# Regularity and Exponential Growth of Pullback Attractors for Semilinear Parabolic Equations Involving the Grushin Operator

Authors: Nguyen Dinh Binh
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

Considered here is the first initial boundary value problem for a semilinear degenerate parabolic equation involving the Grushin operator in a bounded domainΩ. We prove the regularity and exponential growth of a pullback attractor in the space S02(Ω)∩L2p−2(Ω) for the nonautonomous dynamical system associated to the problem. The obtained results seem to be optimal and, in particular, improve and extend some recent results on pullback attractors for reaction-diffusion equations in bounded domains.


---

## Body

## 1. Introduction

LetΩ be a bounded domain in ℝN1×ℝN2(N1,N2≥1), with smooth boundary ∂Ω. In this paper, we consider the following problem:
(1.1)ut-Gsu+f(u)=g(t,x),(t,x)∈Qτ,T=(τ,T]×Ω,u(x,t)=0,x∈∂Ω,t∈(τ,T],u(x,τ)=uτ(x),x∈Ω,
where
(1.2)Gsu=Δx1u+|x1|2sΔx2u,x=(x1,x2)∈Ω⊂RN1×RN2,s⩾0,
is the Grushin operator, uτ∈L2(Ω) is given, the nonlinearity f and the external force g satisfy the following conditions.(H1)
The nonlinearityf∈C1(ℝ,ℝ) satisfies(1.3)f(u)u≥C1|u|p-C2,p≥2,(1.4)|f′(u)|≤C3|u|p-2+C4,(1.5)f′(u)≥-l,
where ℓ, Ci, (i=1,2,3,4) are positive constants. Relation (1.3) and (1.4) imply that(1.6)α1|u|p-α2≤F(u)≤α3|u|p+α4,
where F(s)=∫0sf(τ)dτ, and αi(i=1,2,3,4) are positive constants.(H2)
g∈Wloc1,2(ℝ;L2(Ω)) satisfies(1.7)∫-∞teλ1s(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds<+∞,∀t∈R,
where λ1 is the first eigenvalue of the operator -Gs in Ω with the homogeneous Dirichlet boundary condition.The Grushin operatorGs was first introduced in [1]. Noting that if s>0, then Gs is not elliptic in domains of ℝN1×ℝN2 which intersect the hyperplane {x1=0}. In the last few years, the existence and long-time behavior of solutions to parabolic equations involving the Grushin operator have been studied widely in both autonomous and nonautonomous cases (see, e.g., [2–7]). In particular, the existence of a pullback attractor in S01(Ω)∩Lp(Ω) for the process associated to problem (1.1) is considered in [2].In this paper we continue the study in the paper [2]. First, we will prove the existence of pullback attractors in S02(Ω) (see Section 2 for its definition) and L2p-2(Ω). As we know, if the external force g is only in L2(Ω), then solutions of problem (1.1) are at most in L2p-2(Ω)∩S02(Ω) and have no higher regularity. Therefore, there are no compact embedding results that hold for this case. To overcome the difficulty caused by the lack of embedding results, we exploit the asymptotic a prioriestimate method which was initiated in [8, 9] for autonomous equations and developed recently for nonautonomous equations in the case of pullback attractors in [10]. Noting that, to prove the existence of pullback attractors in S01(Ω)∩Lp(Ω), we only need assumption (H2) of the external force g; however, to prove the existence of pullback attractors in S02(Ω) and L2p-2(Ω), we need an additional assumption of g, namely, (3.18) in Section 3. Next, following the general lines of the approach in [11], we give exponential growth conditions in S02(Ω)∩L2p-2(Ω) for the pullback attractors. It is noticed that, as far as we know, the best known results on the pullback attractors for nonautonomous reaction-diffusion equations are the boundedness and exponential growth in H2(Ω) of the pullback attractors [11, 12]. Therefore, the obtained results seem to be optimal and, in particular when s=0, improve the recent results on pullback attractors for the nonautonomous reaction-diffusion equations in [11–15].The content of the paper is as follows. In Section2, for the convenience of the reader, we recall some concepts and results on function spaces and pullback attractors which we will use. In Section 3, we prove the existence of pullback attractors in the spaces S02(Ω) and L2p-2(Ω) by using the asymptotic a prioriestimate method. In Section 4, under additional assumptions of g, an exponential growth in S02(Ω)∩L2p-2(Ω) for the pullback attractors is deduced.


## 2. Preliminaries

### 2.1. Operator and Function Spaces

In order to study the boundary value problem for equations involving the Grushin operator, we have usually used the natural energy spaceS01(Ω) defined as the completion of C0∞(Ω) in the following norm:
(2.1)‖u‖S01(Ω)=(∫Ω(|∇x1u|2+|x1|2s|∇x2u|2)dx)1/2and the scalar product(2.2)((u,v)):=(∫Ω(∇x1u∇x1v+|x1|2s∇x2u∇x2v)dx)1/2.The following lemma comes from [16].Lemma 2.1.
Assume thatΩ is a bounded domain in ℝN1×ℝN2(N1,N2≥0). Then the following embeddings hold:(i)
S01(Ω)↪L2α*(Ω) continuously;(ii)
S01(Ω)↪Lp(Ω) compactly if p∈[1,2s*), where2s*=2N(s)/(N(s)-2), N(s)=N1+(s+1)N2.Now, we introduce the spaceS02(Ω) defined as the closure of C0∞(Ω) with the norm
(2.3)‖u‖S02(Ω)=(∫Ω(|Δx1u|2+|x1|2s|Δx2u|2)dx)1/2=(∫Ω|Gsu|2dx)1/2.The following lemma comes directly from the definitions ofS01(Ω) and S02(Ω).Lemma 2.2.
Assume thatΩ is a bounded domain in ℝN1×ℝN2(N1,N2≥0), with smooth boundary ∂Ω. Then S02(Ω)⊂S01(Ω) continuously.It is known that (see, e.g., [3]) for the operator A=-Gs, there exist {ej}j≥1such that(2.4)(ej,ek)=δjk,Aej=λjej,j,k=1,2,…,0<λ1≤λ2≤λ3≤⋯,λj⟶+∞asj⟶∞,and{ej}j≥1 is a complete orthonormal system in L2(Ω).


### 2.2. Pullback Attractors

LetX be a Banach space with the norm ∥·∥. ℬ(X) denotes all bounded sets of X. The Hausdorff semidistance between A and B is defined by(2.5)dist⁡(A,B)=sup⁡x∈Ainf⁡y∈B‖x-y‖.Let{U(t,τ):t≥τ,τ∈ℝ} be a process in X, that is, U(t,τ):X→X such that U(τ,τ)=Id and U(t,s)U(s,τ)=U(t,τ) for all t≥s≥τ, τ∈ℝ. The process {U(t,τ)} is said to be norm-to-weak continuous if U(t,τ)xn⇀U(t,τ)x, as xn→x in X, for all t≥τ, τ∈ℝ. The following result is useful for proving the norm-to-weak continuity of a process.Proposition 2.3 (see [9]).
LetX,Y be two Banach spaces, and let X*,Y* be, respectively, their dual spaces. Suppose that X is dense in Y, the injection i:X→Y is continuous, and its adjoint i*:Y*→X* is dense, and {U(t,τ)} is a continuous or weak continuous process on Y. Then {U(t,τ)} is norm-to-weak continuous on X if and only if for t≥τ, τ∈ℝ, U(t,τ) maps compact sets of X into bounded sets of X.Definition 2.4.
The process{U(t,τ)} is said to be pullback asymptotically compact if for any t∈ℝ, any D∈ℬ(X), any sequence τn→-∞, and any sequence {xn}⊂D, the sequence {U(t,τn)xn} is relatively compact in X.Definition 2.5.
A family of bounded setsℬ={B(t):t∈ℝ}⊂X is called a pullback absorbing set for the process {U(t,τ)} if for any t∈ℝ and any D∈ℬ(X), there exist τ0=τ0(D,t)≤t and B(t)∈ℬ such that
(2.6)⋃τ≤τ0U(t,τ)D⊂B(t).Definition 2.6.
The family𝒜={A(t):t∈ℝ}⊂ℬ(X) is said to be a pullback attractor for {U(t,τ)} if (1)
A(t) is compact for all t∈ℝ,(2)
𝒜 is invariant, that is,
(2.7)U(t,τ)A(τ)=A(t),∀t≥τ,(3)
𝒜 is pullback attracting, that is,
(2.8)lim⁡τ→-∞dist⁡(U(t,τ)D,A(t))=0,∀D∈B(X),andallt∈R,(4)
if{C(t):t∈ℝ} is another family of closed pullback attracting sets, then A(t)⊂C(t), for all t∈ℝ.Theorem 2.7 (see [13]).
Let{U(t,τ)} be a norm-to-weak continuous process which is pullback asymptotically compact. If there exists a pullback absorbing set ℬ={B(t):t∈ℝ}, then {U(t,τ)} has a unique pullback attractor 𝒜={A(t):t∈ℝ} and
(2.9)A(t)=⋂s≤t⋃τ≤sU(t,τ)B(τ)¯.In the rest of the paper, we denote by|·|2, (·,·) the norm and inner product in L2(Ω), respectively, and by |·|p the norm in Lp(Ω). By ∥·∥ we denote the norm in S01(Ω). For a Banach space E, ∥·∥E will be the norm. We also denote by C an arbitrary constant, which is different from line to line, and even in the same line.


