently,(3.16)ddt(∥v∥2+∥u∥12+k∥u+∥2+2𝒢(u(x,t),t))+(α-ϱ)∥v∥2+2ϱλ1(λ1-ϱ(α-ϱ)-m)∥u∥12+2ϱk∥u+∥2+(ϱC0-δ2)2𝒢(u(x,t),t)⩽2∥h(t)∥2α-ϱ+2(ϱK2+C2|Ω|).
We introduce the functional as follows:(3.17)y(t)=∥v∥2+∥u∥12+k∥u+∥2+2𝒢(u(x,t),t)+2K1,fort⩾τ.
Settingβ=min{α-ϱ,2ϱλ1-1(λ1-ϱ(α-ϱ)-m),2ϱ,ϱC0-δ2}, we choose proper positive constants m and δ, such that
(3.18)m<λ1-ϱ(α-ϱ),δ2<ϱC0
hold, then β>0.
We definemh(t)=∥h(t)∥2, then
(3.19)ddty(t)+βy(t)⩽C4+C5mh(t),
where C4=2(ϱK2+C2|Ω|)+2βK1, C5=2(α-ϱ)-1.
Analogous to the proof of Lemma 2.1.3 in [10], we can estimate the integral and obtain
(3.20)y(t)⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫0t‍mh(s)e-β(t-s)ds⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫t-1t‍mh(s)e-β(t-s)ds+C5∫t-2t-1‍mh(s)e-β(t-s)ds+⋯⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫t-1t‍mh(s)ds+C5e-β∫t-2t-1‍mh(s)ds+C5e-2β∫t-3t-2‍mh(s)ds+⋯⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5mh(1+e-β+e-2β+⋯)⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5mh(1+β-1)⩽y(τ)e-β(t-τ)+C4β-1+C5mh(1+β-1),fort⩾τ,
where mh=supt⩾τ∫tt+1‍mh(s)ds.
By virtue of (2.7), we can get
(3.21)2𝒢(u,t)⩾-2m∥u∥2-2K1⩾-2mλ1-1∥u∥12-2K1.
Choosing m⩽λ1/4, we obtain from (3.17)
(3.22)y(t)=∥u∥12+∥ut+ϱu∥2+k∥u+∥2+2𝒢(u,t)+2K1⩾12∥u∥12+∥ut+ϱu∥2+k∥u+∥2⩾∥z(t)∥ℰ02.
In consideration of (2.9) and 0<γ<∞, we can see
(3.23)2𝒢(uτ(x),τ)⩽2C3∫Ω‍(|uτ(x)|γ+2+1)dx⩽C6(∥uτ∥1γ+2+1),(3.24)y(τ)=∥u(τ)∥12+∥ut(τ)+ϱu(τ)∥2+k∥(u(τ))+∥2+2𝒢(u(τ),τ)+2K1⩽C7(∥z(τ)∥ℰ0γ+2+1).
Combining (3.20), (3.22), and (3.24), we can deduce that
(3.25)∥z(t)∥ℰ02⩽y(τ)e-β(t-τ)+C4β-1+C5mh(1+β-1)⩽C7(∥z(τ)∥ℰ0γ+2+1)e-β(t-τ)+C4β-1+C5mh(1+β-1)⩽C7∥z(τ)∥ℰ0γ+2e-β(t-τ)+C8,t⩾τ.
Assume that∥z(τ)∥ℰ02⩽R, as t⩾t0=t0(∥B∥ℰ0), we have
(3.26)∥z(t)∥ℰ0⩽μ0.
We complete the proof.


