# On the Dynamics of Nonautonomous Parabolic Systems Involving the Grushin Operators

Authors: Anh Cung The; Toi Vu Manh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We study the long-time behavior of solutions to nonautonomous semilinear parabolic systems involving the Grushin operators in bounded domains. We prove the existence of a pullback𝒟-attractor in
(L2(Ω))m for the corresponding process in the general case. When the system has a special gradient structure, we prove that the obtained pullback
𝒟-attractor is more regular and has a finite fractal dimension. The obtained results, in particular, extend and improve some existing ones for the reaction-diffusion equations and the Grushin equations.


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## Body

## 1. Introduction

Nonautonomous equations are of great importance and interest as they appear in many applications in the natural sciences. One way of studying the long-time behavior of solutions of such equations is using the theory of pullback attractors. This theory has been developed for both the nonautonomous and random dynamical systems and has shown to be very useful in the understanding of the dynamics of such dynamical systems (see [1] and references therein). In recent years, the existence of pullback attractors for reaction-diffusion equations has been studied widely (see, e.g., [2–6]). However, to the best of our knowledge, little seems to be known for the asymptotic behavior of solutions of nonautonomous degenerate equations.One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of the Grushin type [7] (1.1)Gsu=Δxu+|x|2sΔyu,X=(x,y)∈Ω⊂RN1×RN2,s≥0.
The global existence and long-time behavior of solutions to semilinear parabolic equations involving the Grushin operator, in both autonomous and nonautonomous cases, have been studied in some recent works [8–10].In this paper we consider the following nonautonomous semilinear parabolic system:(1.2)∂u∂t-aGsu+f(u)=g(X,t),X∈Ω,t>τ,u(X,t)=0,X∈∂Ω,t>τ,u(X,τ)=uτ(X),X∈Ω,
where X=(x,y)∈Ω⊂ℝN1×ℝN2(N1,N2⩾1), uτ∈(L2(Ω))m is given, u=(u1,…,um) is an unknown vector-function. Here a∈Matm(ℝ), f(u)=(f1(u1,…,um),…,fm(u1,…,um)), and g(X,t)=(g1(X,t),…,gm(X,t)) satisfy the following conditions:(H1)
a∈Matm(ℝ) has a positive symmetric part: (1/2)(a+a*)⩾βIm,β>0;(H2)
f:ℝm→ℝm is a C1-vector function such that:
(1.3)C1|u|p-C0≤(f(u),u)=∑j=1mfj(u)uj,p≥2,(1.4)|f(u)|≤C2(|u|p-1+1),(1.5)-C3|v|2≤(fu(u)v,v)=∑i=1m∑j=1m∂fi∂uj(u)vjvi,
where C0, C1, C2, and C3 are positive constants;(H3)
g∈Wloc1,2(ℝ;(L2(Ω))m) such that
(1.6)∫-∞0eλ1βs‖g(s)‖(L2(Ω))m2ds<+∞,∫-∞0∫-∞seλ1βr‖g(r)‖(L2(Ω))m2drds<+∞,∫-∞0eλ1βs‖g′(s)‖(L2(Ω))m2ds<+∞,
where λ1 is the first eigenvalue of the operator Gs in Ω with the homogeneous Dirichlet boundary condition.In order to study problem (1.2), we will use the natural energy space 𝕊01(Ω):=(S01(Ω))m defined as the complete of (C0∞(Ω))m in the following norm: (1.7)‖u‖S01(Ω)=(∫Ω(|∇xu|2+|x|2s|∇yu|2)dX)1/2.
From the results in [11], we know that the embedding 𝕊01(Ω)↪𝕃p(Ω) is continuous if 1⩽p⩽2s*:=(2N(s))/(N(s)-2), where N(s):=N1+(s+1)N2; moreover, this embedding is compact if 1⩽p<2s*.Notations
Denote𝕃p(Ω):=(Lp(Ω))m, and 𝕊-1(Ω) the dual space of 𝕊01(Ω). For functions u,v:ℝN→ℝm, we set
(1.8)((∇u,∇v)):=∑i=1m(∇ui,∇vi)=∑i=1m∑k=1N∂ui∂Xk∂vi∂Xk,
so if a=(aij)i,j=1m∈Matm(ℝ), then
(1.9)((a∇u,∇v))=∑i,j=1maij(∇ui,∇vj)=∑i,j=1m∑k=1Naij∂ui∂Xk∂vj∂Xk,
where (·,·) denotes the inner product in ℝN.Noting that by assumption (H1), we have(1.10)((a∇u,∇u))=12∑i,j=1m(aij+aji)(∇ui,∇uj)≥β∑j=1m(∇uj,∇uj)=β∑j=1m|∇uj|2.
Hence(1.11)∫Ω[((a∇xu,∇xu))+|x|2s((a∇yu,∇yu))]dX≥β‖u‖S01(Ω)2,(1.12)∫Ω(aGsu,Gsu)dX≥β‖Gsu‖L2(Ω)2.The aim of this paper is to study the long-time behavior of solutions to problem (1.2) by using the theory of pullback 𝒟-attractors. We first prove, under assumptions (H1)–(H3), the existence of a pullback 𝒟-attractor in 𝕃2(Ω) for the process U(t,τ) associated to problem (1.2). Then, with an additional condition that the system has a special gradient structure, namely, a=βIm and there exists a function F:ℝm→ℝ such that f(u)=graduF(u), we prove the existence of a pullback 𝒟-attractor in the space 𝕊01(Ω)∩𝕃p(Ω) for the process U(t,τ). Moreover, we prove the boundedness of the pulback 𝒟-attractor in 𝕃2p-2(Ω) and in 𝕊02(Ω), and give estimates of the fractal dimension of the pulback 𝒟-attractor. It is worth noticing that our results, in particular, extend and improve some recent results on the existence of pullback 𝒟-attractors for the reaction-diffusion equations [3–5] and for the Grushin equations [8].Let us explain the methods used in the paper. We first prove the existence of a family of pullback𝒟-absorbing sets in 𝕊01(Ω). Thanks to the compactness of the embedding 𝕊01(Ω)↪𝕃2(Ω), we immediately get the existence of a pullback 𝒟-attractor in 𝕃2(Ω). When the system has a special gradient structure, we are able to prove the existence of a pullback 𝒟-attractor in 𝕊01(Ω)∩𝕃p(Ω). To do this, we follow the general lines of the approach used in [8], with some modifications so that we can improve conditions imposed on the external force g. In particular, we use the asymptotic a priori estimate method initiated in [12] to testify the pullback asymptotic compactness of the corresponding process. Moreover, in this case we also prove the regularity of the pullback 𝒟-attractor in the spaces 𝕃2p-2(Ω) and 𝕊01(Ω). Finally, using the recent results in [13], we give an estimate of the fractal dimension of the pullback 𝒟-attractor. It is noticed that we do not impose the restriction on the exponent p in (H2) as in [13].The rest of the paper is organized as follows. In Section2, for the convenience of the reader, we recall some concepts and results on pullback 𝒟-attractors which we will use. In Section 3, we prove the existence of a pullback 𝒟-attractor in 𝕃2(Ω) in the general case. In Section 4, under the additional assumption that the system has a gradient structure, we prove the regularity and fractal dimension estimates of the pullback 𝒟-attractor.


## 2. Preliminaries

### 2.1. Pullback Attractors

For convenience of the reader, we recall in this section some concepts and results on the theory of pullback𝒟-attractors, which will be used in the paper.LetX be a metric space with metric d. Denote by ℬ(X) the set of all bounded subsets of X. For A,B⊂X, the Hausdorff semidistance between A and B is defined by (2.1)dist⁡(A,B)=sup⁡x∈Ainf⁡y∈Bd(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 verifying the norm-to-weak continuity of a process.Proposition 2.1 (see [14]).
LetX,Y be two Banach spaces, X*,Y* be, respectively, their dual spaces. Assume 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 a compact set of X to be a bounded set of X.Suppose that𝒟 is a nonempty class of parameterized sets 𝒟̂={D(t):t∈ℝ}⊂ℬ(X).Definition 2.2.
The process{U(t,τ)} is said to be pullback 𝒟-asymptotically compact if for any t∈ℝ, any 𝒟̂∈𝒟, and any sequence {τn}n with τn≤t for all n, and τn→-∞, any sequence xn∈D(τn), the sequence {U(t,τn)xn} is relatively compact in X.Definition 2.3.
A process{U(t,τ)} is called pullback ω-𝒟-limit compact if for any ɛ>0, any t∈ℝ, and 𝒟̂∈𝒟, there exists a τ0=τ0(𝒟̂,ɛ,t)≤t such that
(2.2)α(⋃τ≤τ0U(t,τ)D(τ))≤ɛ,
where α is the Kuratowski measure of noncompactness of B∈ℬ(X),
(2.3)α(B)=inf⁡{δ>0∣Bhasafiniteopencoverofsetsofdiameter≤δ}.Lemma 2.4 (see [3]).
A process{U(t,τ)} is pullback 𝒟-asymptotically compact if and only if it is ω-𝒟-limit compact.Definition 2.5.
A family of bounded setsℬ̂∈𝒟 is called pullback 𝒟-absorbing for the process {U(t,τ)} if for any t∈ℝ and any 𝒟̂∈𝒟, there exists τ0=τ0(𝒟̂,t)≤t such that
(2.4)⋃τ≤τ0U(t,τ)D(τ)⊂B(t).Definition 2.6.
A family𝒜̂={A(t):t∈ℝ}⊂ℬ(X) is said to be a pullback 𝒟-attractor for {U(t,τ)} if (1)
A(t) is compact for all t∈ℝ;(2)
𝒜̂ is invariant, that is, U(t,τ)A(τ)=A(t), for all t≥τ;(3)
𝒜̂ is pullback 𝒟-attracting, that is,
(2.5)lim⁡τ→-∞dist⁡(U(t,τ)D(τ),A(t))=0,
for all 𝒟̂∈𝒟 and all t∈ℝ;(4)
if{C(t):t∈ℝ} is another family of closed attracting sets, then A(t)⊂C(t), for all t∈ℝ.Theorem 2.7 (see [3]).
Let{U(t,τ)} be a norm-to-weak continuous process such that {U(t,τ)} is pullback 𝒟-asymptotically compact. If there exists a family of pullback 𝒟-absorbing sets ℬ̂={B(t):t∈ℝ}∈𝒟, then {U(t,τ)} has a unique pullback 𝒟-attractor 𝒜̂={A(t):t∈ℝ} and
(2.6)A(t)=⋂s≤t⋃τ≤sU(t,τ)B(τ)¯.