## 2.1. Operator and Function Spaces

In order to study the boundary value problem for equations involving the Grushin operator, we have usually used the natural energy spaceS01(Ω) defined as the completion of C0∞(Ω) in the following norm:
(2.1)‖u‖S01(Ω)=(∫Ω(|∇x1u|2+|x1|2s|∇x2u|2)dx)1/2and the scalar product(2.2)((u,v)):=(∫Ω(∇x1u∇x1v+|x1|2s∇x2u∇x2v)dx)1/2.The following lemma comes from [16].Lemma 2.1.
Assume thatΩ is a bounded domain in ℝN1×ℝN2(N1,N2≥0). Then the following embeddings hold:(i)
S01(Ω)↪L2α*(Ω) continuously;(ii)
S01(Ω)↪Lp(Ω) compactly if p∈[1,2s*), where2s*=2N(s)/(N(s)-2), N(s)=N1+(s+1)N2.Now, we introduce the spaceS02(Ω) defined as the closure of C0∞(Ω) with the norm
(2.3)‖u‖S02(Ω)=(∫Ω(|Δx1u|2+|x1|2s|Δx2u|2)dx)1/2=(∫Ω|Gsu|2dx)1/2.The following lemma comes directly from the definitions ofS01(Ω) and S02(Ω).Lemma 2.2.
Assume thatΩ is a bounded domain in ℝN1×ℝN2(N1,N2≥0), with smooth boundary ∂Ω. Then S02(Ω)⊂S01(Ω) continuously.It is known that (see, e.g., [3]) for the operator A=-Gs, there exist {ej}j≥1such that(2.4)(ej,ek)=δjk,Aej=λjej,j,k=1,2,…,0<λ1≤λ2≤λ3≤⋯,λj⟶+∞asj⟶∞,and{ej}j≥1 is a complete orthonormal system in L2(Ω).


## 2.2. Pullback Attractors

LetX be a Banach space with the norm ∥·∥. ℬ(X) denotes all bounded sets of X. The Hausdorff semidistance between A and B is defined by(2.5)dist⁡(A,B)=sup⁡x∈Ainf⁡y∈B‖x-y‖.Let{U(t,τ):t≥τ,τ∈ℝ} be a process in X, that is, U(t,τ):X→X such that U(τ,τ)=Id and U(t,s)U(s,τ)=U(t,τ) for all t≥s≥τ, τ∈ℝ. The process {U(t,τ)} is said to be norm-to-weak continuous if U(t,τ)xn⇀U(t,τ)x, as xn→x in X, for all t≥τ, τ∈ℝ. The following result is useful for proving the norm-to-weak continuity of a process.Proposition 2.3 (see [9]).
LetX,Y be two Banach spaces, and let X*,Y* be, respectively, their dual spaces. Suppose that X is dense in Y, the injection i:X→Y is continuous, and its adjoint i*:Y*→X* is dense, and {U(t,τ)} is a continuous or weak continuous process on Y. Then {U(t,τ)} is norm-to-weak continuous on X if and only if for t≥τ, τ∈ℝ, U(t,τ) maps compact sets of X into bounded sets of X.Definition 2.4.
The process{U(t,τ)} is said to be pullback asymptotically compact if for any t∈ℝ, any D∈ℬ(X), any sequence τn→-∞, and any sequence {xn}⊂D, the sequence {U(t,τn)xn} is relatively compact in X.Definition 2.5.
A family of bounded setsℬ={B(t):t∈ℝ}⊂X is called a pullback absorbing set for the process {U(t,τ)} if for any t∈ℝ and any D∈ℬ(X), there exist τ0=τ0(D,t)≤t and B(t)∈ℬ such that
(2.6)⋃τ≤τ0U(t,τ)D⊂B(t).Definition 2.6.
The family𝒜={A(t):t∈ℝ}⊂ℬ(X) is said to be a pullback attractor for {U(t,τ)} if (1)
A(t) is compact for all t∈ℝ,(2)
𝒜 is invariant, that is,
(2.7)U(t,τ)A(τ)=A(t),∀t≥τ,(3)
𝒜 is pullback attracting, that is,
(2.8)lim⁡τ→-∞dist⁡(U(t,τ)D,A(t))=0,∀D∈B(X),andallt∈R,(4)
if{C(t):t∈ℝ} is another family of closed pullback attracting sets, then A(t)⊂C(t), for all t∈ℝ.Theorem 2.7 (see [13]).
Let{U(t,τ)} be a norm-to-weak continuous process which is pullback asymptotically compact. If there exists a pullback absorbing set ℬ={B(t):t∈ℝ}, then {U(t,τ)} has a unique pullback attractor 𝒜={A(t):t∈ℝ} and
(2.9)A(t)=⋂s≤t⋃τ≤sU(t,τ)B(τ)¯.In the rest of the paper, we denote by|·|2, (·,·) the norm and inner product in L2(Ω), respectively, and by |·|p the norm in Lp(Ω). By ∥·∥ we denote the norm in S01(Ω). For a Banach space E, ∥·∥E will be the norm. We also denote by C an arbitrary constant, which is different from line to line, and even in the same line.