#### 3.2.2. A Priori Estimates inℰ1

Lemma 3.4.
Assuming thatz(t) is a strong solution of (2.10) with initial data z0∈B. If the nonlinearity g(u,t) satisfies (2.3)–(2.6), h0∈Lc*2(ℝτ;V),  h∈ℋ(h0), k>0, then there is a positive constant μ2 such that for any bounded (in  ℰ1) subset B, there exists t1=t1(∥B∥ℰ1) such that
(3.27)∥z(t)∥ℰ12=12(∥u∥22+∥ut∥12)⩽μ22,t⩾t1=t1(∥B∥ℰ1).Proof.
Now we will prove thatz=(u,ut) are bounded in ℰ1=D(A)×V.
We assume thatϱ is positive and satisfies
(3.28)0<ϱ(α-ϱ)<λ1.
Multiplying (2.10) by Av(t)=Aut(t)+ϱAu(t) and integrating over Ω, we have
(3.29)12ddt(∥v∥12+∥u∥22)+ϱ∥u∥22+(α-ϱ)∥v∥12-ϱ(α-ϱ)〈u,v〉1+k〈u+,Av〉+〈g(u,t),Av〉=〈h(t),Av〉,
where A=Δ2.
We can deduce that(3.30)ϱ(α-ϱ)〈u,v〉1⩽(α-ϱ)∥v∥124+(α-ϱ)ϱ2∥u∥12,〈h(t),Av〉⩽(α-ϱ)∥v∥124+∥h(t)∥12α-ϱ.
Then, substituting (3.30) into (3.29), we have
(3.31)ddt(∥v∥12+∥u∥22)+2ϱ∥u∥22+(α-ϱ)∥v∥12-2ϱ2(α-ϱ)∥u∥12+2k〈u+,Av〉+2〈g,Av〉⩽2∥h(t)∥12α-ϱ.
In view of (2.5) and Theorem 3.3, we can see that
(3.32)〈g,Av〉=〈g,Aut+ϱAu〉=ddt〈g(u,t),Au〉-〈gu(u,t)ut,Au〉-〈gs(u,t),Au〉+ϱ〈g(u,t),Au〉⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-∫Ω‍|gu(u,t)|·|ut|·|Au|dx-∫Ω‍|gs(u,t)|·|Au|dx⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-∫Ω‍C1(1+u|γ)·|ut|·|Au|dx-∫Ω‍C1(1+|u|y+1)·|Au|dx⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-C∥u∥2⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-ϱ8∥u∥22-C.
Exploiting ∥(u+)t∥⩽∥ut∥ and Theorem 3.3, we can obtain
(3.33)k〈u+,Av〉=〈ku+,Aut+ϱAu〉=ddtk〈u+,Au〉-k〈(u+)t,Au〉+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-k∥(u+)t∥·∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-k∥ut∥·∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-kμ0∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-ϱ8∥u∥22-C+ϱk〈u+,Au〉.
Consequently,
(3.34)ddt(∥v∥12+∥u∥22+2k〈u+,Au〉+2〈g(u,t),Au〉)+(α-ϱ)∥v∥12+2ϱλ1(34λ1-ϱ(α-ϱ))∥u∥22⩽+2ϱk〈u+,Au〉+2ϱ〈g(u,t),Au〉⩽2∥h(t)∥12α-ϱ+C.
Choose ϱ small enough such that
(3.35)32-2ϱ(α-ϱ)λ1⩾1.
This leads to
(3.36)ddt(∥v∥12+∥Au+ku++g(u,t)∥2)+(α-ϱ)∥v∥12+ϱ∥Au+ku++g(u,t)∥2⩽2∥h(t)∥12α-ϱ+C+2∫Ω‍g(u,t)(gu(u,t)ut+gs(u,t))dx+2k∫Ω‍u+·(u+)tdx+2k〈gu(u,t)ut+gs(u,t),u+〉+2k〈g(u,t),u+(u+)t〉+ϱ∥g(u)∥2+ϱk2∥u+∥2+2ϱk〈g(u,t),u+〉.
By (2.5), (2.9), the Hölder inequality, and Theorem 3.3, we have
(3.37)ddt(∥v∥12+∥Au+ku++g(u,t)∥2)+(α-ϱ)∥v∥12+ϱ∥Au+ku++g(u,t)∥2⩽2∥h(t)∥12α-ϱ+C.
We introduce the functional as follows:(3.38)𝒱(t)=∥v∥12+∥Au+ku++g(u,t)∥2,fort⩾τ.
Settingβ=min{α-ϱ,ϱ}, we define mh*(t)=∥h(t)∥12, then
(3.39)ddt𝒱(t)+β𝒱(t)⩽C+C5mh*(t),
where C5=2(α-ϱ)-1.
Analogous to the proof of Theorem3.3, we can estimate the integral and obtain
(3.40)𝒱(t)⩽𝒱(τ)e-β(t-τ)+Cβ-1(1-e-βt)+C5∫0t‍mh*(s)e-β(t-s)ds⩽𝒱(τ)e-β(t-τ)+Cβ-1+C5mh*(1+β-1),fort⩾τ,
where mh*=supt⩾τ∫tt+1‍mh*(s)ds.
Assuming that∥𝒱(τ)∥ℰ1⩽R, as t⩾t1=t1(∥B∥ℰ1), we have
(3.41)∥𝒱(t)∥ℰ1⩽μ12.
Applying (2.9), the Hölder inequality, the Cauchy inequality, and Theorem 3.3, we can deduce from (3.41) that
(3.42)∥Au∥2+∥ut∥12⩽μ22,
where μ2 depends on ϱ, α, k, ∥h∥12, μ0, and μ1.
We complete the proof.And then, combining Theorem3.2 with Lemma 3.4, we can get the result as follows.Theorem 3.5 (bounded uniformly absorbing set inℰ1).
Presuming thatg0∈Lc2(ℝτ;M) and h0∈Lc*2(ℝτ;V). Let g∈ℋ(g0) satisfy (2.3)–(2.6), h∈ℋ(h0), and {Uσ(t,τ)},  σ∈ℋ(σ0)=ℋ(g0)×ℋ(h0) be the family of processes corresponding to (2.10) in ℰ1, then {Uσ(t,τ)} has a uniformly (w.r.t. σ∈ℋ(σ0)) absorbing set B1=Bℰ1(0,μ2) in ℰ1. That is, for any bounded subset B⊂ℰ1, there exists t1=t1(∥B∥ℰ1) such that
(3.43)∪σ∈ℋ(σ0)Uσ(t,τ)B⊂B1,∀t⩾t1.


### 3.3. The Existence of Uniform Attractor

We will show the existence of uniform attractor to problem (2.10) in ℰ1.Theorem 3.6 (uniform attractor).
Let{Uσ(t,τ)} be the family of processes corresponding to problem (2.10). If g0∈Lc2(ℝτ;ℳ) satisfyies (2.3)–(2.6), h0∈Lc*2(ℝτ;V), and σ0=(g0,h0), then {Uσ(t,τ)} possesses a compact uniform (w.r.t. σ∈ℋ(σ0)) attractor 𝒜ℋ(σ0) in ℰ1, which attracts any bounded set in ℰ1 with ∥·∥ℰ1, satisfying
(3.44)𝒜ℋ(σ0)=ω0,ℋ(σ0)(B1)=ωτ,ℋ(σ0)(B1),
where B1 is the uniformly (w.r.t. σ∈ℋ(σ0)) absorbing set in ℰ1.Proof.
From Theorems2.4 and 3.5, we merely need to prove that the family of processes {Uσ(t,τ)}, σ∈ℋ(σ0) satisfies uniform (w.r.t. σ∈ℋ(σ0)) condition (C) in ℰ1. We assume that λ~i,  i=1,2,… are eigenvalue of operator A in D(A), satisfying
(3.45)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); at the same time they are also a group of canonical basis in V or H, and satisfy
(3.46)Aω~i=λ~iω~i,∀i∈ℕ.
LetVm=span{ω~1,ω~2,…,ω~m},  Pm:V→Vm is an orthogonal projector. For any (u,ut)∈ℰ1, we write
(3.47)(u,ut)=(u1,u1t)+(u2,u2t),
where (u1,u1t)=(Pmu,Pmut).
Choose0<ϱ<1, and 0<ϱ(α-ϱ)<(1/2)λ1. Taking the scalar product with Av2(t)=Au2t(t)+ϱAu2(t) for (2.10) in H, we have
(3.48)12ddt(∥v2∥12+∥u2∥22)+ϱ∥u2∥22-ϱ(α-ϱ)〈u,v2〉1+(α-ϱ)∥v2∥12+k〈u+,Av2〉+〈g(u,t),Av2〉=〈h(t),Av2〉,
where
(3.49)〈h(t),Av2〉⩽(α-ϱ)∥v2∥128+2(α-ϱ)-1∥(I-Pm)h(t)∥12,(3.50)-〈g(u,t),Av2〉⩽(α-ϱ)∥v2∥128+2(α-ϱ)-1∥(I-Pm)g(u,t)∥12.
Clearly, we can get that(3.51)ϱ(α-ϱ)〈u,v2〉1⩽(α-ϱ)∥v2∥124+(α-ϱ)ϱ2∥u2∥12,(3.52)k〈u+,Av2〉=〈ku+,Aut+ϱAu〉=ddtk〈u+,Au〉-k〈(u+)t,Au〉+ϱk〈u+,Au〉⩾12ddtk∥(u2)+∥12+ϱk∥(u2)+∥12-ϱ2∥u∥22-C.
Combining (3.49)–(3.52), we obtain from (3.48)
(3.53)12ddt(∥v2∥12+∥u2∥22+k∥(u2)+∥12)+12ϱ∥u2∥22+12(α-ϱ)∥v2∥12+ϱk∥(u2)+∥12-(α-ϱ)ϱ2∥u2∥12⩽12ddt(∥v2∥12+∥u2∥22+k∥(u2)+∥12)+ϱλ1-1(12λ1-(α-ϱ)ϱ)∥u2∥22+12(α-ϱ)∥v2∥12+ϱk∥(u2)+∥12⩽2(α-ϱ)-1∥(I-Pm)g(u,t)∥12+2(α-ϱ)-1∥(I-Pm)h(t)∥12⩽2C(α-ϱ)-1∥(I-Pm)g(u,t)∥ℳ2(1+∥u2∥12γ+2)+2(α-ϱ)-1∥(I-Pm)h(t)∥12.
We define the functional(3.54)ℒ(t)=12(∥v2∥12+∥u2∥22+k∥(u2)+∥12),
and set ω=min{2ϱλ1-1((1/2)λ1-(α-ϱ)ϱ),α-ϱ,2ϱ}, then
(3.55)ddtℒ(t)+ωℒ(t)⩽2C(α-ϱ)-1∥(I-Pm)g(u,t)∥ℳ2(1+(2μ1)2γ+2)+2(α-ϱ)-1∥(I-Pm)h(t)∥12,fort⩾t1.
By Gronwall Lemma, we obtain
(3.56)ℒ(t)⩽ℒ(t1)e-ω(t-t1)+2α-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)h(s)∥12ds+2Cα-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)g(u,s)∥ℳ2ds,fort⩾t1.
Obviously, there exists a constantC~, such that
(3.57)∥z2(t)∥ℰ12⩽ℒ(t)⩽C~∥z2(t)∥ℰ12,
so
(3.58)∥z2(t)∥ℰ12⩽C~∥z2(t1)∥ℰ12e-ω(t-t1)+2α-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)h(s)∥12ds+2Cα-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)g(u,s)∥ℳ2ds.
Sinceg∈Lc2(ℝτ,ℳ)⊂Lc*2(ℝτ,ℳ), h∈Lc*2(ℝτ,H), from Lemma 2.7, we can know for any ϵ1>0, there exists a constant m large enough such that
(3.59)2α-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)h(s)∥12ds⩽ϵ13,∀h∈ℋ(h0),2Cα-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)g(u,s)∥ℳ2ds⩽ϵ13,∀g∈ℋ(g0),
where t⩾τ.
Lett2=1/ωln(3C~μ12/ϵ1)+t1, then
(3.60)C~∥z2(t1)∥ℰ12e-ω(t-t1)⩽ϵ13,∀t⩾t2.
So for everyσ∈ℋ(σ0), we can get
(3.61)∥z2(t)∥ℰ12⩽ϵ1,∀t⩾t2,
where ∥z2(t)∥ℰ12=(1/2)(∥u2∥22+∥u2t∥12).
Therefore, the family of processesUσ(t,τ),  σ∈ℋ(σ0) satisfies uniformly (w.r.t. σ∈ℋ(σ0)) condition (C) in ℰ1. Applying Theorem 2.4, we can obtain the existence of uniform (w.r.t. σ∈ℋ(σ0)) attractor of the family of processes Uσ(t,τ),  σ∈ℋ(σ0) in ℰ1, which satisfies (3.44).
We complete the proof.