### 2.2. Fractal Dimension of Pullback Attractors

Given a compactK⊂X and ɛ>0, we denote by N(K,ɛ) the minimum number of open balls in X with radius ɛ which are necessary to cover K.Definition 2.8.
For any nonempty compact setK⊂X, the fractal dimension of K is the number
(2.7)dim⁡f(K):=lim⁡ɛ→0log⁡N(K,ɛ)log⁡1/ɛ.Definition 2.9.
A bounded subsetB0⊂H is called a uniformly pullback absorbing set for process U(t,τ) if for every B⊂H is bounded, there exists a τ0≥0 such that
(2.8)U(t,t-τ0)B∈B0,∀τ≥τ0,
here, τ0 does not depend on the choice of t.Theorem 2.10 (see [13]).
LetU(t,τ) be a process in a separable Hilbert space H, B be a uniformly pullback absorbing set in H, Â={A(t):t∈ℝ} be a pullback attractor for U(t,τ), if there exists a finite dimensional projection P in the space H such that
(2.9)‖P(U(t,t-T0)u1-U(t,t-T0)u2)‖H≤l(T0)‖u1-u2‖H
for all u1,u2∈B and some T0,l(T0)>0 and
(2.10)‖(I-P)(U(t,t-T0)u1-U(t,t-T0)u2)‖H≤δ‖u1-u2‖
for all u1,u2∈B, where δ<1, T0 and l(T0) are independent of the choice of t. Then the family of pullback attractors Â={A(t):t∈ℝ} possesses a finite fractal dimension especifically
(2.11)dim⁡f(A(t))≤dim⁡Plog⁡(1+8l(T0)1-δ)[log⁡21+δ]-1,∀t∈R.


## 2.1. Pullback Attractors

For convenience of the reader, we recall in this section some concepts and results on the theory of pullback𝒟-attractors, which will be used in the paper.LetX be a metric space with metric d. Denote by ℬ(X) the set of all bounded subsets of X. For A,B⊂X, the Hausdorff semidistance between A and B is defined by (2.1)dist⁡(A,B)=sup⁡x∈Ainf⁡y∈Bd(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 verifying the norm-to-weak continuity of a process.Proposition 2.1 (see [14]).
LetX,Y be two Banach spaces, X*,Y* be, respectively, their dual spaces. Assume 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 a compact set of X to be a bounded set of X.Suppose that𝒟 is a nonempty class of parameterized sets 𝒟̂={D(t):t∈ℝ}⊂ℬ(X).Definition 2.2.
The process{U(t,τ)} is said to be pullback 𝒟-asymptotically compact if for any t∈ℝ, any 𝒟̂∈𝒟, and any sequence {τn}n with τn≤t for all n, and τn→-∞, any sequence xn∈D(τn), the sequence {U(t,τn)xn} is relatively compact in X.Definition 2.3.
A process{U(t,τ)} is called pullback ω-𝒟-limit compact if for any ɛ>0, any t∈ℝ, and 𝒟̂∈𝒟, there exists a τ0=τ0(𝒟̂,ɛ,t)≤t such that
(2.2)α(⋃τ≤τ0U(t,τ)D(τ))≤ɛ,
where α is the Kuratowski measure of noncompactness of B∈ℬ(X),
(2.3)α(B)=inf⁡{δ>0∣Bhasafiniteopencoverofsetsofdiameter≤δ}.Lemma 2.4 (see [3]).
A process{U(t,τ)} is pullback 𝒟-asymptotically compact if and only if it is ω-𝒟-limit compact.Definition 2.5.
A family of bounded setsℬ̂∈𝒟 is called pullback 𝒟-absorbing for the process {U(t,τ)} if for any t∈ℝ and any 𝒟̂∈𝒟, there exists τ0=τ0(𝒟̂,t)≤t such that
(2.4)⋃τ≤τ0U(t,τ)D(τ)⊂B(t).Definition 2.6.
A family𝒜̂={A(t):t∈ℝ}⊂ℬ(X) is said to be a pullback 𝒟-attractor for {U(t,τ)} if (1)
A(t) is compact for all t∈ℝ;(2)
𝒜̂ is invariant, that is, U(t,τ)A(τ)=A(t), for all t≥τ;(3)
𝒜̂ is pullback 𝒟-attracting, that is,
(2.5)lim⁡τ→-∞dist⁡(U(t,τ)D(τ),A(t))=0,
for all 𝒟̂∈𝒟 and all t∈ℝ;(4)
if{C(t):t∈ℝ} is another family of closed attracting sets, then A(t)⊂C(t), for all t∈ℝ.Theorem 2.7 (see [3]).
Let{U(t,τ)} be a norm-to-weak continuous process such that {U(t,τ)} is pullback 𝒟-asymptotically compact. If there exists a family of pullback 𝒟-absorbing sets ℬ̂={B(t):t∈ℝ}∈𝒟, then {U(t,τ)} has a unique pullback 𝒟-attractor 𝒜̂={A(t):t∈ℝ} and
(2.6)A(t)=⋂s≤t⋃τ≤sU(t,τ)B(τ)¯.


## 2.2. Fractal Dimension of Pullback Attractors

Given a compactK⊂X and ɛ>0, we denote by N(K,ɛ) the minimum number of open balls in X with radius ɛ which are necessary to cover K.Definition 2.8.
For any nonempty compact setK⊂X, the fractal dimension of K is the number
(2.7)dim⁡f(K):=lim⁡ɛ→0log⁡N(K,ɛ)log⁡1/ɛ.Definition 2.9.
A bounded subsetB0⊂H is called a uniformly pullback absorbing set for process U(t,τ) if for every B⊂H is bounded, there exists a τ0≥0 such that
(2.8)U(t,t-τ0)B∈B0,∀τ≥τ0,
here, τ0 does not depend on the choice of t.Theorem 2.10 (see [13]).
LetU(t,τ) be a process in a separable Hilbert space H, B be a uniformly pullback absorbing set in H, Â={A(t):t∈ℝ} be a pullback attractor for U(t,τ), if there exists a finite dimensional projection P in the space H such that
(2.9)‖P(U(t,t-T0)u1-U(t,t-T0)u2)‖H≤l(T0)‖u1-u2‖H
for all u1,u2∈B and some T0,l(T0)>0 and
(2.10)‖(I-P)(U(t,t-T0)u1-U(t,t-T0)u2)‖H≤δ‖u1-u2‖
for all u1,u2∈B, where δ<1, T0 and l(T0) are independent of the choice of t. Then the family of pullback attractors Â={A(t):t∈ℝ} possesses a finite fractal dimension especifically
(2.11)dim⁡f(A(t))≤dim⁡Plog⁡(1+8l(T0)1-δ)[log⁡21+δ]-1,∀t∈R.


## 3. Existence of Pullback𝒟-Attractors in 𝕃2(Ω)

Denote(3.1)V:=Lp(τ,T;Lp(Ω))∩L2(τ,T;S01(Ω)),V*:=L2(τ,T;S-1(Ω))+Lp′(τ,T;Lp′(Ω)),
where p′ is the conjugate of p (i.e., 1/p+1/p′=1).Definition 3.1.
LetT>0 and uτ∈𝕃2(Ω) be given. A function u is called a weak solution of problem (1.2) on (τ,T) if