## 3. Existence of Pullback Attractors inS02(Ω)∩L2p-2(Ω)

It is well known (see, e.g., [2] or [14]) that under conditions (H1)-(H2), problem (1.1) defines a process(3.1)U(t,τ):L2(Ω)⟶S01(Ω)∩Lp(Ω),∀t≥τ,where U(t,τ)uτ is the unique weak solution of (1.1) with initial datum uτ at time τ. The process {U(t,τ)} has a pullback attractor in S01(Ω)∩Lp(Ω).In this section, we will prove that the pullback attractor is in fact inS02(Ω)∩L2p-2(Ω).Lemma 3.1.
Assuming thatf and g satisfy (H 1)-(H 2), u(t) is a weak solution of (1.1). Then the following inequality holds for t>τ:
(3.2)‖u‖2+|u|pp≤C(e-λ1(t-τ)|uτ|22+1+e-λ1t∫-∞teλ1s|g(s)|22ds),
where C is a positive constant.Proof.
Multiplying (1.1) by u and then integrating over Ω, we get
(3.3)12ddt|u|22+‖u‖2+∫Ωf(u)udx=∫Ωg(t)udx≤1λ1|g(t)|22+λ14|u|22.
Using hypothesis (H1) and the inequality ∥u∥2≥λ1|u|22, we have
(3.4)ddt|u|22+λ1|u|22+C(‖u‖2+|u|pp)≤C(1+|g(t)|22).
Letting F(s)=∫0sf(τ)dτ, by (H1), we have
(3.5)α1|u|p-α2≤F(u)≤α3|u|p+α4.
Now multiplying (3.4) by eλ1t and using (3.5), we get
(3.6)ddt(eλ1t|u(t)|22)+Ceλ1t(‖u‖2+2∫ΩF(u(t))dx)≤Ceλ1t(1+|g(t)|22).
Integrating (3.6) from τ to s∈[τ,t-1] and s to s+1, respectively, we obtain
(3.7)eλ1s|u(s)|22≤eλ1τ|uτ|22+Ceλ1s+C∫τseλ1r|g(r)|22dr,∀s∈[τ,t-1],(3.8)C∫ss+1eλ1r(‖u(r)‖2+2∫ΩF(u(x,r))dx)dr≤eλ1s|u(s)|22+C∫ss+1eλ1r(1+|g(r)|22)dr≤eλ1τ|uτ|22+Ceλ1s+C∫τseλ1r|g(r)|22dr+Ceλ1(s+1)+C∫ss+1eλ1r|g(r)|22dr≤C(eλ1τ|uτ|22+eλ1t+∫τteλ1r|g(r)|22dr).
Multiplying (1.1) by ut and integrating over Ω, we have
(3.9)|ut(s)|22+12dds(‖u(s)‖2+2∫ΩF(u(x,s))dx)=∫Ωg(s)ut(s)≤12|g(s)|22+12|ut(s)|22.
Thus
(3.10)eλ1s|ut(s)|22+dds[eλ1s(‖u(s)‖2+2∫ΩF(u(x,s))dx)]≤λ1eλ1s(‖u(s)‖2+2∫ΩF(u(x,s))dx)+eλ1s|g(s)|22.
Combining (3.8) and (3.10), and using the uniform Gronwall inequality, we have
(3.11)eλ1t(‖u(t)‖2+2∫ΩF(u(x,t))dx)≤C(eλ1τ|uτ|2+eλ1t+∫-∞teλ1s|g(s)|22ds).
Using (H1) once again and thanks to eλ1τ|uτ|22→0 as τ→-∞, we get the desired result from (3.11).Lemma 3.2.
Assume that (H 1), (H 2) hold. Then for anyt∈ℝ and any D⊂L2(Ω) that is bounded, there exists τ0≤t-1 such that
(3.12)|ut(t)|22≤C(1+e-λ1t∫-∞teλ1s(|g(s)|22+|g′(s)|22)ds),
for any τ≤τ0 and any uτ∈D, where ut(s)=(d/dt)(U(t,τ)uτ)|t=s.Proof.
Integrating (3.10) from r to r+1,r∈[τ,t-1] and using (3.8) and (3.11), in particular we find
(3.13)∫rr+1eλ1s|ut|22ds≤eλ1r(‖u(r)‖2+2∫ΩF(u(x,r))dx)+λ1∫rr+1eλ1s(‖u(s)‖2+2∫ΩF(u(x,s))dx)ds+∫rr+1eλ1s|g(s)|22ds≤C(eλ1τ|uτ|22+eλ1t+∫-∞teλ1s|g(s)|22ds).
On the other hand, differentiating (1.1) and denoting  v=ut, we have
(3.14)vt-Gsv+f′(u)v=g′(t).
Taking the inner product of (3.14) with v in L2(Ω), we get
(3.15)12ddt|v|22+‖v‖2+(f′(u)v,v)=(g′(t),v).
Using (1.5) and Young’s inequality, after a few computations, we see that
(3.16)ddt(eλ1t|v|22)+2eλ1t‖v‖2≤Ceλ1t|v|22+Ceλ1t|g′(t)|22.
Combining (3.16) and (3.13) and using the uniform Gronwall inequality, we obtain
(3.17)eλ1t|v(t)|22≤C(eλ1τ|uτ|22+eλ1t+∫-∞teλ1s(|g(s)|22+|g′(s)|22)ds).
The proof is now complete because eλ1τ|uτ|22→0 as τ→-∞.