## 3.1. Existence and Uniqueness of Strong Solutions

At first, we give the concept of strong solutions for the initial-boundary value problem (2.10).Definition 3.1.
SetI=[τ,T], for T>τ⩾0. We suppose that k>0, h∈Lb2(ℝτ;V), g∈C1(ℝ×ℝ;ℝ) satisfying (2.3)–(2.6) and g(0,0)=0. The function z=(u,ut)∈L∞(I;ℰ1) is said to be a strong solution to problem (2.10) in the time interval I, with initial data z(τ)=zτ=(u1,u2)∈ℰ1, provided
(3.2)〈utt,v-〉+α〈ut,v-〉+∫Ω‍ΔuΔv-dx+∫Ω‍g(u,t)v-dx+k〈u+,v-〉=∫Ω‍h(x,t)v-dx,
for all v-∈V and a.e. t∈I.Then, by using the methods in [18] (Galerkin approximation method), we can get the following result about the existence and uniqueness of strong solutions.Theorem 3.2 (existence and uniqueness of strong solutions).
DefineI=[τ,T], for all T>τ. Let k>0, h∈Lb2(ℝτ;V), g∈C1(ℝ×ℝ;ℝ) satisfying (2.3)–(2.6). Then for any given zτ∈ℰ1, there is a unique solution z=(u,ut) for problem (2.10) in ℰ1. Furthermore, for i=1,2, let {zτi,hi}(zτi∈ℰ1  and  hi∈Lb2(ℝτ;V)) be two initial conditions, and denote by zi corresponding solutions to problem (2.10). Then the estimates hold as follows: for all τ⩽t⩽T+τ,
(3.3)∥z1(t)-z2(t)∥ℰ12⩽Q(∥zτi∥ℰ1,T)(∥zτ1-zτ2∥ℰ12+∥h1-h2∥Lb2(ℝτ;V)2).Thus, (2.10) will be written as an evolutionary system introduced z(t)=(u(t),ut(t)) and zτ=z(τ)=(u1,u2) for brevity, as ∥z∥ℰ12=(1/2)(∥u∥22+∥ut∥12), the system (2.10) can be written in the operator form
(3.4)∂tz=Aσ(t)(z),z|t=τ=zτ,
where σ(s)=(g(u,s),h(x,s)) is the symbol of (3.4). If zτ∈ℰ1, then problem (3.4) has a unique solution z(t)∈L∞(ℝτ,ℰ1). This implies that the process {Uσ(t,τ)} given by the formula Uσ(t,τ)zτ=z(t) is defined in ℰ1.Now we define the symbol space. A fixed symbolσ0(s)=(g0(u,s),h0(x,s)) can be given, where h0(x,s) is in Lc*2(ℝτ;V), the function g0(u,s)∈Lc2(ℝτ;ℳ) satisfying (2.3)–(2.6), and ℳ is a Banach space,
(3.5)ℳ={g∈C1(ℝ×ℝ,ℝ)∥g(u)∥1+∥gs(u)∥1∥u∥1γ+1+1+∥gu(u)∥1∥u∥1γ+1<∞},
endowed with the following norm:
(3.6)∥g∥ℳ=supu∈ℝ{∥g(u)∥1+∥gs(u)∥1∥u∥1γ+1+1+∥gu(u)∥1∥u∥1γ+1}.
Obviously, the function σ0(s)=(g0(u,s),h0(x,s)) is in Lc2(ℝτ;ℳ)×Lc*2(ℝτ;V). we define ℋ(σ0)=ℋ(g0)×ℋ(h0)=[g0(u,s+l)∣l∈ℝ]Lloc2,w(ℝτ;ℳ)×[h0(x,s+l)∣l∈ℝ]Lloc2,w(ℝτ;V), where [] denotes the closure of a set in topological space Lloc2,w(ℝτ;ℳ)  (or Lloc2,w(ℝτ;V)). So, if (g,h)∈ℋ(σ0), then g(u,t) and h(x,t) all satisfy condition (C*).Applying Propositions2.8 and 2.9 and Theorem 3.2, we can easily know that the family of processes {Uσ(t,τ)}:ℰ1→ℰ1,  σ∈ℋ(σ0), t⩾τ are defined. Furthermore, the translation semigroup {T(l)∣l∈ℝ+} satisfies that for all l∈ℝ+, T(l)ℋ(σ0)=ℋ(σ0), and the following translation identity:
(3.7)Uσ(t+l,τ+l)=UT(l)σ(t,τ),∀σ∈ℋ(σ0),fort⩾τ⩾0,l⩾0
holds.Then for anyσ∈ℋ(σ0), the problem (3.4) with σ instead of σ0 possesses a corresponding to process {Uσ(t,τ)} acting on ℰ1.Consequently, for eachσ∈ℋ(σ0), σ0(s)=(g0(u,s),h0(x,s)) (here h0(x,s)∈Lc*2(ℝτ;V), g0(u,s)∈Lc2(ℝτ;ℳ) satisfying (2.3)–(2.6)), we can define a process
(3.8)Uσ(t,τ):ℰ1→ℰ1,zτ=(u1,u2)→(u(t),ut(t))=Uσ(t,τ)zτ,
and {Uσ(t,τ)},  σ∈ℋ(σ0) is a family of processes on ℰ1.