(3.2)u∈V,∂u∂t∈V*,u|t=τ=uτa.e.inΩ,∫τT∫Ω[(ut,φ)+((a∇xu,∇xφ))+|x|2s((a∇yu,∇yφ))+(f(u),φ)]dXdt=∫τT∫Ω(g(t),φ)dXdt
for all test functions φ∈V.One can prove that ifu∈V and ∂u/∂t∈V*, then u∈C([0,T];𝕃2(Ω)) (see [10]). This makes the initial condition in (1.2) meaningful.Theorem 3.2.
Under assumptions (H1)–(H3), for anyτ∈ℝ, T>τ, uτ∈𝕃2(Ω) given, problem (1.2) has a unique weak solution u on (τ,T). Moreover, the solution u exists on the interval (τ,+∞) and the following inequality holds:
(3.3)‖u‖L2(Ω)2≤e-λ1β(t-τ)‖uτ‖L2(Ω)2+2C0|Ω|λ1β+e-λ1βt∫-∞teλ1βs‖g(s)‖L2(Ω)2ds,∀t≥τ.Proof.
The existence and uniqueness of a weak solution to problem (1.2) are proved similarly to the scalar case in [10], so it is omitted here.
We now prove inequality (3.3). Multiplying (1.2) by u, integrating over Ω, and using (1.11), we have(3.4)12ddt‖u‖L2(Ω)2+β‖u‖S01(Ω)2+∫Ω(f(u),u)dX=∫Ω(g(t),u)dX.
Using condition (1.3) and the Cauchy inequality, we obtain
(3.5)ddt‖u‖L2(Ω)2+2β‖u‖S01(Ω)2+2C1‖u‖Lp(Ω)p-2C0|Ω|≤2βλ1‖g(t)‖L2(Ω)2+βλ12‖u‖L2(Ω)2.
Because ∥u∥𝕊01(Ω)2≥λ1∥u∥𝕃2(Ω)2, so (3.5) becomes
(3.6)ddt‖u‖L2(Ω)2+λ1β‖u‖L2(Ω)2≤2C0|Ω|+1λ1β‖g(t)‖L2(Ω)2.
Applying the Gronwall inequality we get (3.3).Now, we can define the family of two-parameter mappings(3.7)U(t,τ):L2(Ω)⟶S01(Ω)∩Lp(Ω),uτ⟼U(t,τ)uτ,
where U(t,τ)uτ=u(t) is the unique weak solution of (1.2) with the initial datum uτ at time τ. Then U defines a continuous process on 𝕃2(Ω).Letℛ be the set of all functions r:ℝ→(0,+∞) such that limt→-∞eλ1βtr2(t)=0 and denote by 𝒟 the class of all families 𝒟̂={D(t):t∈ℝ}⊂ℬ(𝕊01(Ω)) such that D(t)⊂B̂(r(t)) for some r(t)∈ℛ, where B̂(r(t)) is the closed ball in 𝕊01(Ω) with radius r(t).Lemma 3.3.
Under assumptions (H1)–(H3), there exists a constantC>0 such that the solution u of problem (1.2) satisfies the following inequality for all t>τ:
(3.8)‖u‖S01(Ω)2≤C((1+(t-τ)+1t-τ)e-α(t-τ)‖uτ‖L2(Ω)2+(1+1t-τ)+(1+1t-τ)e-αt∫-∞teαs‖g(s)‖L2(Ω)2ds+(1+1t-τ)e-αt∫-∞t∫-∞seαr‖g(r)‖L2(Ω)2drds),
where α=βλ1. This implies that there exists a family of pullback 𝒟-absorbing sets in 𝕊01(Ω) for the process {U(t,τ)}.Proof.
We multiply (1.2) by -Gsu and integrate over Ω. After some standard transformations we obtain
(3.9)12ddt‖u(t)‖S01(Ω)2+β‖Gsu(t)‖L2(Ω)2⩽∫Ω[(f(u(t)),Δxu(t))+|x|2s(f(u),Δyu(t))]dX+∫Ω(g,Gsu(t))dX.
Without loss of generality, we may assume that f(0)=0. Otherwise we can replace f(u) by f̃(u)=f(u)-f(0). The function f̃ satisfies the same conditions with modified constants Ci(i=0,1,2,3), because |f(0)|⩽C2 (see (1.4)). Hence, since f(u(t))|∂Ω=0, we get
(3.10)∫Ω(f(u),Δxu(t))dX=∑k=1N1∑i=1m∫Ωfi(u)∂2ui∂Xk2dX=-∑k=1N1∑i=1m∑j=1m∫Ω∂fi∂uj(u)∂uj∂Xk∂ui∂XkdX=-∑k=1N1∫Ω(fu(u)∂u∂Xk,∂u∂Xk)dX≤C3∑k=1N1∫Ω|∂u∂Xk|2dX=C3∫Ω|∇xu|2dX,
where we have used condition (1.5). Similarly, we have
(3.11)∫Ω|x|2s(f(u),Δyu(t))dX≤C3∫Ω|x|2s|∇yu|2dX.
Hence
(3.12)∫Ω(f(u),Δxu(t))dX+∫Ω|x|2s(f(u),Δyu(t))≤C3‖u(t)‖S01(Ω)2.
By the Cauchy inequality we have
(3.13)∫Ω(g,Gsu(t))dX≤12β‖g‖L2(Ω)2+β2‖Gsu(t)‖L2(Ω)2.
From (3.9)–(3.13) we obtain
(3.14)ddt‖u(t)‖S01(Ω)2+β‖Gsu(t)‖L2(Ω)2≤2C3‖u(t)‖S01(Ω)2+1β‖g‖L2(Ω)2,
thus,
(3.15)ddt‖u(t)‖S01(Ω)2+α‖u(t)‖S01(Ω)2⩽2C3‖u(t)‖S01(Ω)2+1β‖g‖L2(Ω)2,
where α=βλ1. Multiplying (3.15) by (t-τ)eαt and integrating from τ to t, we obtain
(3.16)(t-τ)eαt‖u‖S01(Ω)2≤(2C3(t-τ)+1)∫τteαs‖u(s)‖S01(Ω)2ds+t-τβ∫-∞teαs‖g(s)‖L2(Ω)2ds.
Multiplying (3.3) by αeαt and integrating from τ to t, we have
(3.17)α∫τteαs‖u(s)‖L2(Ω)2ds≤α(t-τ)eατ‖uτ‖L2(Ω)2+2C0|Ω|αeαt+∫-∞t∫-∞seαr‖g(r)‖L2(Ω)2drds.
Now, from (3.5) we get
(3.18)ddt‖u‖L2(Ω)2+β‖u‖S01(Ω)2≤1α‖g(t)‖L2(Ω)2+2C1|Ω|.
Multiplying this equation by eαt and integrating from τ to t, we deduce that
(3.19)eαt‖u(t)‖L2(Ω)2+β∫τteαs‖u(s)‖S01(Ω)2ds≤eατ‖uτ‖L2(Ω)2+2C1|Ω|αeαt+1α∫τteαs‖g(s)‖L2(Ω)2ds+α∫τteαs‖u(s)‖L2(Ω)2ds.
Using (3.17), (3.19) becomes
(3.20)eαt‖u(t)‖L2(Ω)2+β∫τteαs‖u(s)‖S01(Ω)2ds≤eατ‖uτ‖L2(Ω)2+(C0+C1)|Ω|αeαt+α(t-τ)eατ‖uτ‖L2(Ω)2+1α∫-∞teατ‖g(s)‖L2(Ω)2ds+∫-∞t∫-∞seαr‖g(r)‖L2(Ω)2drds.
Substituting (3.20) into (3.16) we obtain
(3.21)‖u(t)‖S01(Ω)2≤e-α(t-τ)(2C3+1λ1+2C3λ1(t-τ)+1β(t-τ))‖uτ‖L2(Ω)2+(2C3+1t-τ)C0+C1αβ|Ω|+1αβ(2C3+α+1t-τ)e-αt∫-∞teαs‖g(s)‖L2(Ω)2ds+1β(2C3+1t-τ)e-αt∫-∞t∫-∞seαr‖g(r)‖L2(Ω)2drds.
Hence we get (3.8) with C=C(β,C0,C1,C3,λ1).
Letr02(t) be the right-hand side of (3.8), and let B̂0(r0(t)) be the closed ball in 𝕊01(Ω) centered at 0 with radius r0(t). Obviously for any 𝒟̂∈𝒟 and any t∈ℝ, by (3.8) there exists τ0=τ0(𝒟̂)≤t such that the solution u with initial datum uτ∈𝒟(τ) at time τ satisfies ∥u(t)∥𝕊01(Ω)≤r0(t) for all τ≤τ0; that is, ℬ̂={B̂0(r0(t)):t∈ℝ} is a family of bounded pullback 𝒟-absorbing sets in 𝕊01(Ω).From the above lemma we deduce that the process{U(t,τ)} maps a compact set of 𝕊01(Ω) to be a bounded set of 𝕊01(Ω), and thus by Proposition 2.1, the process {U(t,τ)} is norm-to-weak continuous in 𝕊01(Ω). Since {U(t,τ)} has a family of pullback 𝒟-absorbing sets in 𝕊01(Ω) and the embedding 𝕊01(Ω)↪𝕃2(Ω) is compact, we immediately get the following.Theorem 3.4.
Under assumptions (H1)–(H3), the process{U(t,τ)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕃2(Ω).