### 3.1. Existence of a Pullback Attractor inL2p-2(Ω)

In this section, following the general lines of the method introduced in [9], we prove the existence of a pullback attractor in L2p-2(Ω). In order to do this, we need an additional condition of g(3.18)∫-∞teλ1t‖g′(t)‖Lm(Ω)m′dt<+∞,∀t∈R,
where m, m′ are defined as in (3.30).Lemma 3.3.
The process{U(t,τ)} associated to problem (1.1) has a pullback absorbing set in L2p-2(Ω).Proof.
Multiplying (1.1) by |u|p-2u and integrating over Ω, we get
(3.19)∫Ωut|u|p-2udx+(p-1)∫Ω(|∇x1u|2+|x1|2s|∇x2u|2)|u|p-2dx+∫Ωf(u)||u|p-2udx|=∫Ω|g(t)u|p-2udx.
From (1.3) and the fact that Lp(Ω)⊂Lp-2(Ω) continuously, we have
(3.20)∫Ωf(u)|u|p-2udx≥∫Ω(C1|u|p-C2)|u|p-2dx≥C1‖u‖L2p-2(Ω)2p-2-C2|u|pp.
On the other hand, by Cauchy’s inequality, we see that
(3.21)|∫Ωut|u|p-2udx|≤C14‖u‖L2p-2(Ω)2p-2+1C1|ut|22,(3.22)|∫Ωg(t)|u|p-2udx|≤C14‖u‖L2p-2(Ω)2p-2+14|g(t)|22.
Combining (3.19)–(3.22) imply that
(3.23)‖u‖L2p-2(Ω)2p-2≤C(|ut|22+|u|pp+|g(t)|22).
Applying (3.2) and Lemma 3.2, we conclude the existence of a pullback absorbing set in L2p-2(Ω) for the process U(t,τ).Lemma 3.4.
For anys∈ℝ, any 2≤p<∞, and any bounded set B⊂L2(Ω), there exists τ0 such that
(3.24)∫Ω|ut(s)|pdx≤M,∀τ≤τ0,uτ∈B,
where M depends on s, p but not on B, and ut(s)=(d/dt)(U(t,τ)uτ)|t=s.Proof.
We will prove the lemma by induction argument. Lettingβ=N(s)/(N(s)-2)>1 and denoting v=ut we prove that for k=0,1,2,…, there exist τk and Mk(s) such that
(Pk)eλ1s∫Ω|v(s)|2βkdx≤Mk(s)foranyuτ∈B,τ≤τk,(Qk)∫ss+1(eλ1r∫Ω|v(r)|2βk+1dx)1/βdr≤Mk(s)foranyuτ∈B,τ≤τk,
where τk depends on k and B and Mk depends only on k.
Fork=0, we have (P0) from (3.17). Integrating (3.16) and using S01(Ω)↪L2β(Ω) continuously, we get (Q0).
Assuming that (Pk), (Qk) hold, we prove so are (Pk+1) and (Qk+1). Multiplying (3.14) by |v|2βk+1-2v and integrating over Ω, we obtain(3.25)Cddt∫Ω|v|2βk+1dx+C∫Ω(|∇x1v|2+|x1|2s|∇x2v|2)|v|β2k+1-2dx≤l∫Ω|v|2βk+1dx+(g′(t),|v|2βk+1-2v).
Using the imbedding S01(Ω)↪L2β(Ω) once again, we get
(3.26)∫Ω(|∇x1v|2+|x1|2s|∇x2v|2)|v|β2k+1dx≥∫Ω|v|2|v|2β2k+1-2dx=‖vβk+1‖L2β(Ω)2=(∫Ω|v|2βk+2dx)1/β.
Combining Holder’s and Young’s inequalities, we see that
(3.27)∫Ωg′(t)|v|2βk+1-2vdx≤(∫Ω|g′(t)|mdx)1/m(∫Ω|v|(2βk+1-1)ndx)1/n≤(∫Ω|g′(t)|mdx)m′/mm′+(∫Ω|v|(2βk+1-1)ndx)n′/nn′,
where 1/m+1/n=1/m′+1/n′=1. Choose n, n′ such that
(3.28)(2βk+1-1)n=2βk+2,n′n=1β,
thus
(3.29)n=2βk+22βk+1-1,n′=2βk+12βk+1-1.
Hence
(3.30)m=nn-1=2βk+1-12βk+2-2βk+1+1,m′=2βk+1.
Then from (3.27), we infer that
(3.31)∫Ωg′(t)|v|2βk+1-2vdx≤1m′‖g′(t)‖Lm(Ω)m′+1n′(∫Ω|v|2βk+2dx)1/β.
Applying (3.26) and (3.31) in (3.25), we find that
(3.32)ddt(eλ1t∫Ω|v|2βk+1dx)+Ceλ1t(∫Ω|v|2βk+2dx)1/β≤Ceλ1t∫Ω|v|2βk+1dx+Ceλ1t‖g′(t)‖Lm(Ω)m′.
Combining (Qk) and (3.32), using the uniform Gronwall inequality and taking into account assumption (3.18), we get (Pk+1). On the other hand, integrating (3.32) from t to t+1, we find (Qk+1). Now since β>1, and taking k≥logβp/2, we get the desired estimate.We will use the following lemma.Lemma 3.5 (see [15]).
If there existsσ>0 such that ∫-∞teσs|φ(s)|2ds<∞, for all t∈ℝ, then
(3.33)lim⁡γ→+∞∫-∞te-γ(t-s)|φ(s)|2ds=0,t∈R.LetHm=span{e1,e2,…,em} in L2(Ω), and let Pm:L2(Ω)→Hm be the orthogonal projection, where {ei}i=1∞ are the eigenvectors of operator A=-Gs. For any u∈L2(Ω), we write(3.34)u=Pmu+(I-Pm)u=u1+u2.Lemma 3.6.
For anyt∈ℝ, any B⊂L2(Ω) and any ɛ, there exist τ0(t,B,ɛ) and m0∈ℕ such that
(3.35)|(I-Pm)v|22<ɛ,∀τ≤τ0,∀uτ∈B,m≥m0.Proof.
Multiplying (3.14) by v2=(I-Pm)v and then integrating over Ω, using |∇v2|22≥λm|v2|22 and Cauchy’s inequality we get
(3.36)ddt|v2|22+λm|v2|22≤C∫Ω|f′(u)v|2dx+C|g′(t)|22.
We multiply (3.36) by eλmt and use assumption (1.4). We get
(3.37)ddt(eλmt|v2|22)≤Ceλmt∫Ω|u|2(p-2)|v|2dx+Ceλmt|g′(t)|22.
Integrating (3.37) from s to t,
(3.38)eλmt|v2(t)|22≤eλms|v2(s)|22+C∫steλmr∫Ω|u|2(p-2)|v|2dxdr+C∫steλmr|g′(r)|22dr≤eλms|v(s)|22+C∫-∞teλmr∫Ω|u|2(p-2)|v|2dxdr+C∫-∞teλmr|g′(r)|22dr.
Now integrating (3.38) with respect to s from τ to t, we infer that(3.39)(t-τ)eλmt|v2(t)|22≤∫τteλmr|v(r)|22dr+C(t-τ)∫-∞teλmr∫Ω|u|2(p-2)|v|2dxdr+C(t-τ)∫-∞teλmr|g′(r)|22dr.
Thus
(3.40)|v2(t)|22≤1t-τ∫-∞te-λm(t-r)|v(r)|22dr+C∫-∞te-λm(t-r)∫Ω|u|2(p-2)|v|2dxdr+C∫-∞te-λm(t-r)|g′(r)|22dr.
By Lemma 3.5 and since λm→+∞ as m→+∞, there exist τ1 and m1 such that
(3.41)1t-τ∫-∞te-λm(t-r)C∫-∞te-λm(t-r)|g′(r)|22dr<ɛ3,|v(r)|22dr<ɛ3,
for all τ≤τ1 and m≥m1. For the second term of the right-hand side of (3.40), using Holder’s inequality we have
(3.42)∫-∞te-λm(t-r)∫Ω|u|2(p-2)|v|2dxdr≤∫-∞t(∫Ωe((-p-1)/(p-2)λm)(t-r)|u|2p-2dx)(p-2)/(p-1)(∫Ωe-(p-1)λm(t-r)|v|2p-2dx)1/(p-1)dr≤(∫-∞te((-p-1)/(p-2)λm)(t-r)‖u‖L2p-2(Ω)2p-2dr)(p-2)/(p-1)(∫-∞te-(p-1)λm(t-r)dr∫Ω|v|2p-2dx)1/(p-1).
From Lemmas (3.5)–(3.7), we see that there exist τ2 and m2∈ℕ such that
(3.43)C∫-∞te-λm(t-r)∫Ω|u|2(p-2)|v|2dxdr<ɛ3,∀τ≤τ0,m≥m2.
Let τ0=min{τ1,τ2} and m0=max{m1,m2}, from (3.40), taking into account (3.41) and (3.43), we obtain (3.35).Lemma 3.7 (see [9]).
LetB be a bounded subset in Lq(Ω)(q≥1). If B has a finite ɛ-net in Lq(Ω), then there exists an M=M(B,ɛ), such that for any u∈B, the following estimate is valid:
(3.44)∫Ω(|u|≥M)|u|qdx<ɛ.Using Lemma3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set {ut(s):s≤t,uτ∈B} has a finite ϵ-net in L2(Ω). Therefore, we get the following result.Lemma 3.8.
For anyt∈ℝ, any B⊂L2(Ω) that is bounded, and any ɛ>0, there exists τ0≤t and M0>0 such that(3.45)∫Ω(|u|≥M)|ut(t)|2dx<ɛ,∀τ<τ0,M>M0,uτ∈B.Lemma 3.9 (see [9]).
For anyt∈ℝ, any bounded set B⊂L2(Ω), and any ɛ>0, there exist τ0 and M0>0 such that
(3.46)mes(Ω(u(t)≥M))<ɛ∀τ≤τ0,M≥M0,uτ∈B,
where mes is the Lebesgue measure in ℝN and Ω(u(t)≥M)={x∈Ω:u(t,x)≥M}.Lemma 3.10 (see [2]).
Let{U(t,τ)} be a norm-to-weak continuous process in L2(Ω) and Lq(Ω),q≥2. Then {U(t,τ)} is pullback asymptotically compact in Lq(Ω) if
(i)
{U(t,τ)} is pullback asymptotically compact in L2(Ω);(ii)
for anyt∈ℝ, any bounded set D⊂L2(Ω), and any ϵ>0, there exist M>0 andτ0≤t(3.47)sup⁡τ≤τ0sup⁡uτ∈D(∫Ω(|U(t,τ)uτ|≥M)|U(t,τ)uτ|qdx)≤Cϵ,
where C is independent of M, τ, uτ, and ϵ.We are now ready to prove the existence of a pullback attractor inL2p-2(Ω).Theorem 3.11.
Assume that assumptions (1.3)–(1.7) and (3.18) hold. Then the process {U(t,τ)} associated to problem (1.1) possesses a pullback attractor 𝒜2p-2={A2p-2(t)}t∈ℝ in L2p-2(Ω).Proof.
Because of Lemma3.10, since {U(t,τ)} has a pullback absorbing set in L2p-2(Ω), we only have to prove that for any t∈ℝ, any B⊂L2(Ω), and any ɛ>0, there exist τ2≤t and M2>0 such that
(3.48)∫Ω(|u|≥M)|u|2p-2dx≤Cɛ,∀τ≤τ2,M≥M2,uτ∈B.
Taking the inner product of (1.1) with (u-M)+p-1 in L2(Ω), where
(3.49)(u-M)+={u-Mifu≥M,0ifu<M,
we have
(3.50)∫Ωut(u-M)+p-1dx+(p-1)∫Ω(|∇x1u|2dx+|x1|2s|∇x2u|2)(u-M)+p-2dx+∫Ωf(u)(u-M)+p-1dx≤∫Ωg(t)(u-M)+p-1dx.
Some standard computations give us
(3.51)∫Ωf(u)(u-M)+p-1dx≥C∫Ω(u≥M)|u|2p-2dx+C∫Ω(u≥M)|u|pdx,(3.52)-∫Ωut(u-M)+p-1dx≤C4∫Ω(u≥M)|u|2p-2dx+1C∫Ω(u≥M)|ut|2dx,(3.53)∫Ωg(t)(u-M)+p-1dx≤C4∫Ω(u≥M)|u|2p-2dx+1C∫Ω(u≥M)|g(t)|2dx.
Combining (3.50)–(3.53), we find
(3.54)∫Ω(u≥M)|u|2p-2dx≤C(∫Ω(u≥M)|ut|2dx+∫Ω(u≥M)|g(t)|2dx+C∫Ω(u≥M)|u|pdx).
Applying Lemmas 3.7 and 3.8 to (3.54) we find there exist τ0 and M0 such that
(3.55)∫Ω(u≥M)|u|2p-2dx<ɛ∀τ≤τ0,M≥M0.
Repeating the above arguments with |(u+M)-|p-2(u+M)- in place of (u-M)+p-1, we have
(3.56)∫Ω(u≤-M)|u|2p-2dx<ɛ∀τ≤τ1,M≥M1,
for some τ1≤t and M1>0, where
(3.57)(u+M)-={u+Mifu≤-M,0ifu>M.
Letting τ2=min{τ0,τ1} and M2=max{M0,M1} we have
(3.58)∫Ω(|u|≥M2)|u|2p-2dx<Cɛ,∀τ≤τ2,M≥M2.
This completes the proof.