## 3.2. A Priori Estimates

### 3.2.1. A Priori Estimates inℰ0

Theorem 3.3.
Assume thatz(t) is a solution of (2.10) with initial data z0∈B. If the nonlinearity g(u,t) satisfies (2.3)–(2.6), h0∈Lc*2(ℝτ;H),  h∈ℋ(h0), k>0, then there is a positive constant μ0 such that for any bounded (in  ℰ0) subset B, there exists t0=t0(∥B∥ℰ0) such that
(3.9)∥z(t)∥ℰ02=12(∥u∥12+∥ut∥2)⩽μ02,t⩾t0=t0(∥B∥ℰ0).Proof.
Now we will prove thatz=(u,ut) are bounded in ℰ0=V×H.
We assume thatϱ is positive and satisfies
(3.10)0<ϱ(α-ϱ)<λ1.
Multiplying (2.10) by v(t)=ut(t)+ϱu(t) and integrating over Ω, we have
(3.11)12ddt(∥v∥2+∥u∥12)+ϱ∥u∥12+(α-ϱ)∥v∥2-ϱ(α-ϱ)〈u,v〉+k〈u+,v〉+〈g(u,t),v〉=〈h(t),v〉.
We can easily see that(3.12)ϱ(α-ϱ)〈u,v〉⩽(α-ϱ)∥v∥24+(α-ϱ)ϱ2∥u∥2,(3.13)〈h(t),v〉⩽(α-ϱ)∥v∥24+∥h(t)∥2α-ϱ.
Then, substituting (3.12)-(3.13) into (3.11), we can obtain that
(3.14)ddt(∥v∥2+∥u∥12)+2ϱ∥u∥12+(α-ϱ)∥v∥2-2ϱ2(α-ϱ)∥u∥2+2k〈u+,v〉+2〈g,v〉⩽2∥h(t)∥2α-ϱ.
In view of (2.6) and (2.8), we can know
(3.15)〈g,v〉=〈g,ut+ϱu〉=ddt∫Ω‍G(u(x,t),t)dx+ϱ〈g(u,t),u〉-∫Ω‍Gs(u(x,t),t)dx=ddt∫Ω‍G(u(x,t),t)dx+ϱ∫Ω‍g(u(x,t),t)u(x,t)dx-ϱC0∫Ω‍G(u(x,t),t)dx+ϱC0∫Ω‍G(u(x,t),t)dx-∫Ω‍Gs(u(x,t),t)dx⩾ddt𝒢(u(x,t),t)+ϱC0𝒢(u(x,t),t)-ϱ(m∥u∥2+K2)-δ2𝒢(u(x,t),t)-C2|Ω|,k〈u+,v〉=12ddtk∥u+∥2+ϱk∥u+∥2.
Consequently,(3.16)ddt(∥v∥2+∥u∥12+k∥u+∥2+2𝒢(u(x,t),t))+(α-ϱ)∥v∥2+2ϱλ1(λ1-ϱ(α-ϱ)-m)∥u∥12+2ϱk∥u+∥2+(ϱC0-δ2)2𝒢(u(x,t),t)⩽2∥h(t)∥2α-ϱ+2(ϱK2+C2|Ω|).
We introduce the functional as follows:(3.17)y(t)=∥v∥2+∥u∥12+k∥u+∥2+2𝒢(u(x,t),t)+2K1,fort⩾τ.
Settingβ=min{α-ϱ,2ϱλ1-1(λ1-ϱ(α-ϱ)-m),2ϱ,ϱC0-δ2}, we choose proper positive constants m and δ, such that
(3.18)m<λ1-ϱ(α-ϱ),δ2<ϱC0
hold, then β>0.
We definemh(t)=∥h(t)∥2, then
(3.19)ddty(t)+βy(t)⩽C4+C5mh(t),
where C4=2(ϱK2+C2|Ω|)+2βK1, C5=2(α-ϱ)-1.
Analogous to the proof of Lemma 2.1.3 in [10], we can estimate the integral and obtain
(3.20)y(t)⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫0t‍mh(s)e-β(t-s)ds⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫t-1t‍mh(s)e-β(t-s)ds+C5∫t-2t-1‍mh(s)e-β(t-s)ds+⋯⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫t-1t‍mh(s)ds+C5e-β∫t-2t-1‍mh(s)ds+C5e-2β∫t-3t-2‍mh(s)ds+⋯⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5mh(1+e-β+e-2β+⋯)⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5mh(1+β-1)⩽y(τ)e-β(t-τ)+C4β-1+C5mh(1+β-1),fort⩾τ,
where mh=supt⩾τ∫tt+1‍mh(s)ds.
By virtue of (2.7), we can get
(3.21)2𝒢(u,t)⩾-2m∥u∥2-2K1⩾-2mλ1-1∥u∥12-2K1.
Choosing m⩽λ1/4, we obtain from (3.17)
(3.22)y(t)=∥u∥12+∥ut+ϱu∥2+k∥u+∥2+2𝒢(u,t)+2K1⩾12∥u∥12+∥ut+ϱu∥2+k∥u+∥2⩾∥z(t)∥ℰ02.
In consideration of (2.9) and 0<γ<∞, we can see
(3.23)2𝒢(uτ(x),τ)⩽2C3∫Ω‍(|uτ(x)|γ+2+1)dx⩽C6(∥uτ∥1γ+2+1),(3.24)y(τ)=∥u(τ)∥12+∥ut(τ)+ϱu(τ)∥2+k∥(u(τ))+∥2+2𝒢(u(τ),τ)+2K1⩽C7(∥z(τ)∥ℰ0γ+2+1).
Combining (3.20), (3.22), and (3.24), we can deduce that
(3.25)∥z(t)∥ℰ02⩽y(τ)e-β(t-τ)+C4β-1+C5mh(1+β-1)⩽C7(∥z(τ)∥ℰ0γ+2+1)e-β(t-τ)+C4β-1+C5mh(1+β-1)⩽C7∥z(τ)∥ℰ0γ+2e-β(t-τ)+C8,t⩾τ.
Assume that∥z(τ)∥ℰ02⩽R, as t⩾t0=t0(∥B∥ℰ0), we have
(3.26)∥z(t)∥ℰ0⩽μ0.
We complete the proof.