## 4. Some Further Results in the Gradient Case

In this section, instead of (H1)–(H3), we assume that(H1bis)
a=βIm, where Im is the unit matrix and β>0;(H2bis)
f satisfies (H2) and f(u)=graduF(u)=((∂F/∂u1)(u),…,(∂F/∂um)(u)), where F:ℝm→ℝ is a potential function satisfying
(4.1)C¯1|u|p-C¯0≤F(u)≤C¯2|u|p+C¯0,∀u∈Rm,
with C¯1, C¯2, C¯0 being positive constants(H3bis)
g∈Wloc1,2(ℝ,𝕃2(Ω)) satisfies
(4.2)∫-∞0eαt(‖g(t)‖L2(Ω)2+‖g′(t)‖L2(Ω)2)dt<+∞,
where α=βλ1.The aim of this section is to prove that the pullback𝒟-attractor obtained in Section 3 is more regular and has a finite fractal dimension.


### 4.1. Existence of a Pullback𝒟-Attractor in 𝕊01(Ω)∩𝕃p(Ω)

Denote byℛ the set of all functions r:ℝ→(0,+∞) such that limt→-∞eλ1βtr2(t)=0 and denote by 𝒟 the class of all families 𝒟̂={D(t):t∈ℝ}⊂ℬ(𝕊01(Ω)∩𝕃p(Ω)) such that D(t)⊂B̂(r(t)) for some r(t)∈ℛ, where B̂(r(t)) is the closed ball in 𝕊01(Ω)∩𝕃p(Ω) with radius r(t). Thanks to the above gradient structure, one can prove the existence of a pullback 𝒟-attractor, not only in 𝕃2(Ω), but also in the space 𝕊01(Ω)∩𝕃p(Ω) for the process {U(t,τ)}.We first prove the following.Lemma 4.1.
Under assumptions (H1bis)–(H3bis), the solutionu of problem (1.2) satisfies the following inequality for all t>τ:
(4.3)‖u‖S01(Ω)2+‖u‖Lp(Ω)p≤C(e-α(t-τ)‖uτ‖L2(Ω)2+1+e-αt∫-∞teαr‖g(r)‖L2(Ω)2dr),
where C=C(C0,C1,C¯1,C¯0,β,λ1). This implies that there exists a family of pullback 𝒟-absorbing sets in 𝕊01(Ω)∩𝕃p(Ω) for the process {U(t,τ)}.Proof.
Using (3.5) with α=λ1β and the fact that ∥u∥𝕊01(Ω)2≥λ1∥u∥𝕃2(Ω)2, we have
(4.4)ddt‖u‖L2(Ω)2+α‖u‖L2(Ω)2+β2‖u‖S01(Ω)2+2C1‖u‖Lp(Ω)p≤2C0|Ω|+2α‖g(t)‖L2(Ω)2,
thus
(4.5)ddt(eαt‖u‖L2(Ω)2)+Ceαt(β‖u‖S01(Ω)2+2‖u‖Lp(Ω)p)≤2C0|Ω|eαt+2αeαt‖g(t)‖L2(Ω)2.
Integrating from τ to s,τ≤s≤t+1, and in particular, we have
(4.6)eαs‖u(s)‖L2(Ω)2≤eατ‖uτ‖L2(Ω)2+2C0α|Ω|eαs+2α∫τseαr‖g(r)‖L2(Ω)2dr.
Furthermore, multiplying (4.5) from s to s+1 and using (4.6) we obtain
(4.7)∫ss+1eαr(β‖u(r)‖S01(Ω)2+2‖u(r)‖Lp(Ω)p)dr≤eαs‖u(s)‖L2(Ω)2+2C0|Ω|αeαs+2α∫ss+1eαr‖g(r)‖L2(Ω)2dr≤C(eατ‖uτ‖L2(Ω)2+eαs+∫ss+1eαr‖u(r)‖L2(Ω)2+∫τseαr‖u(r)‖L2(Ω)2)≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖u(r)‖L2(Ω)2).
By assumption (H2bis), then (4.7) becomes
(4.8)∫ss+1eαr(β‖u(r)‖S01(Ω)2+2F(u(r)))dr≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖u(r)‖L2(Ω)2).
Multiplying (1.2) by ∂u/∂t and integrating over Ω, we have
(4.9)‖ut‖L2(Ω)2+12ddt(β‖u‖S01(Ω)2+∫ΩF(u)dX)=∫Ω(g(t),ut)dX≤12‖g(t)‖L2(Ω)2+12‖ut‖L2(Ω)2,
thus
(4.10)ddteαt(β‖u(t)‖S01(Ω)+2∫ΩF(u(t))dX)≤eαt(β‖u(t)‖S01(Ω)+2∫ΩF(u(t))dX)+eαt‖g(t)‖L2(Ω)2.
Using (4.8), (4.10), and the uniform Gronwall inequality, we get
(4.11)eαt(β‖u(t)‖S01(Ω)2+2∫ΩF(u(t))dX)≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖g(r)‖L2(Ω)2dr).
Now, using (H2bis) once again we have from (4.11) that
(4.12)eαt(β‖u(t)‖S01(Ω)2+2‖u(t)‖Lp(Ω)p)≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖g(r)‖L2(Ω)2dr).
Thus we obtain (4.3) with a suitable positive constant
(4.13)C=C(C0,C1,C¯1,C¯0,β,λ1).
Hence, by the argument as in the end of the proof of Lemma 3.3, we obtain a family of bounded pullback 𝒟-absorbing sets in 𝕊01(Ω)∩𝕃p(Ω).To prove that the process{U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕃p(Ω), we need the following lemma.Lemma 4.2 (see [8, Lemma 3.6]).
Let{U(t,τ)} be a norm-to-weak continuous process in 𝕃2(Ω) and 𝕃p(Ω), and let {U(t,τ)} satisfy the following two conditions: (i)
{U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕃2(Ω);(ii)
for anyɛ>0, ℬ̂∈𝒟, there exist constants M(ɛ,ℬ̂) and τ0(ɛ,ℬ̂)≤t such that
(4.14)(∫Ω(|U(t,τ)uτ|≥M)|U(t,τ)uτ|p)1/p<ɛ,foranyuτ∈B(τ),τ≤τ0. Then {U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕃p(Ω).Theorem 4.3.
Under assumptions (H1bis)–(H3bis), the process{U(t,τ)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕃p(Ω).Proof.
It is sufficient to show that the process{U(t,τ)} satisfies the condition (ii) in Lemma 4.2. We will give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
LetM be a positive number, we will write u≥M (or u≤-M) as any component of u is greater than or equal to M (or as any component of u is less than or equal to -M).
Using (1.3), (1.4), and for u≥M large enough, we have(4.15)(f(u),u-M)≥(f(u),u)-Mm|f(u)|≥C̃3|u|p-C̃2|u|p-1≥C42|u-M|p+αp|u-M|2,with0<C42<C̃3,
because lim|u|→+∞(C̃3|u|p-C̃2|u|p-1)/((C4/2)|u-M|p+(α/p)|u-M|2)>1.
Multiplying (1.2) by (u-M)+|(u-M)+|p-2 and integrating over Ω we obtain(4.16)1pddt∫Ω(u≥M)|u-M|pdX+∫Ω(u≥M)((a∇x(u-M),∇x((u-M)|u-M|p-2)))dX+∫Ω(u≥M)|x|2s((a∇y(u-M),∇y((u-M)|u-M|p-2)))dX+∫Ω(u≥M)|u-M|p-2(f(u),u-M)dX=∫Ω(g(t),u-M)dX,
where
(4.17)(u-M)+:={u-Mifu≥M,0,inothercases.
On the other hand, by the Cauchy inequality, we have
(4.18)|(g(t),(u-M))|u-M|p-2|≤|u-M|p-1|g(t)|≤C42|u-M|2p-2+12C4|g|2,
which implies that