### 3.2. Existence of a Pullback Attractor inS02(Ω)

In this section, we prove the existence of a pullback attractor inS02(Ω).Lemma 3.12.
The process{U(t,τ)} associated to (1.1) has a pullback absorbing set in S02(Ω).Proof.
We multiply (1.1) by -Gsu; then, using f(0)=0, we have
(3.59)‖u‖S02(Ω)2=∫ΩutGsudx-∫Ωf′(u)(|∇x1u|2+|x1|2s|∇x2u|2)dx-∫Ωg(t)Gsudx.
Using f′(u)≥-ℓ, Cauchy’s inequality, and argument as in Lemma 3.3, from (3.59) we have
(3.60)‖u‖S02(Ω)2≤2(|ut|22+l‖u‖2+|g(t)|22).
Taking into account (3.11), the proof is complete.In order to prove the existence of the pullback attractor inS02(Ω), we will verify so-called “(PDC) condition”, which is defined as followDefinition 3.13.
A process{U(t,τ)} is said to satisfy (PDC) condition in X if for any t∈ℝ, any bounded set B⊂L2(Ω) and any ɛ>0, there exists τ0≤t and a finite dimensional subspace X1 of X such that (i)
P(⋃τ≤τ0U(t,τ)B) is bounded in X; and(ii)
∥(IX-P)U(t,τ)uτ∥X<ɛ, for all τ≤τ0 and uτ∈B, where P:X→X1 is a canonical projection and IX is the identity.Lemma 3.14 (see [13]).
If a process{U(t,τ)} satisfies (PDC) condition in X then it is pullback asymptotically compact in X. Moreover, if X is convex then the converse is true.Lemma 3.15 (see [9]).
Assume thatf satisfies (1.3) and (1.5). Then for any subset A⊂L2p-2(Ω), if κ(A)<ɛ in L2p-2(Ω), then we have
(3.61)κ(f(A))<CɛinL2(Ω),
where the Kuratowski noncompactness measure κ(B) in a Banach space X defined as
(3.62)κ(B)=inf⁡{δ:BhasafinitecoveringbyballsinXwithradiiδ}.Theorem 3.16.
Assume thatf satisfies (1.3)–(1.5), g satisfies (1.7) and (3.18). Then the process {U(t,τ)} generated by (1.1) has a pullback attractor 𝒜={A(t):t∈ℝ} in S02(Ω).Proof.
We consider a complete trajectoryu(t) lies on pullback attractor 𝒜2p-2 in L2p-2(Ω) for U(t,τ), that is, u(t)∈𝒜2p-2(t) and U(t,τ)uτ=u(t), for all t≥τ. Denoting A=-Gs and multiplying (1.1) by Au2=A(I-Pm)u=(I-Pm)Au we have
(3.63)(ut,Au2)+‖u2‖S02(Ω)2+∫Ωf(u)Au2dx=(g(t),Au2).
Using Holder’s inequality we get
(3.64)‖u2‖S02(Ω)2≤C(|(I-Pm)ut|22+|(I-Pm)g(t)|22+∫Ω(f(u))2dx).
Thanks to Lemmas 3.6 and 3.15 and the fact that g∈Cloc(ℝ;L2(Ω)), we see that {U(t,τ)} satisfies condition (PDC) in S02(Ω). Now from Lemmas 3.3 and 3.14 we get the desired result.