### 3.2.2. A Priori Estimates inℰ1

Lemma 3.4.
Assuming thatz(t) is a strong solution of (2.10) with initial data z0∈B. If the nonlinearity g(u,t) satisfies (2.3)–(2.6), h0∈Lc*2(ℝτ;V),  h∈ℋ(h0), k>0, then there is a positive constant μ2 such that for any bounded (in  ℰ1) subset B, there exists t1=t1(∥B∥ℰ1) such that
(3.27)∥z(t)∥ℰ12=12(∥u∥22+∥ut∥12)⩽μ22,t⩾t1=t1(∥B∥ℰ1).Proof.
Now we will prove thatz=(u,ut) are bounded in ℰ1=D(A)×V.
We assume thatϱ is positive and satisfies
(3.28)0<ϱ(α-ϱ)<λ1.
Multiplying (2.10) by Av(t)=Aut(t)+ϱAu(t) and integrating over Ω, we have
(3.29)12ddt(∥v∥12+∥u∥22)+ϱ∥u∥22+(α-ϱ)∥v∥12-ϱ(α-ϱ)〈u,v〉1+k〈u+,Av〉+〈g(u,t),Av〉=〈h(t),Av〉,
where A=Δ2.
We can deduce that(3.30)ϱ(α-ϱ)〈u,v〉1⩽(α-ϱ)∥v∥124+(α-ϱ)ϱ2∥u∥12,〈h(t),Av〉⩽(α-ϱ)∥v∥124+∥h(t)∥12α-ϱ.
Then, substituting (3.30) into (3.29), we have
(3.31)ddt(∥v∥12+∥u∥22)+2ϱ∥u∥22+(α-ϱ)∥v∥12-2ϱ2(α-ϱ)∥u∥12+2k〈u+,Av〉+2〈g,Av〉⩽2∥h(t)∥12α-ϱ.
In view of (2.5) and Theorem 3.3, we can see that
(3.32)〈g,Av〉=〈g,Aut+ϱAu〉=ddt〈g(u,t),Au〉-〈gu(u,t)ut,Au〉-〈gs(u,t),Au〉+ϱ〈g(u,t),Au〉⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-∫Ω‍|gu(u,t)|·|ut|·|Au|dx-∫Ω‍|gs(u,t)|·|Au|dx⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-∫Ω‍C1(1+u|γ)·|ut|·|Au|dx-∫Ω‍C1(1+|u|y+1)·|Au|dx⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-C∥u∥2⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-ϱ8∥u∥22-C.
Exploiting ∥(u+)t∥⩽∥ut∥ and Theorem 3.3, we can obtain
(3.33)k〈u+,Av〉=〈ku+,Aut+ϱAu〉=ddtk〈u+,Au〉-k〈(u+)t,Au〉+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-k∥(u+)t∥·∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-k∥ut∥·∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-kμ0∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-ϱ8∥u∥22-C+ϱk〈u+,Au〉.
Consequently,
(3.34)ddt(∥v∥12+∥u∥22+2k〈u+,Au〉+2〈g(u,t),Au〉)+(α-ϱ)∥v∥12+2ϱλ1(34λ1-ϱ(α-ϱ))∥u∥22⩽+2ϱk〈u+,Au〉+2ϱ〈g(u,t),Au〉⩽2∥h(t)∥12α-ϱ+C.
Choose ϱ small enough such that
(3.35)32-2ϱ(α-ϱ)λ1⩾1.
This leads to
(3.36)ddt(∥v∥12+∥Au+ku++g(u,t)∥2)+(α-ϱ)∥v∥12+ϱ∥Au+ku++g(u,t)∥2⩽2∥h(t)∥12α-ϱ+C+2∫Ω‍g(u,t)(gu(u,t)ut+gs(u,t))dx+2k∫Ω‍u+·(u+)tdx+2k〈gu(u,t)ut+gs(u,t),u+〉+2k〈g(u,t),u+(u+)t〉+ϱ∥g(u)∥2+ϱk2∥u+∥2+2ϱk〈g(u,t),u+〉.
By (2.5), (2.9), the Hölder inequality, and Theorem 3.3, we have
(3.37)ddt(∥v∥12+∥Au+ku++g(u,t)∥2)+(α-ϱ)∥v∥12+ϱ∥Au+ku++g(u,t)∥2⩽2∥h(t)∥12α-ϱ+C.
We introduce the functional as follows:(3.38)𝒱(t)=∥v∥12+∥Au+ku++g(u,t)∥2,fort⩾τ.
Settingβ=min{α-ϱ,ϱ}, we define mh*(t)=∥h(t)∥12, then
(3.39)ddt𝒱(t)+β𝒱(t)⩽C+C5mh*(t),
where C5=2(α-ϱ)-1.
Analogous to the proof of Theorem3.3, we can estimate the integral and obtain
(3.40)𝒱(t)⩽𝒱(τ)e-β(t-τ)+Cβ-1(1-e-βt)+C5∫0t‍mh*(s)e-β(t-s)ds⩽𝒱(τ)e-β(t-τ)+Cβ-1+C5mh*(1+β-1),fort⩾τ,
where mh*=supt⩾τ∫tt+1‍mh*(s)ds.
Assuming that∥𝒱(τ)∥ℰ1⩽R, as t⩾t1=t1(∥B∥ℰ1), we have
(3.41)∥𝒱(t)∥ℰ1⩽μ12.
Applying (2.9), the Hölder inequality, the Cauchy inequality, and Theorem 3.3, we can deduce from (3.41) that
(3.42)∥Au∥2+∥ut∥12⩽μ22,
where μ2 depends on ϱ, α, k, ∥h∥12, μ0, and μ1.
We complete the proof.And then, combining Theorem3.2 with Lemma 3.4, we can get the result as follows.Theorem 3.5 (bounded uniformly absorbing set inℰ1).
Presuming thatg0∈Lc2(ℝτ;M) and h0∈Lc*2(ℝτ;V). Let g∈ℋ(g0) satisfy (2.3)–(2.6), h∈ℋ(h0), and {Uσ(t,τ)},  σ∈ℋ(σ0)=ℋ(g0)×ℋ(h0) be the family of processes corresponding to (2.10) in ℰ1, then {Uσ(t,τ)} has a uniformly (w.r.t. σ∈ℋ(σ0)) absorbing set B1=Bℰ1(0,μ2) in ℰ1. That is, for any bounded subset B⊂ℰ1, there exists t1=t1(∥B∥ℰ1) such that
(3.43)∪σ∈ℋ(σ0)Uσ(t,τ)B⊂B1,∀t⩾t1.