(4.19)(g(t),u-M)|u-M|p-2≥-C42|u-M|2p-2-12C4|g|2.
Hence, from (4.15) and (4.19), we have
(4.20)|u-M|p-2[(f(u),u-M)+(g(t),u-M)]≥αp|u-M|p-12C4|g|2.
From (4.16), using (4.20) and noting that
(4.21)∫Ω(u≥M)((a∇x(u-M),∇x((u-M)|u-M|p-2)))dX+∫Ω(u≥M)|x|2s((a∇y(u-M),∇y((u-M)|u-M|p-2)))dX≥0,a=βIm,
we have
(4.22)ddt‖u-M‖Lp(Ω(u≥M))p+α‖u-M‖Lp(Ω(u≥M)))p≤p2C4‖g‖L2(Ω)2.
Now, multiplying the above inequality by (t-τ)eαt and integrating from τ to t, we get
(4.23)(t-τ)eαt‖u-M‖Lp(Ω(u≥M))p≤∫τteαs‖u-M‖Lp(Ω(u≥M))pds+p2C4(t-τ)∫τteαs‖g(s)‖L2(Ω)2ds≤∫τteαs‖u‖Lp(Ω)pds+p2C4(t-τ)∫τteαs‖g(s)‖L2(Ω)2ds.
Then
(4.24)‖(u-M)+‖Lp(Ω)p≤1t-τe-αt∫τteαs‖u‖Lp(Ω)pds+pe-αt2C4∫-∞teαs‖g(s)‖L2(Ω)2ds.
On the other hand, integrating (4.5) from τ to t, we have
(4.25)∫τteαs‖u(s)‖Lp(Ω)pds≤eατ‖uτ‖L2(Ω)2+2C0α|Ω|eαt+1α∫τteαs‖g(s)‖L2(Ω)2ds.
Therefore, substituting (4.25) into (4.24), we obtain
(4.26)‖(u-M)+‖Lp(Ω)p≤C(e-α(t-τ)t-τ‖uτ‖L2(Ω)2+1t-τ+1t-τe-αt∫-∞teαs‖g(s)‖L2(Ω)2ds).
Hence, for any ɛ>0, there exists M1>0 and τ1<t such that for any τ<τ1 and any M≥M1, we have
(4.27)∫Ω(u(t)≥M)|u-M|pdx≤ɛ.
Repeating the same step above, just taking (u+M)- instead of (u-M)+, we deduce that there exist M2>0 and τ2<t such that for any τ<τ2 and any M≥M2,
(4.28)∫Ω(u(t)≤-M)|u+M|pdx≤ɛ,
where
(4.29)(u+M)-={u+M,u≤-M,0,inothercases.
Let M0=max{M1,M2} and τ0=min{τ1,τ2}, we obtain
(4.30)∫Ω(|u|≥M)(|u|-M)pdx≤ɛforτ≤τ0,M≥M0.
So, we have
(4.31)∫Ω(|u|≥2M)|u|pdx=∫Ω(|u|≥2M)((|u|-M)+M)pdx≤2p-1(∫Ω(|u|≥2M)(|u|-M)pdx+∫Ω(|u|≥2M)Mpdx)≤2p-1(∫Ω(|u|≥M)(|u|-M)pdx+∫Ω(|u|≥M)(|u|-M)pdx)≤2pɛ.
This completes the proof.To prove the existence of a pullback𝒟-attractor in 𝕊01(Ω)∩𝕃p(Ω), we need the folowing lemma.Lemma 4.4.
Under assumptions (H1bis)–(H3bis), for anyt∈ℝ and any bounded subset B⊂𝕃2(Ω), there exists a positive constant T=T(B,t)≤t such that
(4.32)‖ut(t)‖L2(Ω)2≤C(1+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds),
for all τ≤T(B,t) and all uτ∈B, where C>0 is independent of t and B.Proof.
We give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
Differentiating (1.2) in time and setting v=ut, we get(4.33)vt-aGsv+f′(u)v=g′(r).
Multiplying this inequality by eαrv and integrating over Ω and using (1.11), we get
(4.34)12ddr(eαr‖v‖L2(Ω)2)+βeαr‖v‖S01(Ω)2+eαr∫Ω(f′(u)v,v)dX≤α2eαr‖v‖L2(Ω)2+eαr∫Ω(g′(r),v)dX.
By the Cauchy inequality and using (1.5), we obtain that
(4.35)ddr(eαr‖v‖L2(Ω)2)≤(2C3+α+1)eαr‖v‖L2(Ω)2+eαr‖g′(r)‖L2(Ω)2.
Let τ≤s≤t-1. Using (3.5), we have
(4.36)dds(eαs‖u‖L2(Ω)2)+β‖u‖S01(Ω)2+2C1eαs‖u‖Lp(Ω)p≤1αeαs‖g(s)‖L2(Ω)2+C0eαs|Ω|.
By (H2bis) we then infer from the above inequality that
(4.37)dds(eαs‖u‖L2(Ω)2)+C(βeαs‖u‖S01(Ω)2+2eαs∫ΩF(u)dX)≤C(eαs‖g(s)‖L2(Ω)2+eαs).
Integrating this inequality from r to r+1, we obtain
(4.38)∫rr+1(βeαs‖u‖S01(Ω)2+2eαs∫ΩF(u)dX)ds≤C(eαr‖u(r)‖L2(Ω)2+∫rr+1(eαs‖g(s)‖L2(Ω)2+eαs)ds).
On the other hand, integrating (4.36) from τ to t, we obtain
(4.39)eαt‖u‖L2(Ω)2≤eατ‖uτ‖L2(Ω)2+1α∫-∞teαs‖g(s)‖L2(Ω)2ds+C0|Ω|αeαt.
So, substituting (4.39) into (4.38), we deduce
(4.40)∫rr+1(βeαs‖u‖S01(Ω)2+2eαs∫ΩF(u)dX)ds≤C(eατ‖uτ‖L2(Ω)2+∫-∞teαs‖g(s)‖L2(Ω)2ds+eαt)<∞,∀r∈[τ,t-1].
Now multiplying (1.2) by eαrv and integrating over Ω, we have
(4.41)eαr‖v‖L2(Ω)2+ddr(βeαr‖u‖S01(Ω)2+2eαr∫ΩF(u)dX)≤α(βeαr‖u‖S01(Ω)2+2eαr∫ΩF(u)dX)+eαr‖g(r)‖L2(Ω)2.
So applying the uniform Gronwall inequality, we get
(4.42)βeαr‖u‖S01(Ω)2+2eαr∫ΩF(u)dX≤C(eατ‖uτ‖L2(Ω)2+eαt+∫-∞teαs‖g(s)‖L2(Ω)2ds).
Integrating (4.41) from r to r+1 and by (4.40)–(4.42), we have
(4.43)∫rr+1eαs‖v‖L2(Ω)2ds≤C(eατ‖uτ‖L2(Ω)2+eαt+∫-∞teαs‖g(s)‖L2(Ω)2ds).
Therefore, by (4.35), (4.43), using the uniform Gronwall inequality once again, we get
(4.44)eαt‖v‖L2(Ω)2≤C(eατ‖uτ‖L2(Ω)2+eαt+∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds).
Hence we get (4.32).Theorem 4.5.
Under assumptions (H1bis)–(H3bis), the process{U(t,τ)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕊01(Ω)∩𝕃p(Ω).Proof.
By Lemma4.1, {U(t,τ)} has a family of bounded pullback 𝒟-absorbing sets in 𝕊01(Ω)∩𝕃p(Ω). It remains to show that {U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕊01(Ω)∩𝕃p(Ω), that is, for any t∈ℝ, any ℬ̂∈𝒟, and any sequence τn→-∞, any sequence uτn∈B(τn), the sequence {U(t,τn)uτn} is precompact in 𝕊01(Ω)∩𝕃p(Ω). Thanks to Theorem 4.3, we need only to show that the sequence {U(t,τn)uτn} is precompact in 𝕊01(Ω).
Letun(t)=U(t,τn)uτn. By Theorem 3.4, we can assume that {un(t)} is a Cauchy sequence in 𝕃2(Ω). We have(4.45)‖un(t)-um(t)‖S01(Ω)2=-〈Gsun(t)-Gsum(t),un(t)-um(t)〉=-〈dundt(t)-dumdt(t),un(t)-um(t)〉-〈f(un(t))-f(um(t)),un(t)-um(t)〉≤‖ddtun(t)-ddtum(t)‖L2(Ω)2‖un(t)-um(t)‖L2(Ω)2+C3‖un(t)-um(t)‖L2(Ω)2,
where we have used condition (1.5). Because {un(t)} is a Cauchy sequence in 𝕃2(Ω) and by Lemma 4.4, one gets
(4.46)‖un(t)-um(t)‖S01(Ω)⟶0,asm,n⟶∞.
The proof is complete.