## 3.1. Existence of a Pullback Attractor inL2p-2(Ω)

In this section, following the general lines of the method introduced in [9], we prove the existence of a pullback attractor in L2p-2(Ω). In order to do this, we need an additional condition of g(3.18)∫-∞teλ1t‖g′(t)‖Lm(Ω)m′dt<+∞,∀t∈R,
where m, m′ are defined as in (3.30).Lemma 3.3.
The process{U(t,τ)} associated to problem (1.1) has a pullback absorbing set in L2p-2(Ω).Proof.
Multiplying (1.1) by |u|p-2u and integrating over Ω, we get
(3.19)∫Ωut|u|p-2udx+(p-1)∫Ω(|∇x1u|2+|x1|2s|∇x2u|2)|u|p-2dx+∫Ωf(u)||u|p-2udx|=∫Ω|g(t)u|p-2udx.
From (1.3) and the fact that Lp(Ω)⊂Lp-2(Ω) continuously, we have
(3.20)∫Ωf(u)|u|p-2udx≥∫Ω(C1|u|p-C2)|u|p-2dx≥C1‖u‖L2p-2(Ω)2p-2-C2|u|pp.
On the other hand, by Cauchy’s inequality, we see that
(3.21)|∫Ωut|u|p-2udx|≤C14‖u‖L2p-2(Ω)2p-2+1C1|ut|22,(3.22)|∫Ωg(t)|u|p-2udx|≤C14‖u‖L2p-2(Ω)2p-2+14|g(t)|22.
Combining (3.19)–(3.22) imply that
(3.23)‖u‖L2p-2(Ω)2p-2≤C(|ut|22+|u|pp+|g(t)|22).
Applying (3.2) and Lemma 3.2, we conclude the existence of a pullback absorbing set in L2p-2(Ω) for the process U(t,τ).Lemma 3.4.
For anys∈ℝ, any 2≤p<∞, and any bounded set B⊂L2(Ω), there exists τ0 such that
(3.24)∫Ω|ut(s)|pdx≤M,∀τ≤τ0,uτ∈B,
where M depends on s, p but not on B, and ut(s)=(d/dt)(U(t,τ)uτ)|t=s.Proof.
We will prove the lemma by induction argument. Lettingβ=N(s)/(N(s)-2)>1 and denoting v=ut we prove that for k=0,1,2,…, there exist τk and Mk(s) such that
(Pk)eλ1s∫Ω|v(s)|2βkdx≤Mk(s)foranyuτ∈B,τ≤τk,(Qk)∫ss+1(eλ1r∫Ω|v(r)|2βk+1dx)1/βdr≤Mk(s)foranyuτ∈B,τ≤τk,
where τk depends on k and B and Mk depends only on k.
Fork=0, we have (P0) from (3.17). Integrating (3.16) and using S01(Ω)↪L2β(Ω) continuously, we get (Q0).
Assuming that (Pk), (Qk) hold, we prove so are (Pk+1) and (Qk+1). Multiplying (3.14) by |v|2βk+1-2v and integrating over Ω, we obtain(3.25)Cddt∫Ω|v|2βk+1dx+C∫Ω(|∇x1v|2+|x1|2s|∇x2v|2)|v|β2k+1-2dx≤l∫Ω|v|2βk+1dx+(g′(t),|v|2βk+1-2v).
Using the imbedding S01(Ω)↪L2β(Ω) once again, we get
(3.26)∫Ω(|∇x1v|2+|x1|2s|∇x2v|2)|v|β2k+1dx≥∫Ω|v|2|v|2β2k+1-2dx=‖vβk+1‖L2β(Ω)2=(∫Ω|v|2βk+2dx)1/β.
Combining Holder’s and Young’s inequalities, we see that
(3.27)∫Ωg′(t)|v|2βk+1-2vdx≤(∫Ω|g′(t)|mdx)1/m(∫Ω|v|(2βk+1-1)ndx)1/n≤(∫Ω|g′(t)|mdx)m′/mm′+(∫Ω|v|(2βk+1-1)ndx)n′/nn′,
where 1/m+1/n=1/m′+1/n′=1. Choose n, n′ such that
(3.28)(2βk+1-1)n=2βk+2,n′n=1β,
thus
(3.29)n=2βk+22βk+1-1,n′=2βk+12βk+1-1.
Hence
(3.30)m=nn-1=2βk+1-12βk+2-2βk+1+1,m′=2βk+1.
Then from (3.27), we infer that
(3.31)∫Ωg′(t)|v|2βk+1-2vdx≤1m′‖g′(t)‖Lm(Ω)m′+1n′(∫Ω|v|2βk+2dx)1/β.
Applying (3.26) and (3.31) in (3.25), we find that
(3.32)ddt(eλ1t∫Ω|v|2βk+1dx)+Ceλ1t(∫Ω|v|2βk+2dx)1/β≤Ceλ1t∫Ω|v|2βk+1dx+Ceλ1t‖g′(t)‖Lm(Ω)m′.
Combining (Qk) and (3.32), using the uniform Gronwall inequality and taking into account assumption (3.18), we get (Pk+1). On the other hand, integrating (3.32) from t to t+1, we find (Qk+1). Now since β>1, and taking k≥logβp/2, we get the desired estimate.We will use the following lemma.Lemma 3.5 (see [15]).
If there existsσ>0 such that ∫-∞teσs|φ(s)|2ds<∞, for all t∈ℝ, then
(3.33)lim⁡γ→+∞∫-∞te-γ(t-s)|φ(s)|2ds=0,t∈R.LetHm=span{e1,e2,…,em} in L2(Ω), and let Pm:L2(Ω)→Hm be the orthogonal projection, where {ei}i=1∞ are the eigenvectors of operator A=-Gs. For any u∈L2(Ω), we write(3.34)u=Pmu+(I-Pm)u=u1+u2.Lemma 3.6.
For anyt∈ℝ, any B⊂L2(Ω) and any ɛ, there exist τ0(t,B,ɛ) and m0∈ℕ such that
(3.35)|(I-Pm)v|22<ɛ,∀τ≤τ0,∀uτ∈B,m≥m0.Proof.
Multiplying (3.14) by v2=(I-Pm)v and then integrating over Ω, using |∇v2|22≥λm|v2|22 and Cauchy’s inequality we get
(3.36)ddt|v2|22+λm|v2|22≤C∫Ω|f′(u)v|2dx+C|g′(t)|22.
We multiply (3.36) by eλmt and use assumption (1.4). We get
(3.37)ddt(eλmt|v2|22)≤Ceλmt∫Ω|u|2(p-2)|v|2dx+Ceλmt|g′(t)|22.
Integrating (3.37) from s to t,
(3.38)eλmt|v2(t)|22≤eλms|v2(s)|22+C∫steλmr∫Ω|u|2(p-2)|v|2dxdr+C∫steλmr|g′(r)|22dr≤eλms|v(s)|22+C∫-∞teλmr∫Ω|u|2(p-2)|v|2dxdr+C∫-∞teλmr|g′(r)|22dr.
Now integrating (3.38) with respect to s from τ to t, we infer that(3.39)(t-τ)eλmt|v2(t)|22≤∫τteλmr|v(r)|22dr+C(t-τ)∫-∞teλmr∫Ω|u|2(p-2)|v|2dxdr+C(t-τ)∫-∞teλmr|g′(r)|22dr.
Thus
(3.40)|v2(t)|22≤1t-τ∫-∞te-λm(t-r)|v(r)|22dr+C∫-∞te-λm(t-r)∫Ω|u|2(p-2)|v|2dxdr+C∫-∞te-λm(t-r)|g′(r)|22dr.
By Lemma 3.5 and since λm→+∞ as m→+∞, there exist τ1 and m1 such that
(3.41)1t-τ∫-∞te-λm(t-r)C∫-∞te-λm(t-r)|g′(r)|22dr<ɛ3,|v(r)|22dr<ɛ3,
for all τ≤τ1 and m≥m1. For the second term of the right-hand side of (3.40), using Holder’s inequality we have
(3.42)∫-∞te-λm(t-r)∫Ω|u|2(p-2)|v|2dxdr≤∫-∞t(∫Ωe((-p-1)/(p-2)λm)(t-r)|u|2p-2dx)(p-2)/(p-1)(∫Ωe-(p-1)λm(t-r)|v|2p-2dx)1/(p-1)dr≤(∫-∞te((-p-1)/(p-2)λm)(t-r)‖u‖L2p-2(Ω)2p-2dr)(p-2)/(p-1)(∫-∞te-(p-1)λm(t-r)dr∫Ω|v|2p-2dx)1/(p-1).
From Lemmas (3.5)–(3.7), we see that there exist τ2 and m2∈ℕ such that
(3.43)C∫-∞te-λm(t-r)∫Ω|u|2(p-2)|v|2dxdr<ɛ3,∀τ≤τ0,m≥m2.
Let τ0=min{τ1,τ2} and m0=max{m1,m2}, from (3.40), taking into account (3.41) and (3.43), we obtain (3.35).Lemma 3.7 (see [9]).
LetB be a bounded subset in Lq(Ω)(q≥1). If B has a finite ɛ-net in Lq(Ω), then there exists an M=M(B,ɛ), such that for any u∈B, the following estimate is valid:
(3.44)∫Ω(|u|≥M)|u|qdx<ɛ.Using Lemma3.7 and taking into account Lemmas 3.2 and 3.6 we conclude that the set {ut(s):s≤t,uτ∈B} has a finite ϵ-net in L2(Ω). Therefore, we get the following result.Lemma 3.8.
For anyt∈ℝ, any B⊂L2(Ω) that is bounded, and any ɛ>0, there exists τ0≤t and M0>0 such that(3.45)∫Ω(|u|≥M)|ut(t)|2dx<ɛ,∀τ<τ0,M>M0,uτ∈B.Lemma 3.9 (see [9]).
For anyt∈ℝ, any bounded set B⊂L2(Ω), and any ɛ>0, there exist τ0 and M0>0 such that
(3.46)mes(Ω(u(t)≥M))<ɛ∀τ≤τ0,M≥M0,uτ∈B,
where mes is the Lebesgue measure in ℝN and Ω(u(t)≥M)={x∈Ω:u(t,x)≥M}.Lemma 3.10 (see [2]).
Let{U(t,τ)} be a norm-to-weak continuous process in L2(Ω) and Lq(Ω),q≥2. Then {U(t,τ)} is pullback asymptotically compact in Lq(Ω) if
(i)
{U(t,τ)} is pullback asymptotically compact in L2(Ω);(ii)
for anyt∈ℝ, any bounded set D⊂L2(Ω), and any ϵ>0, there exist M>0 andτ0≤t(3.47)sup⁡τ≤τ0sup⁡uτ∈D(∫Ω(|U(t,τ)uτ|≥M)|U(t,τ)uτ|qdx)≤Cϵ,
where C is independent of M, τ, uτ, and ϵ.We are now ready to prove the existence of a pullback attractor inL2p-2(Ω).Theorem 3.11.
Assume that assumptions (1.3)–(1.7) and (3.18) hold. Then the process {U(t,τ)} associated to problem (1.1) possesses a pullback attractor 𝒜2p-2={A2p-2(t)}t∈ℝ in L2p-2(Ω).Proof.
Because of Lemma3.10, since {U(t,τ)} has a pullback absorbing set in L2p-2(Ω), we only have to prove that for any t∈ℝ, any B⊂L2(Ω), and any ɛ>0, there exist τ2≤t and M2>0 such that
(3.48)∫Ω(|u|≥M)|u|2p-2dx≤Cɛ,∀τ≤τ2,M≥M2,uτ∈B.
Taking the inner product of (1.1) with (u-M)+p-1 in L2(Ω), where
(3.49)(u-M)+={u-Mifu≥M,0ifu<M,
we have
(3.50)∫Ωut(u-M)+p-1dx+(p-1)∫Ω(|∇x1u|2dx+|x1|2s|∇x2u|2)(u-M)+p-2dx+∫Ωf(u)(u-M)+p-1dx≤∫Ωg(t)(u-M)+p-1dx.
Some standard computations give us
(3.51)∫Ωf(u)(u-M)+p-1dx≥C∫Ω(u≥M)|u|2p-2dx+C∫Ω(u≥M)|u|pdx,(3.52)-∫Ωut(u-M)+p-1dx≤C4∫Ω(u≥M)|u|2p-2dx+1C∫Ω(u≥M)|ut|2dx,(3.53)∫Ωg(t)(u-M)+p-1dx≤C4∫Ω(u≥M)|u|2p-2dx+1C∫Ω(u≥M)|g(t)|2dx.
Combining (3.50)–(3.53), we find
(3.54)∫Ω(u≥M)|u|2p-2dx≤C(∫Ω(u≥M)|ut|2dx+∫Ω(u≥M)|g(t)|2dx+C∫Ω(u≥M)|u|pdx).
Applying Lemmas 3.7 and 3.8 to (3.54) we find there exist τ0 and M0 such that
(3.55)∫Ω(u≥M)|u|2p-2dx<ɛ∀τ≤τ0,M≥M0.
Repeating the above arguments with |(u+M)-|p-2(u+M)- in place of (u-M)+p-1, we have
(3.56)∫Ω(u≤-M)|u|2p-2dx<ɛ∀τ≤τ1,M≥M1,
for some τ1≤t and M1>0, where
(3.57)(u+M)-={u+Mifu≤-M,0ifu>M.
Letting τ2=min{τ0,τ1} and M2=max{M0,M1} we have
(3.58)∫Ω(|u|≥M2)|u|2p-2dx<Cɛ,∀τ≤τ2,M≥M2.
This completes the proof.