## 3.2.1. A Priori Estimates inℰ0

Theorem 3.3.
Assume thatz(t) is a solution of (2.10) with initial data z0∈B. If the nonlinearity g(u,t) satisfies (2.3)–(2.6), h0∈Lc*2(ℝτ;H),  h∈ℋ(h0), k>0, then there is a positive constant μ0 such that for any bounded (in  ℰ0) subset B, there exists t0=t0(∥B∥ℰ0) such that
(3.9)∥z(t)∥ℰ02=12(∥u∥12+∥ut∥2)⩽μ02,t⩾t0=t0(∥B∥ℰ0).Proof.
Now we will prove thatz=(u,ut) are bounded in ℰ0=V×H.
We assume thatϱ is positive and satisfies
(3.10)0<ϱ(α-ϱ)<λ1.
Multiplying (2.10) by v(t)=ut(t)+ϱu(t) and integrating over Ω, we have
(3.11)12ddt(∥v∥2+∥u∥12)+ϱ∥u∥12+(α-ϱ)∥v∥2-ϱ(α-ϱ)〈u,v〉+k〈u+,v〉+〈g(u,t),v〉=〈h(t),v〉.
We can easily see that(3.12)ϱ(α-ϱ)〈u,v〉⩽(α-ϱ)∥v∥24+(α-ϱ)ϱ2∥u∥2,(3.13)〈h(t),v〉⩽(α-ϱ)∥v∥24+∥h(t)∥2α-ϱ.
Then, substituting (3.12)-(3.13) into (3.11), we can obtain that
(3.14)ddt(∥v∥2+∥u∥12)+2ϱ∥u∥12+(α-ϱ)∥v∥2-2ϱ2(α-ϱ)∥u∥2+2k〈u+,v〉+2〈g,v〉⩽2∥h(t)∥2α-ϱ.
In view of (2.6) and (2.8), we can know
(3.15)〈g,v〉=〈g,ut+ϱu〉=ddt∫Ω‍G(u(x,t),t)dx+ϱ〈g(u,t),u〉-∫Ω‍Gs(u(x,t),t)dx=ddt∫Ω‍G(u(x,t),t)dx+ϱ∫Ω‍g(u(x,t),t)u(x,t)dx-ϱC0∫Ω‍G(u(x,t),t)dx+ϱC0∫Ω‍G(u(x,t),t)dx-∫Ω‍Gs(u(x,t),t)dx⩾ddt𝒢(u(x,t),t)+ϱC0𝒢(u(x,t),t)-ϱ(m∥u∥2+K2)-δ2𝒢(u(x,t),t)-C2|Ω|,k〈u+,v〉=12ddtk∥u+∥2+ϱk∥u+∥2.
Consequently,(3.16)ddt(∥v∥2+∥u∥12+k∥u+∥2+2𝒢(u(x,t),t))+(α-ϱ)∥v∥2+2ϱλ1(λ1-ϱ(α-ϱ)-m)∥u∥12+2ϱk∥u+∥2+(ϱC0-δ2)2𝒢(u(x,t),t)⩽2∥h(t)∥2α-ϱ+2(ϱK2+C2|Ω|).
We introduce the functional as follows:(3.17)y(t)=∥v∥2+∥u∥12+k∥u+∥2+2𝒢(u(x,t),t)+2K1,fort⩾τ.
Settingβ=min{α-ϱ,2ϱλ1-1(λ1-ϱ(α-ϱ)-m),2ϱ,ϱC0-δ2}, we choose proper positive constants m and δ, such that
(3.18)m<λ1-ϱ(α-ϱ),δ2<ϱC0
hold, then β>0.
We definemh(t)=∥h(t)∥2, then
(3.19)ddty(t)+βy(t)⩽C4+C5mh(t),
where C4=2(ϱK2+C2|Ω|)+2βK1, C5=2(α-ϱ)-1.
Analogous to the proof of Lemma 2.1.3 in [10], we can estimate the integral and obtain
(3.20)y(t)⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫0t‍mh(s)e-β(t-s)ds⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫t-1t‍mh(s)e-β(t-s)ds+C5∫t-2t-1‍mh(s)e-β(t-s)ds+⋯⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5∫t-1t‍mh(s)ds+C5e-β∫t-2t-1‍mh(s)ds+C5e-2β∫t-3t-2‍mh(s)ds+⋯⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5mh(1+e-β+e-2β+⋯)⩽y(τ)e-β(t-τ)+C4β-1(1-e-βt)+C5mh(1+β-1)⩽y(τ)e-β(t-τ)+C4β-1+C5mh(1+β-1),fort⩾τ,
where mh=supt⩾τ∫tt+1‍mh(s)ds.
By virtue of (2.7), we can get
(3.21)2𝒢(u,t)⩾-2m∥u∥2-2K1⩾-2mλ1-1∥u∥12-2K1.
Choosing m⩽λ1/4, we obtain from (3.17)
(3.22)y(t)=∥u∥12+∥ut+ϱu∥2+k∥u+∥2+2𝒢(u,t)+2K1⩾12∥u∥12+∥ut+ϱu∥2+k∥u+∥2⩾∥z(t)∥ℰ02.
In consideration of (2.9) and 0<γ<∞, we can see
(3.23)2𝒢(uτ(x),τ)⩽2C3∫Ω‍(|uτ(x)|γ+2+1)dx⩽C6(∥uτ∥1γ+2+1),(3.24)y(τ)=∥u(τ)∥12+∥ut(τ)+ϱu(τ)∥2+k∥(u(τ))+∥2+2𝒢(u(τ),τ)+2K1⩽C7(∥z(τ)∥ℰ0γ+2+1).
Combining (3.20), (3.22), and (3.24), we can deduce that
(3.25)∥z(t)∥ℰ02⩽y(τ)e-β(t-τ)+C4β-1+C5mh(1+β-1)⩽C7(∥z(τ)∥ℰ0γ+2+1)e-β(t-τ)+C4β-1+C5mh(1+β-1)⩽C7∥z(τ)∥ℰ0γ+2e-β(t-τ)+C8,t⩾τ.
Assume that∥z(τ)∥ℰ02⩽R, as t⩾t0=t0(∥B∥ℰ0), we have
(3.26)∥z(t)∥ℰ0⩽μ0.
We complete the proof.