### 4.2.𝕃2p-2(Ω) and 𝕊02(Ω)-Boundedness of the Pullback 𝒟-Attractor

First, we prove the existence of a family of pullback𝒟-absorbing sets for process U(t,τ) in 𝕃2p-2(Ω).Proposition 4.6.
Under assumptions (H1bis)–(H3bis), then for anyt∈ℝ and any bounded subset B⊂𝕃2(Ω), there exists a positive constant τ0=τ0(B,t)≤t such that
(4.47)‖u‖L2p-2(Ω)2p-2≤C(1+‖g(t)‖L2(Ω)2+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)
for all τ≤τ0 and all uτ∈B, where C>0 is independent of t and B.Proof.
Multiplying (1.2) by |u|p-2u and integrating over Ω we obtain
(4.48)∫Ω[((a∇xu,∇x(|u|p-2u)))+|x|2s((a∇yu,∇y(|u|p-2u)))]dX+∫Ω(f(u),u)|u|p-2dX=-∫Ω(ut,|u|p-2u)dX+∫Ω(g(t),|u|p-2u)dX.
By the Cauchy inequality, (1.3) and note that
(4.49)∫Ω[((a∇xu,∇x(|u|p-2u)))+|x|2s((a∇yu,∇y(|u|p-2u)))]dX≥0;herea=βIm,
then we get
(4.50)C1‖u‖L2p-2(Ω)2p-2≤1C1‖ut‖L2(Ω)2+1C1‖g(t)‖L22+C12‖u‖L2p-22p-2.
Hence, by (4.32) we deduce from (4.16) that
(4.51)C12‖u‖L2p-2(Ω)2p-2≤1C1C(1+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)+1C1‖g(t)‖L2(Ω)2.
Therefore, we get (4.47) and the proof is complete.And now, we denote by𝕊02(Ω) the closure of (C0∞(Ω))m in the norm (4.52)‖u‖S02(Ω)=(∫Ω(|Δxu|2+|x|2sΔyu|2)dX)1/2.
It is easy to see that 𝕊02(Ω) is a Banach space endowed with the above norm. We now prove the 𝕊02(Ω)-boundedness of the pullback 𝒟-attractor.First, we recall a lemma (see [15]) which is necessary for our proof.Lemma 4.7.
LetX,Y be Banach spaces such that X is reflexive, and the inclusion X⊂Y is continuous. Assume that {un} is a bounded sequence in L∞(τ,T;X) such that un⇀u weakly in Lq(τ,T;X) for some q∈[1,+∞) and u∈C([τ,T];Y). Then, u(t)∈X for all t∈[τ,T] and ∥u(t)∥X≤supn≥1∥un∥L∞(τ,T;X), for all t∈[τ,T].Letun(t) be the Galerkin approximations of the solution u(t) of (1.2) then by Lemma 4.7 with noticing that un=U(t,τ)unτ⇀u=U(t,τ)uτ in L2(τ,T;𝕊01(Ω)) and the inclusion 𝕊02(Ω)⊂𝕊01(Ω) is continuous, we only need the estimation on u(t)=U(t,τ)uτ.Theorem 4.8.
Under assumptions (H1bis)–(H3bis), the pullback𝒟-attractor 𝒜̂={A(t):t∈ℝ} in 𝕊01(Ω)∩𝕃p(Ω) of the process {U(t,τ)} is bounded in 𝕊02(Ω). More precisely, for any τ<T1<T2, the set ⋃t∈[T1,T2]A(t) is a bounded subset of 𝕊02(Ω).Proof.
Let us fix a bounded setB⊂𝕃2(Ω), τ∈ℝ, ɛ>0, t>τ+ɛ and uτ∈B. Multiplying the first equation in (1.2) by Gsu and integrating over Ω, we have
(4.53)∫Ω(aGsu(r),Gsu(r))dX=∫Ω(ur′,Gsu(r))dX+∫Ω(f(u(r)),Gsu(r))dX-∫Ω(g(r),Gsu(r))dX.
By the Cauchy inequality we have
(4.54)-∫Ω(g(r),Gsu(r))dX≤2β‖g(r)‖L2(Ω)2+β8‖Gsu(r)‖L2(Ω)2,∫Ω(ur′,Gsu(r))dX≤2β‖ur′‖S01(Ω)2+β8‖Gsu(r)‖L2(Ω)2.
Using (3.12), (1.12), and (4.54), then from (4.53) we get
(4.55)β‖Gsu(r)‖L2(Ω)2≤C3‖u(r)‖S01(Ω)2+2β(‖ur′‖L2(Ω)2+‖g(r)‖L2(Ω)2)+β4‖Gsu(r)‖L2(Ω)2=C3∫Ω(u(r),-Gsu(r))dX+2β(‖ur′‖L2(Ω)2+‖g(r)‖L2(Ω)2)+β4‖Gsu(r)‖L2(Ω)2≤2C32β‖u(r)‖L2(Ω)2+2β‖ur′‖L2(Ω)2+2β‖g(r)‖L2(Ω)2+β2‖Gsu(r)‖L2(Ω)2.
Hence,
(4.56)β2‖Gsu(r)‖L2(Ω)2≤2C32β‖u(r)‖L2(Ω)2+2β‖ur′‖L2(Ω)2+2β‖g(r)‖L2(Ω)2.
Differentiating the first equation in (1.2) in time t and setting v(r)=u′(r), then multiplying by v(r) and using (1.11) we get
(4.57)12ddr‖v(r)‖L2(Ω)2+β‖v(r)‖S01(Ω)2≤-∫Ω(fu′(u)v(r),v(r))dX+∫Ω(g′(r),v(r))dX≤C3‖v(r)‖L2(Ω)2+12‖v(r)‖L2(Ω)2+12‖g′(r)‖L2(Ω)2.
Hence,
(4.58)ddr‖v(r)‖L2(Ω)2≤(2C3+1)‖v(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2.
Integrating the above inequality, we have
(4.59)‖v(r)‖L2(Ω)2≤‖v(s)‖L2(Ω)2+(2C3+1)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2+∫τ+ɛ/2t‖g′(θ)‖L2(Ω)2dθ,
for all τ+ɛ/2≤s≤r≤t.
Now, integrating with respect tos between τ+ɛ/2 and r, we get(4.60)(r-τ-ɛ2)‖v(r)‖L2(Ω)2≤∫τ+ɛ/2r‖v(s)‖L2(Ω)2ds+(2C3+1)(r-τ-ɛ2)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+(r-τ-ɛ2)∫τ+ɛ/2t‖g′(θ)‖L2(Ω)2dθ≤((2C3+1)(t-τ-ɛ2)+1)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+(r-τ-ɛ2)∫τ+ɛ/2t‖g′(θ)‖L2(Ω)2dθ,
for all τ+ɛ/2≤r≤t, and in particular, for all r∈[τ+ɛ,t] we have that (from the above estimate)
(4.61)‖v(r)‖L2(Ω)2≤2ɛ((2C3+1)(t-τ-ɛ2)+1)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+∫τt‖g′(θ)‖L2(Ω)2dθ.
On the other hand, multiplying the first equation in (1.2) by v(r) and integrating over Ω, we deduce that
(4.62)‖v(r)‖L2(Ω)2+β2ddr‖u(r)‖S01(Ω)2+∫Ω(f(u,v))dX≤∫Ω(g,v)dX,
where we have used (1.11). Using the Cauchy inequality and condition (H2bis), then (4.62) becomes
(4.63)‖v(r)‖L2(Ω)2+ddr(β‖u(r)‖S01(Ω)2+2∫ΩF(u(r))dX)≤‖g(r)‖L2(Ω)2.
Integrating from τ+ɛ/2 to t we have
(4.64)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+β‖u(t)‖S01(Ω)2+2∫ΩF(u(t))dX≤β‖u(τ+ɛ2)‖S01(Ω)2+2∫ΩF(u(τ+ɛ2))dX+∫τ+ɛ/2t‖g(θ)‖L2(Ω)2,
and hence because of (4.1), we get
(4.65)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ≤β‖u(τ+ɛ2)‖S01(Ω)2+2C¯2‖u(τ+ɛ2)‖Lp(Ω)p+4C¯0|Ω|+∫τt‖g(θ)‖L2(Ω)2.
Now, substituting (4.65) into (4.61) we deduce
(4.66)‖v(θ)‖L2(Ω)2≤2ɛ((2C3+1)(t-τ-ɛ2)+1)×(β‖u(τ+ɛ2)‖S01(Ω)2+2C¯2‖u(τ+ɛ2)‖Lp(Ω)2+4C¯0|Ω|+∫τt‖g(θ)‖L2(Ω)2dθ)+∫τt‖g′(θ)‖L2(Ω)2dθ,
for all r∈[τ+ɛ,t]. Finally, from (4.66) and (4.56) we obtain
(4.67)‖Gsu(r)‖L2(Ω)2≤8β2ɛ((2C3+1)(t-τ-ɛ2)+1)×(β‖u(τ+ɛ2)‖S01(Ω)2+2C¯2‖u(τ+ɛ2)‖Lp(Ω)p+4C¯0|Ω|+∫τt‖g(θ)‖L2(Ω)2dθ)+2β∫τt‖g′(θ)‖L2(Ω)2dθ+2β2(C32‖u(r)‖L2(Ω)2+2‖g(r)‖L2(Ω)2),∀r∈[τ+ɛ,t].
Because ∥u∥𝕊02(Ω)2=∥Gsu∥𝕃2(Ω)2 then from (4.67), the proof is complete.


### 4.3. Fractal Dimensional Estimates of the Pullback𝒟-Attractor

Theorem 4.9.
Under assumptions (H1bis)–(H3bis), the processU(t,τ) possesses a pullback 𝒟-attractor 𝒜𝕃2(Ω) which has a finite fractal dimension in 𝕃2(Ω) and
(4.68)dim⁡f(A(t))≤klog⁡(1+8⋅e2C31-δ)[log⁡21+δ]-1,∀t∈R,
where δ<1,k∈ℕ, and C3 in (1.5).Proof.
LetHk=span{e1,e2,…,ek}⊂𝕃2(Ω) and Pk:𝕃2(Ω)→Hk be the orthogonal projection, where e1,e2,…,ej,… are the eigenvectors of the operator -Gs corresponding to eigenvalues {λj}j=1∞ such that 0<λ1<λ2≤λ3≤⋯≤λj≤⋯ and λj→+∞ as j→+∞.
From (4.3), we can easily show that there exists a uniformly pullback absorbing set ℬ of process U(t,τ) in 𝕊01(Ω). We set u1(t)=U(t,τ)u1τ and u2(t)=U(t,τ)u2τ to be solutions associated to problem (1.2) with initial datum u1τ,u2τ∈ℬ.
Letw=u1-u2, because u1, u2 being two solutions of (1.2) then we have(4.69)∂w∂t-aGsw+f(u1)-f(u2)=0.
Multiplying (4.69) with w and integrating over Ω then we have
(4.70)12ddt‖w(t)‖L2(Ω)2+β‖w‖S01(Ω)2+∫Ω(f(u1)-f(u2),w)dX≤0,
here, we have used (1.11).
Using (1.5) then we have(4.71)ddt‖w(t)‖L2(Ω)2≤2C3‖w(t)‖L2(Ω)2.
Thus,
(4.72)‖w(t)‖L2(Ω)2≤e2C3(t-τ)‖w(τ)‖L2(Ω)2.
Let w(t)=w1(t)+w2(t) where w1(t):=Pkw(t) and w2(t):=(I-Pk)w(t). Therefore, by (4.72) we have
(4.73)‖w1(t)‖L2(Ω)2≤e2C3(t-τ)‖w(τ)‖L2(Ω)2.
Now, taking the inner product of (4.69) with w2 in 𝕃2(Ω), we have
(4.74)12ddt‖w2(t)‖L2(Ω)2+β‖w2‖S01(Ω)2≤-∫(f(u1)-f(u2),w2)dX.
Using the Hölder inequality and (1.4), we have
(4.75)-∫Ω(f(u1)-f(u2),w2)dX≤∫Ω|f(u1)-u(u2)||w2|dX≤(∫Ω|f(u1)-f(u2)|2dX)1/2(∫Ω|w2|2dX)1/2≤C(∫Ω(1+|u1|2p-2+|u2|2p-2)dX)1/2‖w2‖L2(Ω)≤C(1+‖u1‖L2p-2(Ω)2p-2+‖u2‖L2p-2(Ω)2p-2)1/2‖w‖L2(Ω)≤C(1+‖u1‖L2p-2(Ω)2p-2+‖u2‖L2p-2(Ω)2p-2)‖w‖L2(Ω).
Therefore, by (4.74), (4.75), and Proposition 4.6 we obtain
(4.76)12ddt‖w2(t)‖L2(Ω)2+β‖w2‖S01(Ω)2≤C(1+‖g(t)‖L2(Ω)2+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)‖w(t)‖L2(Ω).
Because ∥w2(t)∥𝕊01(Ω)2≥λk∥w2(t)∥𝕃2(Ω)2, then (4.76) implies that
(4.77)ddt‖w2(t)‖L2(Ω)2+2βλk‖w2‖L2(Ω)2≤2C(1+‖g(t)‖L2(Ω)2+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)‖w(t)‖L2(Ω).
Now, multiplying (4.77) by eβλkt and integrating from τ to t, we get
(4.78)‖w2(t)‖L2(Ω)2≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+2Ce-βλkt∫τte-βλks×[1+‖g(s)‖L2(Ω)2+e-αt∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr]×‖w(s)‖L2(Ω)ds.
Using (4.73) we have
(4.79)‖w2(t)‖L2(Ω)2≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+C‖w(τ)‖L2(Ω)e-βλkt∫τteβλkseC3(s-τ)×[1+‖g(s)‖L2(Ω)2+e-αs∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr]ds≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+C‖w(τ)‖L2(Ω)eC3(t-τ)∫-∞te-βλk(t-s)×[1+‖g(s)‖L2(Ω)2+e-αs∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr]ds≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+C‖w(τ)‖L2(Ω)eC3(t-τ)×[1βλk+∫-∞te-βλk(t-s)‖g(s)‖L2(Ω)2ds+∫-∞te-βλk(t-s)e-αs×(∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)ds].
Now, because
(4.80)∫-∞teαs‖g(s)‖L2(Ω)2ds<+∞,
we can see that, for all t∈ℝ (see, e.g., [6, Lemma 3.6]),
(4.81)∫-∞te-βλk(t-s)‖g(s)‖L2(Ω)2ds⟶0ask⟶+∞,
and we have
(4.82)∫-∞te-βλk(t-s)e-βλ1s(∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)ds≤(∫-∞te-βλkt+β(λk-λ1)sds)(∫-∞teαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)=e-αtβλk-α∫-∞teαs(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr.
Thus, for any t∈ℝ, from (4.82) we have
(4.83)∫-∞te-βλk(t-s)e-βλ1s(∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)ds⟶0ask⟶+∞.
Combining (4.81), (4.83) and taking T0=t-τ=1, we get k is sufficient large then from (4.79) we deduce
(4.84)‖w2(t)‖L2(Ω)2≤δ‖w(τ)‖L2(Ω)2,here0<δ<1.
From (4.73) and (4.31), we have
(4.85)‖w1(t)‖L2(Ω)2≤l0‖w(τ)‖L2(Ω)2,‖w2(t)‖L2(Ω)2≤δ‖w(τ)‖L2(Ω)2,∀t∈R.
Here, l0=e2C3; 0<δ<1; T0=1. Thus, the process U(t,τ) associated to (1.2) satisfies all conditions of Theorem 2.10. This completes the proof.