## 3.2. Existence of a Pullback Attractor inS02(Ω)

In this section, we prove the existence of a pullback attractor inS02(Ω).Lemma 3.12.
The process{U(t,τ)} associated to (1.1) has a pullback absorbing set in S02(Ω).Proof.
We multiply (1.1) by -Gsu; then, using f(0)=0, we have
(3.59)‖u‖S02(Ω)2=∫ΩutGsudx-∫Ωf′(u)(|∇x1u|2+|x1|2s|∇x2u|2)dx-∫Ωg(t)Gsudx.
Using f′(u)≥-ℓ, Cauchy’s inequality, and argument as in Lemma 3.3, from (3.59) we have
(3.60)‖u‖S02(Ω)2≤2(|ut|22+l‖u‖2+|g(t)|22).
Taking into account (3.11), the proof is complete.In order to prove the existence of the pullback attractor inS02(Ω), we will verify so-called “(PDC) condition”, which is defined as followDefinition 3.13.
A process{U(t,τ)} is said to satisfy (PDC) condition in X if for any t∈ℝ, any bounded set B⊂L2(Ω) and any ɛ>0, there exists τ0≤t and a finite dimensional subspace X1 of X such that (i)
P(⋃τ≤τ0U(t,τ)B) is bounded in X; and(ii)
∥(IX-P)U(t,τ)uτ∥X<ɛ, for all τ≤τ0 and uτ∈B, where P:X→X1 is a canonical projection and IX is the identity.Lemma 3.14 (see [13]).
If a process{U(t,τ)} satisfies (PDC) condition in X then it is pullback asymptotically compact in X. Moreover, if X is convex then the converse is true.Lemma 3.15 (see [9]).
Assume thatf satisfies (1.3) and (1.5). Then for any subset A⊂L2p-2(Ω), if κ(A)<ɛ in L2p-2(Ω), then we have
(3.61)κ(f(A))<CɛinL2(Ω),
where the Kuratowski noncompactness measure κ(B) in a Banach space X defined as
(3.62)κ(B)=inf⁡{δ:BhasafinitecoveringbyballsinXwithradiiδ}.Theorem 3.16.
Assume thatf satisfies (1.3)–(1.5), g satisfies (1.7) and (3.18). Then the process {U(t,τ)} generated by (1.1) has a pullback attractor 𝒜={A(t):t∈ℝ} in S02(Ω).Proof.
We consider a complete trajectoryu(t) lies on pullback attractor 𝒜2p-2 in L2p-2(Ω) for U(t,τ), that is, u(t)∈𝒜2p-2(t) and U(t,τ)uτ=u(t), for all t≥τ. Denoting A=-Gs and multiplying (1.1) by Au2=A(I-Pm)u=(I-Pm)Au we have
(3.63)(ut,Au2)+‖u2‖S02(Ω)2+∫Ωf(u)Au2dx=(g(t),Au2).
Using Holder’s inequality we get
(3.64)‖u2‖S02(Ω)2≤C(|(I-Pm)ut|22+|(I-Pm)g(t)|22+∫Ω(f(u))2dx).
Thanks to Lemmas 3.6 and 3.15 and the fact that g∈Cloc(ℝ;L2(Ω)), we see that {U(t,τ)} satisfies condition (PDC) in S02(Ω). Now from Lemmas 3.3 and 3.14 we get the desired result.