## 3.2.2. A Priori Estimates inℰ1

Lemma 3.4.
Assuming thatz(t) is a strong solution of (2.10) with initial data z0∈B. If the nonlinearity g(u,t) satisfies (2.3)–(2.6), h0∈Lc*2(ℝτ;V),  h∈ℋ(h0), k>0, then there is a positive constant μ2 such that for any bounded (in  ℰ1) subset B, there exists t1=t1(∥B∥ℰ1) such that
(3.27)∥z(t)∥ℰ12=12(∥u∥22+∥ut∥12)⩽μ22,t⩾t1=t1(∥B∥ℰ1).Proof.
Now we will prove thatz=(u,ut) are bounded in ℰ1=D(A)×V.
We assume thatϱ is positive and satisfies
(3.28)0<ϱ(α-ϱ)<λ1.
Multiplying (2.10) by Av(t)=Aut(t)+ϱAu(t) and integrating over Ω, we have
(3.29)12ddt(∥v∥12+∥u∥22)+ϱ∥u∥22+(α-ϱ)∥v∥12-ϱ(α-ϱ)〈u,v〉1+k〈u+,Av〉+〈g(u,t),Av〉=〈h(t),Av〉,
where A=Δ2.
We can deduce that(3.30)ϱ(α-ϱ)〈u,v〉1⩽(α-ϱ)∥v∥124+(α-ϱ)ϱ2∥u∥12,〈h(t),Av〉⩽(α-ϱ)∥v∥124+∥h(t)∥12α-ϱ.
Then, substituting (3.30) into (3.29), we have
(3.31)ddt(∥v∥12+∥u∥22)+2ϱ∥u∥22+(α-ϱ)∥v∥12-2ϱ2(α-ϱ)∥u∥12+2k〈u+,Av〉+2〈g,Av〉⩽2∥h(t)∥12α-ϱ.
In view of (2.5) and Theorem 3.3, we can see that
(3.32)〈g,Av〉=〈g,Aut+ϱAu〉=ddt〈g(u,t),Au〉-〈gu(u,t)ut,Au〉-〈gs(u,t),Au〉+ϱ〈g(u,t),Au〉⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-∫Ω‍|gu(u,t)|·|ut|·|Au|dx-∫Ω‍|gs(u,t)|·|Au|dx⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-∫Ω‍C1(1+u|γ)·|ut|·|Au|dx-∫Ω‍C1(1+|u|y+1)·|Au|dx⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-C∥u∥2⩾ddt〈g(u,t),Au〉+ϱ〈g(u,t),Au〉-ϱ8∥u∥22-C.
Exploiting ∥(u+)t∥⩽∥ut∥ and Theorem 3.3, we can obtain
(3.33)k〈u+,Av〉=〈ku+,Aut+ϱAu〉=ddtk〈u+,Au〉-k〈(u+)t,Au〉+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-k∥(u+)t∥·∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-k∥ut∥·∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-kμ0∥Au∥+ϱk〈u+,Au〉⩾ddtk〈u+,Au〉-ϱ8∥u∥22-C+ϱk〈u+,Au〉.
Consequently,
(3.34)ddt(∥v∥12+∥u∥22+2k〈u+,Au〉+2〈g(u,t),Au〉)+(α-ϱ)∥v∥12+2ϱλ1(34λ1-ϱ(α-ϱ))∥u∥22⩽+2ϱk〈u+,Au〉+2ϱ〈g(u,t),Au〉⩽2∥h(t)∥12α-ϱ+C.
Choose ϱ small enough such that
(3.35)32-2ϱ(α-ϱ)λ1⩾1.
This leads to
(3.36)ddt(∥v∥12+∥Au+ku++g(u,t)∥2)+(α-ϱ)∥v∥12+ϱ∥Au+ku++g(u,t)∥2⩽2∥h(t)∥12α-ϱ+C+2∫Ω‍g(u,t)(gu(u,t)ut+gs(u,t))dx+2k∫Ω‍u+·(u+)tdx+2k〈gu(u,t)ut+gs(u,t),u+〉+2k〈g(u,t),u+(u+)t〉+ϱ∥g(u)∥2+ϱk2∥u+∥2+2ϱk〈g(u,t),u+〉.
By (2.5), (2.9), the Hölder inequality, and Theorem 3.3, we have
(3.37)ddt(∥v∥12+∥Au+ku++g(u,t)∥2)+(α-ϱ)∥v∥12+ϱ∥Au+ku++g(u,t)∥2⩽2∥h(t)∥12α-ϱ+C.
We introduce the functional as follows:(3.38)𝒱(t)=∥v∥12+∥Au+ku++g(u,t)∥2,fort⩾τ.
Settingβ=min{α-ϱ,ϱ}, we define mh*(t)=∥h(t)∥12, then
(3.39)ddt𝒱(t)+β𝒱(t)⩽C+C5mh*(t),
where C5=2(α-ϱ)-1.
Analogous to the proof of Theorem3.3, we can estimate the integral and obtain
(3.40)𝒱(t)⩽𝒱(τ)e-β(t-τ)+Cβ-1(1-e-βt)+C5∫0t‍mh*(s)e-β(t-s)ds⩽𝒱(τ)e-β(t-τ)+Cβ-1+C5mh*(1+β-1),fort⩾τ,
where mh*=supt⩾τ∫tt+1‍mh*(s)ds.
Assuming that∥𝒱(τ)∥ℰ1⩽R, as t⩾t1=t1(∥B∥ℰ1), we have
(3.41)∥𝒱(t)∥ℰ1⩽μ12.
Applying (2.9), the Hölder inequality, the Cauchy inequality, and Theorem 3.3, we can deduce from (3.41) that
(3.42)∥Au∥2+∥ut∥12⩽μ22,
where μ2 depends on ϱ, α, k, ∥h∥12, μ0, and μ1.
We complete the proof.And then, combining Theorem3.2 with Lemma 3.4, we can get the result as follows.Theorem 3.5 (bounded uniformly absorbing set inℰ1).
Presuming thatg0∈Lc2(ℝτ;M) and h0∈Lc*2(ℝτ;V). Let g∈ℋ(g0) satisfy (2.3)–(2.6), h∈ℋ(h0), and {Uσ(t,τ)},  σ∈ℋ(σ0)=ℋ(g0)×ℋ(h0) be the family of processes corresponding to (2.10) in ℰ1, then {Uσ(t,τ)} has a uniformly (w.r.t. σ∈ℋ(σ0)) absorbing set B1=Bℰ1(0,μ2) in ℰ1. That is, for any bounded subset B⊂ℰ1, there exists t1=t1(∥B∥ℰ1) such that
(3.43)∪σ∈ℋ(σ0)Uσ(t,τ)B⊂B1,∀t⩾t1.