## 4.1. Existence of a Pullback𝒟-Attractor in 𝕊01(Ω)∩𝕃p(Ω)

Denote byℛ the set of all functions r:ℝ→(0,+∞) such that limt→-∞eλ1βtr2(t)=0 and denote by 𝒟 the class of all families 𝒟̂={D(t):t∈ℝ}⊂ℬ(𝕊01(Ω)∩𝕃p(Ω)) such that D(t)⊂B̂(r(t)) for some r(t)∈ℛ, where B̂(r(t)) is the closed ball in 𝕊01(Ω)∩𝕃p(Ω) with radius r(t). Thanks to the above gradient structure, one can prove the existence of a pullback 𝒟-attractor, not only in 𝕃2(Ω), but also in the space 𝕊01(Ω)∩𝕃p(Ω) for the process {U(t,τ)}.We first prove the following.Lemma 4.1.
Under assumptions (H1bis)–(H3bis), the solutionu of problem (1.2) satisfies the following inequality for all t>τ:
(4.3)‖u‖S01(Ω)2+‖u‖Lp(Ω)p≤C(e-α(t-τ)‖uτ‖L2(Ω)2+1+e-αt∫-∞teαr‖g(r)‖L2(Ω)2dr),
where C=C(C0,C1,C¯1,C¯0,β,λ1). This implies that there exists a family of pullback 𝒟-absorbing sets in 𝕊01(Ω)∩𝕃p(Ω) for the process {U(t,τ)}.Proof.
Using (3.5) with α=λ1β and the fact that ∥u∥𝕊01(Ω)2≥λ1∥u∥𝕃2(Ω)2, we have
(4.4)ddt‖u‖L2(Ω)2+α‖u‖L2(Ω)2+β2‖u‖S01(Ω)2+2C1‖u‖Lp(Ω)p≤2C0|Ω|+2α‖g(t)‖L2(Ω)2,
thus
(4.5)ddt(eαt‖u‖L2(Ω)2)+Ceαt(β‖u‖S01(Ω)2+2‖u‖Lp(Ω)p)≤2C0|Ω|eαt+2αeαt‖g(t)‖L2(Ω)2.
Integrating from τ to s,τ≤s≤t+1, and in particular, we have
(4.6)eαs‖u(s)‖L2(Ω)2≤eατ‖uτ‖L2(Ω)2+2C0α|Ω|eαs+2α∫τseαr‖g(r)‖L2(Ω)2dr.
Furthermore, multiplying (4.5) from s to s+1 and using (4.6) we obtain
(4.7)∫ss+1eαr(β‖u(r)‖S01(Ω)2+2‖u(r)‖Lp(Ω)p)dr≤eαs‖u(s)‖L2(Ω)2+2C0|Ω|αeαs+2α∫ss+1eαr‖g(r)‖L2(Ω)2dr≤C(eατ‖uτ‖L2(Ω)2+eαs+∫ss+1eαr‖u(r)‖L2(Ω)2+∫τseαr‖u(r)‖L2(Ω)2)≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖u(r)‖L2(Ω)2).
By assumption (H2bis), then (4.7) becomes
(4.8)∫ss+1eαr(β‖u(r)‖S01(Ω)2+2F(u(r)))dr≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖u(r)‖L2(Ω)2).
Multiplying (1.2) by ∂u/∂t and integrating over Ω, we have
(4.9)‖ut‖L2(Ω)2+12ddt(β‖u‖S01(Ω)2+∫ΩF(u)dX)=∫Ω(g(t),ut)dX≤12‖g(t)‖L2(Ω)2+12‖ut‖L2(Ω)2,
thus
(4.10)ddteαt(β‖u(t)‖S01(Ω)+2∫ΩF(u(t))dX)≤eαt(β‖u(t)‖S01(Ω)+2∫ΩF(u(t))dX)+eαt‖g(t)‖L2(Ω)2.
Using (4.8), (4.10), and the uniform Gronwall inequality, we get
(4.11)eαt(β‖u(t)‖S01(Ω)2+2∫ΩF(u(t))dX)≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖g(r)‖L2(Ω)2dr).
Now, using (H2bis) once again we have from (4.11) that
(4.12)eαt(β‖u(t)‖S01(Ω)2+2‖u(t)‖Lp(Ω)p)≤C(eατ‖uτ‖L2(Ω)2+eαt+∫τteαr‖g(r)‖L2(Ω)2dr).
Thus we obtain (4.3) with a suitable positive constant
(4.13)C=C(C0,C1,C¯1,C¯0,β,λ1).
Hence, by the argument as in the end of the proof of Lemma 3.3, we obtain a family of bounded pullback 𝒟-absorbing sets in 𝕊01(Ω)∩𝕃p(Ω).To prove that the process{U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕃p(Ω), we need the following lemma.Lemma 4.2 (see [8, Lemma 3.6]).
Let{U(t,τ)} be a norm-to-weak continuous process in 𝕃2(Ω) and 𝕃p(Ω), and let {U(t,τ)} satisfy the following two conditions: (i)
{U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕃2(Ω);(ii)
for anyɛ>0, ℬ̂∈𝒟, there exist constants M(ɛ,ℬ̂) and τ0(ɛ,ℬ̂)≤t such that
(4.14)(∫Ω(|U(t,τ)uτ|≥M)|U(t,τ)uτ|p)1/p<ɛ,foranyuτ∈B(τ),τ≤τ0. Then {U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕃p(Ω).Theorem 4.3.
Under assumptions (H1bis)–(H3bis), the process{U(t,τ)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕃p(Ω).Proof.
It is sufficient to show that the process{U(t,τ)} satisfies the condition (ii) in Lemma 4.2. We will give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
LetM be a positive number, we will write u≥M (or u≤-M) as any component of u is greater than or equal to M (or as any component of u is less than or equal to -M).
Using (1.3), (1.4), and for u≥M large enough, we have(4.15)(f(u),u-M)≥(f(u),u)-Mm|f(u)|≥C̃3|u|p-C̃2|u|p-1≥C42|u-M|p+αp|u-M|2,with0<C42<C̃3,
because lim|u|→+∞(C̃3|u|p-C̃2|u|p-1)/((C4/2)|u-M|p+(α/p)|u-M|2)>1.
Multiplying (1.2) by (u-M)+|(u-M)+|p-2 and integrating over Ω we obtain(4.16)1pddt∫Ω(u≥M)|u-M|pdX+∫Ω(u≥M)((a∇x(u-M),∇x((u-M)|u-M|p-2)))dX+∫Ω(u≥M)|x|2s((a∇y(u-M),∇y((u-M)|u-M|p-2)))dX+∫Ω(u≥M)|u-M|p-2(f(u),u-M)dX=∫Ω(g(t),u-M)dX,
where
(4.17)(u-M)+:={u-Mifu≥M,0,inothercases.
On the other hand, by the Cauchy inequality, we have
(4.18)|(g(t),(u-M))|u-M|p-2|≤|u-M|p-1|g(t)|≤C42|u-M|2p-2+12C4|g|2,
which implies that
(4.19)(g(t),u-M)|u-M|p-2≥-C42|u-M|2p-2-12C4|g|2.
Hence, from (4.15) and (4.19), we have
(4.20)|u-M|p-2[(f(u),u-M)+(g(t),u-M)]≥αp|u-M|p-12C4|g|2.
From (4.16), using (4.20) and noting that
(4.21)∫Ω(u≥M)((a∇x(u-M),∇x((u-M)|u-M|p-2)))dX+∫Ω(u≥M)|x|2s((a∇y(u-M),∇y((u-M)|u-M|p-2)))dX≥0,a=βIm,
we have
(4.22)ddt‖u-M‖Lp(Ω(u≥M))p+α‖u-M‖Lp(Ω(u≥M)))p≤p2C4‖g‖L2(Ω)2.
Now, multiplying the above inequality by (t-τ)eαt and integrating from τ to t, we get
(4.23)(t-τ)eαt‖u-M‖Lp(Ω(u≥M))p≤∫τteαs‖u-M‖Lp(Ω(u≥M))pds+p2C4(t-τ)∫τteαs‖g(s)‖L2(Ω)2ds≤∫τteαs‖u‖Lp(Ω)pds+p2C4(t-τ)∫τteαs‖g(s)‖L2(Ω)2ds.
Then
(4.24)‖(u-M)+‖Lp(Ω)p≤1t-τe-αt∫τteαs‖u‖Lp(Ω)pds+pe-αt2C4∫-∞teαs‖g(s)‖L2(Ω)2ds.
On the other hand, integrating (4.5) from τ to t, we have
(4.25)∫τteαs‖u(s)‖Lp(Ω)pds≤eατ‖uτ‖L2(Ω)2+2C0α|Ω|eαt+1α∫τteαs‖g(s)‖L2(Ω)2ds.
Therefore, substituting (4.25) into (4.24), we obtain
(4.26)‖(u-M)+‖Lp(Ω)p≤C(e-α(t-τ)t-τ‖uτ‖L2(Ω)2+1t-τ+1t-τe-αt∫-∞teαs‖g(s)‖L2(Ω)2ds).
Hence, for any ɛ>0, there exists M1>0 and τ1<t such that for any τ<τ1 and any M≥M1, we have
(4.27)∫Ω(u(t)≥M)|u-M|pdx≤ɛ.
Repeating the same step above, just taking (u+M)- instead of (u-M)+, we deduce that there exist M2>0 and τ2<t such that for any τ<τ2 and any M≥M2,
(4.28)∫Ω(u(t)≤-M)|u+M|pdx≤ɛ,
where
(4.29)(u+M)-={u+M,u≤-M,0,inothercases.
Let M0=max{M1,M2} and τ0=min{τ1,τ2}, we obtain
(4.30)∫Ω(|u|≥M)(|u|-M)pdx≤ɛforτ≤τ0,M≥M0.
So, we have
(4.31)∫Ω(|u|≥2M)|u|pdx=∫Ω(|u|≥2M)((|u|-M)+M)pdx≤2p-1(∫Ω(|u|≥2M)(|u|-M)pdx+∫Ω(|u|≥2M)Mpdx)≤2p-1(∫Ω(|u|≥M)(|u|-M)pdx+∫Ω(|u|≥M)(|u|-M)pdx)≤2pɛ.
This completes the proof.To prove the existence of a pullback𝒟-attractor in 𝕊01(Ω)∩𝕃p(Ω), we need the folowing lemma.Lemma 4.4.
Under assumptions (H1bis)–(H3bis), for anyt∈ℝ and any bounded subset B⊂𝕃2(Ω), there exists a positive constant T=T(B,t)≤t such that
(4.32)‖ut(t)‖L2(Ω)2≤C(1+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds),
for all τ≤T(B,t) and all uτ∈B, where C>0 is independent of t and B.Proof.
We give some formal calculations, a rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [15].
Differentiating (1.2) in time and setting v=ut, we get(4.33)vt-aGsv+f′(u)v=g′(r).
Multiplying this inequality by eαrv and integrating over Ω and using (1.11), we get
(4.34)12ddr(eαr‖v‖L2(Ω)2)+βeαr‖v‖S01(Ω)2+eαr∫Ω(f′(u)v,v)dX≤α2eαr‖v‖L2(Ω)2+eαr∫Ω(g′(r),v)dX.
By the Cauchy inequality and using (1.5), we obtain that
(4.35)ddr(eαr‖v‖L2(Ω)2)≤(2C3+α+1)eαr‖v‖L2(Ω)2+eαr‖g′(r)‖L2(Ω)2.
Let τ≤s≤t-1. Using (3.5), we have
(4.36)dds(eαs‖u‖L2(Ω)2)+β‖u‖S01(Ω)2+2C1eαs‖u‖Lp(Ω)p≤1αeαs‖g(s)‖L2(Ω)2+C0eαs|Ω|.
By (H2bis) we then infer from the above inequality that
(4.37)dds(eαs‖u‖L2(Ω)2)+C(βeαs‖u‖S01(Ω)2+2eαs∫ΩF(u)dX)≤C(eαs‖g(s)‖L2(Ω)2+eαs).
Integrating this inequality from r to r+1, we obtain
(4.38)∫rr+1(βeαs‖u‖S01(Ω)2+2eαs∫ΩF(u)dX)ds≤C(eαr‖u(r)‖L2(Ω)2+∫rr+1(eαs‖g(s)‖L2(Ω)2+eαs)ds).
On the other hand, integrating (4.36) from τ to t, we obtain
(4.39)eαt‖u‖L2(Ω)2≤eατ‖uτ‖L2(Ω)2+1α∫-∞teαs‖g(s)‖L2(Ω)2ds+C0|Ω|αeαt.
So, substituting (4.39) into (4.38), we deduce
(4.40)∫rr+1(βeαs‖u‖S01(Ω)2+2eαs∫ΩF(u)dX)ds≤C(eατ‖uτ‖L2(Ω)2+∫-∞teαs‖g(s)‖L2(Ω)2ds+eαt)<∞,∀r∈[τ,t-1].
Now multiplying (1.2) by eαrv and integrating over Ω, we have
(4.41)eαr‖v‖L2(Ω)2+ddr(βeαr‖u‖S01(Ω)2+2eαr∫ΩF(u)dX)≤α(βeαr‖u‖S01(Ω)2+2eαr∫ΩF(u)dX)+eαr‖g(r)‖L2(Ω)2.
So applying the uniform Gronwall inequality, we get
(4.42)βeαr‖u‖S01(Ω)2+2eαr∫ΩF(u)dX≤C(eατ‖uτ‖L2(Ω)2+eαt+∫-∞teαs‖g(s)‖L2(Ω)2ds).
Integrating (4.41) from r to r+1 and by (4.40)–(4.42), we have
(4.43)∫rr+1eαs‖v‖L2(Ω)2ds≤C(eατ‖uτ‖L2(Ω)2+eαt+∫-∞teαs‖g(s)‖L2(Ω)2ds).
Therefore, by (4.35), (4.43), using the uniform Gronwall inequality once again, we get
(4.44)eαt‖v‖L2(Ω)2≤C(eατ‖uτ‖L2(Ω)2+eαt+∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds).
Hence we get (4.32).Theorem 4.5.
Under assumptions (H1bis)–(H3bis), the process{U(t,τ)} associated to problem (1.2) has a pullback 𝒟-attractor in 𝕊01(Ω)∩𝕃p(Ω).Proof.
By Lemma4.1, {U(t,τ)} has a family of bounded pullback 𝒟-absorbing sets in 𝕊01(Ω)∩𝕃p(Ω). It remains to show that {U(t,τ)} is pullback 𝒟-asymptotically compact in 𝕊01(Ω)∩𝕃p(Ω), that is, for any t∈ℝ, any ℬ̂∈𝒟, and any sequence τn→-∞, any sequence uτn∈B(τn), the sequence {U(t,τn)uτn} is precompact in 𝕊01(Ω)∩𝕃p(Ω). Thanks to Theorem 4.3, we need only to show that the sequence {U(t,τn)uτn} is precompact in 𝕊01(Ω).
Letun(t)=U(t,τn)uτn. By Theorem 3.4, we can assume that {un(t)} is a Cauchy sequence in 𝕃2(Ω). We have(4.45)‖un(t)-um(t)‖S01(Ω)2=-〈Gsun(t)-Gsum(t),un(t)-um(t)〉=-〈dundt(t)-dumdt(t),un(t)-um(t)〉-〈f(un(t))-f(um(t)),un(t)-um(t)〉≤‖ddtun(t)-ddtum(t)‖L2(Ω)2‖un(t)-um(t)‖L2(Ω)2+C3‖un(t)-um(t)‖L2(Ω)2,
where we have used condition (1.5). Because {un(t)} is a Cauchy sequence in 𝕃2(Ω) and by Lemma 4.4, one gets
(4.46)‖un(t)-um(t)‖S01(Ω)⟶0,asm,n⟶∞.
The proof is complete.