## 4. Exponential Growth inS02(Ω)∩L2p-2(Ω) of Pullback Attractors

In this section, we will give an exponential growth condition inS02(Ω)∩L2p-2(Ω) for the pullback attractor 𝒜(τ).First, we recall a result in [17] which is necessary for the proof of our results.Lemma 4.1.
LetX, Y be Banach spaces such that X is reflexive, and the inclusion X⊂Y is continuous. Assume that {un} is bounded sequence in L∞(t0,T;X) such that un→u weakly in Lq(t0,T;X) for some q∈[1,+∞) and u∈C0([t0,T];Y). Then, u(t)∈X for all t∈[t0,T] and
(4.1)‖u‖X≤sup⁡n≥1‖un‖L∞(t0,T;X),∀t∈[t0,T].In the following theorem, instead of evaluating the functionsun which are differentiable enough and then using Lemma 4.1, we will formally evaluate the function u.Theorem 4.2.
Assume thatf satisfies (1.3)-(1.5), g satisfies (H 2), (3.18) and the following conditions
(4.2)lim⁡τ→-∞eλ1τ∫ττ+1|g′(s)|22ds=0,lim⁡τ→-∞(eλ1τ|g(s)|22)=0.
Then 𝒜(τ) satisfies
(4.3)lim⁡τ→-∞eλ1τ{sup⁡w∈A(τ)‖w‖L2p-2(Ω)2p-2+sup⁡w∈A(τ)‖w‖S02(Ω)2}=0.Proof.
We differentiate with respect to time in (1.1), then multiply by ut, we get(4.4)12ddr|u′(r)|22+‖u(r)‖2=-∫Ωf′(u)u′(r)u′(r)dx+∫Ωg′(r)u′(r)dx≤l|u′(r)|22+12u′(r)22+12|g′(r)|22.
Integrating in the last inequality, in particular, we get
(4.5)|u′(r)|22≤|u′(s)|22+(2l+1)∫τ+ϵ/2t|u′(θ)|22dθ+∫τ+ϵ/2t|g′(θ)|22dθ.
for all τ+ϵ/2≤s≤τ≤t. Now, integrating with respect to s, between τ+ϵ/2 and r(4.6)(r-τ-ϵ2)|u′(r)|22≤[(r-τ-ϵ2)(2l+1)+1]∫τ+ϵ/2t|u′(θ)|22dθ+(r-τ-ϵ2)∫τ+ϵ/2t|g′(θ)|22dθ.
for all τ+ϵ/2≤r≤t, in paricular,
(4.7)|u′(r)|22≤2ϵ-1[(r-τ-ϵ2)(2l+1)+1]∫τ+ϵ/2t|u′(θ)|22dθ+∫τ+ϵ/2t|g′(θ)|22dθ.
for all r∈[τ+ϵ,t].
Multiplying (1.1) by u and then integrating on Ω, we get(4.8)12ddt|u|22+‖u‖2+∫Ωf(u)udx=∫Ωg(t)udx≤12λ1|g(t)|22+λ12|u|22.
Using hypothesis (H1) and the fact that ∥u∥2≥(1/2)∥u∥2+(λ1/2)|u|22, we have
(4.9)ddt|u|22+λ1|u|22+2C1|u|pp-2C2|Ω|≤1λ1|g(t)|22.
Integrating (4.9) from τ to r∈[τ,t], we have
(4.10)|u(r)|22+λ1∫τr‖u(s)‖2ds+2C1∫τr|u(s)|ppds≤|u(τ)|22+1λ1∫τr|g(s)|22ds+2C2|Ω|(t-τ).
Thus,
(4.11)|u(r)|22+∫τr‖u(s)‖2ds+∫τr|u(s)|ppds≤C[|u(τ)|22+∫τr|g(s)|22ds+(t-τ)].
Multiplying (1.1) by ut then integrating over Ω, we have
(4.12)|u′(r)|22+12ddr(‖u(r)‖2+2∫ΩF(u(x,r))dx)≤12|g(r)|22+12|u′(r)|22.
Integrating now between s∈[τ,r] and r≤t, we obtain
(4.13)∫sr|u′(θ)|22dθ+‖u(r)‖2+∫ΩF(u(x,r))dx≤∫sr|g(θ)|22dθ+‖u(s)‖2+∫ΩF(u(x,s))dx.
From (4.13) and using hypothesis (H1), we get
(4.14)∫sr|u′(θ)|22dθ+‖u(r)‖2+α1∫Ω|u(x,r)|pdx-α2|Ω|≤‖u(s)‖2+∫sr|g(θ)|22dθ+α3∫Ω|u(x,r)|pdx+α4|Ω|.
Hence
(4.15)∫sr|u′(θ)|22dθ+‖u(r)‖2+α1u(r)pp≤‖u(s)‖2+∫st|g(θ)|22dθ+α3|u(s)|pp+(α2+α4)|Ω|.
Integrating inequality (4.15) with respect to s from τ to r, we obtain
(4.16)(r-τ)[‖u(r)‖2+|u(r)|pp]≤C[∫τr‖u(s)‖2ds+∫τr|u(s)|ppds]+C(t-τ)∫τt|g(s)|22ds+C|Ω|(t-τ),
for all t≥τ, r∈[τ,t].
From (4.16) and (4.11), we obtain that
(4.17)‖u(r)‖2+|u(r)|pp≤C[|uτ|22+∫ττ+2|g(s)|22ds+1],
for all r∈[τ+1,τ+2].
From (4.7), taking t=τ+3 and ϵ=2 we have(4.18)|u′(r)|22≤(4l+3)∫τ+1τ+3|u′(θ)|22dθ+∫τ+1τ+3|g′(θ)|22dθ
for all r∈[τ+2,τ+3].
Analogously, and if we takes=τ+1 and r=t=τ+3 in inequlity (4.15), then(4.19)∫τ+1τ+3|u′(s)|22ds+‖u(τ+3)‖2+α1|u(τ+3)|pp≤‖u(τ+1)‖2+∫ττ+3|g(s)|22ds+α3|u(τ+1)|pp+(α2+α4)|Ω|.
From (4.18) and (4.19), we obtain
(4.20)|u′(r)|22≤(4l+3)[‖u(τ+1)‖2+α3|u(τ+1)|pp]+(4l+3)[(α2+α4)|Ω|+∫ττ+3|g(s)|22ds]+∫τ+1τ+3|g′(θ)|22dθ
for all r∈[τ+2,τ+3].
Owing to this inequality and (4.17), we have(4.21)|u′(r)|22≤C[|uτ|22+∫ττ+3[|g(s)|22+|g′(s)|22]ds+1],
for all r∈[τ+2,τ+3].
From (3.23), (3.60) and using Young’s inequality, we have(4.22)‖u(r)‖L2p-2(Ω)2p-2+‖u(r)‖S02(Ω)2≤C(|u′(r)|22+‖u(r)‖2+|u(r)|pp+1+|g(r)|22),
for all r≥τ.
From (4.22) and thank to (4.21), we have(4.23)‖u(r)‖L2p-2(Ω)2p-2+‖u(r)‖S02(Ω)2≤C[|uτ|22+∫ττ+3[|g(s)|22+|g′(s)|22]ds+1]+C(‖u(r)‖2+|u(r)|pp+|g(r)|22)
for all r∈[τ+2,τ+3]. From (4.23) and thanks to (4.17), we have
(4.24)‖u(r)‖L2p-2(Ω)2p-2+‖u(r)‖S02(Ω)2≤C[|uτ|+∫ττ+3[|g(s)|22+|g′(s)|22]ds+1+sup⁡r∈[τ+2,τ+3]|g(r)|22],
for all r∈[τ+2,τ+3]. Now, observe that by Cauchy’s inequality
(4.25)|g(r)|≥|g(τ+2)|+(∫τ+2τ+3|g′(s)|22ds)1/2,
for all r∈[τ+2,τ+3], τ∈ℝ,uτ∈L2(Ω).
Thus, from (4.23), we have(4.26)‖U(τ+2,τ)uτ‖L2p-2(Ω)2p-2+‖U(τ+2,τ)uτ‖S02(Ω)2≤C[|uτ|22+∫ττ+3(|g(s)|22+|g′(s)|22)ds+1+|g(τ+2)|22].
for all τ∈ℝ, uτ∈L2(Ω). From this inequlity, and the fact that 𝒜(τ)=U(τ,τ-2)A(τ-2), we obtain
(4.27)‖v‖L2p-2(Ω)2p-2+‖v‖S02(Ω)2≤C[sup⁡w∈A(τ-2)|w|22+∫τ-2τ+1(|g(s)|22+|g′(s)|22)ds+1+|g(τ)|22].
for all v∈A(τ), and any τ∈ℝ. Now, thank to (4.2), (4.3), we obtain (4.3) from (4.27).



---

*Source: 272145-2012-03-11.xml*