## 3.3. The Existence of Uniform Attractor

We will show the existence of uniform attractor to problem (2.10) in ℰ1.Theorem 3.6 (uniform attractor).
Let{Uσ(t,τ)} be the family of processes corresponding to problem (2.10). If g0∈Lc2(ℝτ;ℳ) satisfyies (2.3)–(2.6), h0∈Lc*2(ℝτ;V), and σ0=(g0,h0), then {Uσ(t,τ)} possesses a compact uniform (w.r.t. σ∈ℋ(σ0)) attractor 𝒜ℋ(σ0) in ℰ1, which attracts any bounded set in ℰ1 with ∥·∥ℰ1, satisfying
(3.44)𝒜ℋ(σ0)=ω0,ℋ(σ0)(B1)=ωτ,ℋ(σ0)(B1),
where B1 is the uniformly (w.r.t. σ∈ℋ(σ0)) absorbing set in ℰ1.Proof.
From Theorems2.4 and 3.5, we merely need to prove that the family of processes {Uσ(t,τ)}, σ∈ℋ(σ0) satisfies uniform (w.r.t. σ∈ℋ(σ0)) condition (C) in ℰ1. We assume that λ~i,  i=1,2,… are eigenvalue of operator A in D(A), satisfying
(3.45)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); at the same time they are also a group of canonical basis in V or H, and satisfy
(3.46)Aω~i=λ~iω~i,∀i∈ℕ.
LetVm=span{ω~1,ω~2,…,ω~m},  Pm:V→Vm is an orthogonal projector. For any (u,ut)∈ℰ1, we write
(3.47)(u,ut)=(u1,u1t)+(u2,u2t),
where (u1,u1t)=(Pmu,Pmut).
Choose0<ϱ<1, and 0<ϱ(α-ϱ)<(1/2)λ1. Taking the scalar product with Av2(t)=Au2t(t)+ϱAu2(t) for (2.10) in H, we have
(3.48)12ddt(∥v2∥12+∥u2∥22)+ϱ∥u2∥22-ϱ(α-ϱ)〈u,v2〉1+(α-ϱ)∥v2∥12+k〈u+,Av2〉+〈g(u,t),Av2〉=〈h(t),Av2〉,
where
(3.49)〈h(t),Av2〉⩽(α-ϱ)∥v2∥128+2(α-ϱ)-1∥(I-Pm)h(t)∥12,(3.50)-〈g(u,t),Av2〉⩽(α-ϱ)∥v2∥128+2(α-ϱ)-1∥(I-Pm)g(u,t)∥12.
Clearly, we can get that(3.51)ϱ(α-ϱ)〈u,v2〉1⩽(α-ϱ)∥v2∥124+(α-ϱ)ϱ2∥u2∥12,(3.52)k〈u+,Av2〉=〈ku+,Aut+ϱAu〉=ddtk〈u+,Au〉-k〈(u+)t,Au〉+ϱk〈u+,Au〉⩾12ddtk∥(u2)+∥12+ϱk∥(u2)+∥12-ϱ2∥u∥22-C.
Combining (3.49)–(3.52), we obtain from (3.48)
(3.53)12ddt(∥v2∥12+∥u2∥22+k∥(u2)+∥12)+12ϱ∥u2∥22+12(α-ϱ)∥v2∥12+ϱk∥(u2)+∥12-(α-ϱ)ϱ2∥u2∥12⩽12ddt(∥v2∥12+∥u2∥22+k∥(u2)+∥12)+ϱλ1-1(12λ1-(α-ϱ)ϱ)∥u2∥22+12(α-ϱ)∥v2∥12+ϱk∥(u2)+∥12⩽2(α-ϱ)-1∥(I-Pm)g(u,t)∥12+2(α-ϱ)-1∥(I-Pm)h(t)∥12⩽2C(α-ϱ)-1∥(I-Pm)g(u,t)∥ℳ2(1+∥u2∥12γ+2)+2(α-ϱ)-1∥(I-Pm)h(t)∥12.
We define the functional(3.54)ℒ(t)=12(∥v2∥12+∥u2∥22+k∥(u2)+∥12),
and set ω=min{2ϱλ1-1((1/2)λ1-(α-ϱ)ϱ),α-ϱ,2ϱ}, then
(3.55)ddtℒ(t)+ωℒ(t)⩽2C(α-ϱ)-1∥(I-Pm)g(u,t)∥ℳ2(1+(2μ1)2γ+2)+2(α-ϱ)-1∥(I-Pm)h(t)∥12,fort⩾t1.
By Gronwall Lemma, we obtain
(3.56)ℒ(t)⩽ℒ(t1)e-ω(t-t1)+2α-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)h(s)∥12ds+2Cα-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)g(u,s)∥ℳ2ds,fort⩾t1.
Obviously, there exists a constantC~, such that
(3.57)∥z2(t)∥ℰ12⩽ℒ(t)⩽C~∥z2(t)∥ℰ12,
so
(3.58)∥z2(t)∥ℰ12⩽C~∥z2(t1)∥ℰ12e-ω(t-t1)+2α-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)h(s)∥12ds+2Cα-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)g(u,s)∥ℳ2ds.
Sinceg∈Lc2(ℝτ,ℳ)⊂Lc*2(ℝτ,ℳ), h∈Lc*2(ℝτ,H), from Lemma 2.7, we can know for any ϵ1>0, there exists a constant m large enough such that
(3.59)2α-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)h(s)∥12ds⩽ϵ13,∀h∈ℋ(h0),2Cα-ϱ∫t1t‍e-ω(t-s)∥(I-Pm)g(u,s)∥ℳ2ds⩽ϵ13,∀g∈ℋ(g0),
where t⩾τ.
Lett2=1/ωln(3C~μ12/ϵ1)+t1, then
(3.60)C~∥z2(t1)∥ℰ12e-ω(t-t1)⩽ϵ13,∀t⩾t2.
So for everyσ∈ℋ(σ0), we can get
(3.61)∥z2(t)∥ℰ12⩽ϵ1,∀t⩾t2,
where ∥z2(t)∥ℰ12=(1/2)(∥u2∥22+∥u2t∥12).
Therefore, the family of processesUσ(t,τ),  σ∈ℋ(σ0) satisfies uniformly (w.r.t. σ∈ℋ(σ0)) condition (C) in ℰ1. Applying Theorem 2.4, we can obtain the existence of uniform (w.r.t. σ∈ℋ(σ0)) attractor of the family of processes Uσ(t,τ),  σ∈ℋ(σ0) in ℰ1, which satisfies (3.44).
We complete the proof.



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*Source: 714696-2012-12-13.xml*