## 4.2.𝕃2p-2(Ω) and 𝕊02(Ω)-Boundedness of the Pullback 𝒟-Attractor

First, we prove the existence of a family of pullback𝒟-absorbing sets for process U(t,τ) in 𝕃2p-2(Ω).Proposition 4.6.
Under assumptions (H1bis)–(H3bis), then for anyt∈ℝ and any bounded subset B⊂𝕃2(Ω), there exists a positive constant τ0=τ0(B,t)≤t such that
(4.47)‖u‖L2p-2(Ω)2p-2≤C(1+‖g(t)‖L2(Ω)2+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)
for all τ≤τ0 and all uτ∈B, where C>0 is independent of t and B.Proof.
Multiplying (1.2) by |u|p-2u and integrating over Ω we obtain
(4.48)∫Ω[((a∇xu,∇x(|u|p-2u)))+|x|2s((a∇yu,∇y(|u|p-2u)))]dX+∫Ω(f(u),u)|u|p-2dX=-∫Ω(ut,|u|p-2u)dX+∫Ω(g(t),|u|p-2u)dX.
By the Cauchy inequality, (1.3) and note that
(4.49)∫Ω[((a∇xu,∇x(|u|p-2u)))+|x|2s((a∇yu,∇y(|u|p-2u)))]dX≥0;herea=βIm,
then we get
(4.50)C1‖u‖L2p-2(Ω)2p-2≤1C1‖ut‖L2(Ω)2+1C1‖g(t)‖L22+C12‖u‖L2p-22p-2.
Hence, by (4.32) we deduce from (4.16) that
(4.51)C12‖u‖L2p-2(Ω)2p-2≤1C1C(1+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)+1C1‖g(t)‖L2(Ω)2.
Therefore, we get (4.47) and the proof is complete.And now, we denote by𝕊02(Ω) the closure of (C0∞(Ω))m in the norm (4.52)‖u‖S02(Ω)=(∫Ω(|Δxu|2+|x|2sΔyu|2)dX)1/2.
It is easy to see that 𝕊02(Ω) is a Banach space endowed with the above norm. We now prove the 𝕊02(Ω)-boundedness of the pullback 𝒟-attractor.First, we recall a lemma (see [15]) which is necessary for our proof.Lemma 4.7.
LetX,Y be Banach spaces such that X is reflexive, and the inclusion X⊂Y is continuous. Assume that {un} is a bounded sequence in L∞(τ,T;X) such that un⇀u weakly in Lq(τ,T;X) for some q∈[1,+∞) and u∈C([τ,T];Y). Then, u(t)∈X for all t∈[τ,T] and ∥u(t)∥X≤supn≥1∥un∥L∞(τ,T;X), for all t∈[τ,T].Letun(t) be the Galerkin approximations of the solution u(t) of (1.2) then by Lemma 4.7 with noticing that un=U(t,τ)unτ⇀u=U(t,τ)uτ in L2(τ,T;𝕊01(Ω)) and the inclusion 𝕊02(Ω)⊂𝕊01(Ω) is continuous, we only need the estimation on u(t)=U(t,τ)uτ.Theorem 4.8.
Under assumptions (H1bis)–(H3bis), the pullback𝒟-attractor 𝒜̂={A(t):t∈ℝ} in 𝕊01(Ω)∩𝕃p(Ω) of the process {U(t,τ)} is bounded in 𝕊02(Ω). More precisely, for any τ<T1<T2, the set ⋃t∈[T1,T2]A(t) is a bounded subset of 𝕊02(Ω).Proof.
Let us fix a bounded setB⊂𝕃2(Ω), τ∈ℝ, ɛ>0, t>τ+ɛ and uτ∈B. Multiplying the first equation in (1.2) by Gsu and integrating over Ω, we have
(4.53)∫Ω(aGsu(r),Gsu(r))dX=∫Ω(ur′,Gsu(r))dX+∫Ω(f(u(r)),Gsu(r))dX-∫Ω(g(r),Gsu(r))dX.
By the Cauchy inequality we have
(4.54)-∫Ω(g(r),Gsu(r))dX≤2β‖g(r)‖L2(Ω)2+β8‖Gsu(r)‖L2(Ω)2,∫Ω(ur′,Gsu(r))dX≤2β‖ur′‖S01(Ω)2+β8‖Gsu(r)‖L2(Ω)2.
Using (3.12), (1.12), and (4.54), then from (4.53) we get
(4.55)β‖Gsu(r)‖L2(Ω)2≤C3‖u(r)‖S01(Ω)2+2β(‖ur′‖L2(Ω)2+‖g(r)‖L2(Ω)2)+β4‖Gsu(r)‖L2(Ω)2=C3∫Ω(u(r),-Gsu(r))dX+2β(‖ur′‖L2(Ω)2+‖g(r)‖L2(Ω)2)+β4‖Gsu(r)‖L2(Ω)2≤2C32β‖u(r)‖L2(Ω)2+2β‖ur′‖L2(Ω)2+2β‖g(r)‖L2(Ω)2+β2‖Gsu(r)‖L2(Ω)2.
Hence,
(4.56)β2‖Gsu(r)‖L2(Ω)2≤2C32β‖u(r)‖L2(Ω)2+2β‖ur′‖L2(Ω)2+2β‖g(r)‖L2(Ω)2.
Differentiating the first equation in (1.2) in time t and setting v(r)=u′(r), then multiplying by v(r) and using (1.11) we get
(4.57)12ddr‖v(r)‖L2(Ω)2+β‖v(r)‖S01(Ω)2≤-∫Ω(fu′(u)v(r),v(r))dX+∫Ω(g′(r),v(r))dX≤C3‖v(r)‖L2(Ω)2+12‖v(r)‖L2(Ω)2+12‖g′(r)‖L2(Ω)2.
Hence,
(4.58)ddr‖v(r)‖L2(Ω)2≤(2C3+1)‖v(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2.
Integrating the above inequality, we have
(4.59)‖v(r)‖L2(Ω)2≤‖v(s)‖L2(Ω)2+(2C3+1)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2+∫τ+ɛ/2t‖g′(θ)‖L2(Ω)2dθ,
for all τ+ɛ/2≤s≤r≤t.
Now, integrating with respect tos between τ+ɛ/2 and r, we get(4.60)(r-τ-ɛ2)‖v(r)‖L2(Ω)2≤∫τ+ɛ/2r‖v(s)‖L2(Ω)2ds+(2C3+1)(r-τ-ɛ2)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+(r-τ-ɛ2)∫τ+ɛ/2t‖g′(θ)‖L2(Ω)2dθ≤((2C3+1)(t-τ-ɛ2)+1)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+(r-τ-ɛ2)∫τ+ɛ/2t‖g′(θ)‖L2(Ω)2dθ,
for all τ+ɛ/2≤r≤t, and in particular, for all r∈[τ+ɛ,t] we have that (from the above estimate)
(4.61)‖v(r)‖L2(Ω)2≤2ɛ((2C3+1)(t-τ-ɛ2)+1)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+∫τt‖g′(θ)‖L2(Ω)2dθ.
On the other hand, multiplying the first equation in (1.2) by v(r) and integrating over Ω, we deduce that
(4.62)‖v(r)‖L2(Ω)2+β2ddr‖u(r)‖S01(Ω)2+∫Ω(f(u,v))dX≤∫Ω(g,v)dX,
where we have used (1.11). Using the Cauchy inequality and condition (H2bis), then (4.62) becomes
(4.63)‖v(r)‖L2(Ω)2+ddr(β‖u(r)‖S01(Ω)2+2∫ΩF(u(r))dX)≤‖g(r)‖L2(Ω)2.
Integrating from τ+ɛ/2 to t we have
(4.64)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ+β‖u(t)‖S01(Ω)2+2∫ΩF(u(t))dX≤β‖u(τ+ɛ2)‖S01(Ω)2+2∫ΩF(u(τ+ɛ2))dX+∫τ+ɛ/2t‖g(θ)‖L2(Ω)2,
and hence because of (4.1), we get
(4.65)∫τ+ɛ/2t‖v(θ)‖L2(Ω)2dθ≤β‖u(τ+ɛ2)‖S01(Ω)2+2C¯2‖u(τ+ɛ2)‖Lp(Ω)p+4C¯0|Ω|+∫τt‖g(θ)‖L2(Ω)2.
Now, substituting (4.65) into (4.61) we deduce
(4.66)‖v(θ)‖L2(Ω)2≤2ɛ((2C3+1)(t-τ-ɛ2)+1)×(β‖u(τ+ɛ2)‖S01(Ω)2+2C¯2‖u(τ+ɛ2)‖Lp(Ω)2+4C¯0|Ω|+∫τt‖g(θ)‖L2(Ω)2dθ)+∫τt‖g′(θ)‖L2(Ω)2dθ,
for all r∈[τ+ɛ,t]. Finally, from (4.66) and (4.56) we obtain
(4.67)‖Gsu(r)‖L2(Ω)2≤8β2ɛ((2C3+1)(t-τ-ɛ2)+1)×(β‖u(τ+ɛ2)‖S01(Ω)2+2C¯2‖u(τ+ɛ2)‖Lp(Ω)p+4C¯0|Ω|+∫τt‖g(θ)‖L2(Ω)2dθ)+2β∫τt‖g′(θ)‖L2(Ω)2dθ+2β2(C32‖u(r)‖L2(Ω)2+2‖g(r)‖L2(Ω)2),∀r∈[τ+ɛ,t].
Because ∥u∥𝕊02(Ω)2=∥Gsu∥𝕃2(Ω)2 then from (4.67), the proof is complete.


## 4.3. Fractal Dimensional Estimates of the Pullback𝒟-Attractor

Theorem 4.9.
Under assumptions (H1bis)–(H3bis), the processU(t,τ) possesses a pullback 𝒟-attractor 𝒜𝕃2(Ω) which has a finite fractal dimension in 𝕃2(Ω) and
(4.68)dim⁡f(A(t))≤klog⁡(1+8⋅e2C31-δ)[log⁡21+δ]-1,∀t∈R,
where δ<1,k∈ℕ, and C3 in (1.5).Proof.
LetHk=span{e1,e2,…,ek}⊂𝕃2(Ω) and Pk:𝕃2(Ω)→Hk be the orthogonal projection, where e1,e2,…,ej,… are the eigenvectors of the operator -Gs corresponding to eigenvalues {λj}j=1∞ such that 0<λ1<λ2≤λ3≤⋯≤λj≤⋯ and λj→+∞ as j→+∞.
From (4.3), we can easily show that there exists a uniformly pullback absorbing set ℬ of process U(t,τ) in 𝕊01(Ω). We set u1(t)=U(t,τ)u1τ and u2(t)=U(t,τ)u2τ to be solutions associated to problem (1.2) with initial datum u1τ,u2τ∈ℬ.
Letw=u1-u2, because u1, u2 being two solutions of (1.2) then we have(4.69)∂w∂t-aGsw+f(u1)-f(u2)=0.
Multiplying (4.69) with w and integrating over Ω then we have
(4.70)12ddt‖w(t)‖L2(Ω)2+β‖w‖S01(Ω)2+∫Ω(f(u1)-f(u2),w)dX≤0,
here, we have used (1.11).
Using (1.5) then we have(4.71)ddt‖w(t)‖L2(Ω)2≤2C3‖w(t)‖L2(Ω)2.
Thus,
(4.72)‖w(t)‖L2(Ω)2≤e2C3(t-τ)‖w(τ)‖L2(Ω)2.
Let w(t)=w1(t)+w2(t) where w1(t):=Pkw(t) and w2(t):=(I-Pk)w(t). Therefore, by (4.72) we have
(4.73)‖w1(t)‖L2(Ω)2≤e2C3(t-τ)‖w(τ)‖L2(Ω)2.
Now, taking the inner product of (4.69) with w2 in 𝕃2(Ω), we have
(4.74)12ddt‖w2(t)‖L2(Ω)2+β‖w2‖S01(Ω)2≤-∫(f(u1)-f(u2),w2)dX.
Using the Hölder inequality and (1.4), we have
(4.75)-∫Ω(f(u1)-f(u2),w2)dX≤∫Ω|f(u1)-u(u2)||w2|dX≤(∫Ω|f(u1)-f(u2)|2dX)1/2(∫Ω|w2|2dX)1/2≤C(∫Ω(1+|u1|2p-2+|u2|2p-2)dX)1/2‖w2‖L2(Ω)≤C(1+‖u1‖L2p-2(Ω)2p-2+‖u2‖L2p-2(Ω)2p-2)1/2‖w‖L2(Ω)≤C(1+‖u1‖L2p-2(Ω)2p-2+‖u2‖L2p-2(Ω)2p-2)‖w‖L2(Ω).
Therefore, by (4.74), (4.75), and Proposition 4.6 we obtain
(4.76)12ddt‖w2(t)‖L2(Ω)2+β‖w2‖S01(Ω)2≤C(1+‖g(t)‖L2(Ω)2+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)‖w(t)‖L2(Ω).
Because ∥w2(t)∥𝕊01(Ω)2≥λk∥w2(t)∥𝕃2(Ω)2, then (4.76) implies that
(4.77)ddt‖w2(t)‖L2(Ω)2+2βλk‖w2‖L2(Ω)2≤2C(1+‖g(t)‖L2(Ω)2+e-αt∫-∞teαs(‖g(s)‖L2(Ω)2+‖g′(s)‖L2(Ω)2)ds)‖w(t)‖L2(Ω).
Now, multiplying (4.77) by eβλkt and integrating from τ to t, we get
(4.78)‖w2(t)‖L2(Ω)2≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+2Ce-βλkt∫τte-βλks×[1+‖g(s)‖L2(Ω)2+e-αt∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr]×‖w(s)‖L2(Ω)ds.
Using (4.73) we have
(4.79)‖w2(t)‖L2(Ω)2≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+C‖w(τ)‖L2(Ω)e-βλkt∫τteβλkseC3(s-τ)×[1+‖g(s)‖L2(Ω)2+e-αs∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr]ds≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+C‖w(τ)‖L2(Ω)eC3(t-τ)∫-∞te-βλk(t-s)×[1+‖g(s)‖L2(Ω)2+e-αs∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr]ds≤e-βλk(t-τ)‖w(τ)‖L2(Ω)2+C‖w(τ)‖L2(Ω)eC3(t-τ)×[1βλk+∫-∞te-βλk(t-s)‖g(s)‖L2(Ω)2ds+∫-∞te-βλk(t-s)e-αs×(∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)ds].
Now, because
(4.80)∫-∞teαs‖g(s)‖L2(Ω)2ds<+∞,
we can see that, for all t∈ℝ (see, e.g., [6, Lemma 3.6]),
(4.81)∫-∞te-βλk(t-s)‖g(s)‖L2(Ω)2ds⟶0ask⟶+∞,
and we have
(4.82)∫-∞te-βλk(t-s)e-βλ1s(∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)ds≤(∫-∞te-βλkt+β(λk-λ1)sds)(∫-∞teαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)=e-αtβλk-α∫-∞teαs(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr.
Thus, for any t∈ℝ, from (4.82) we have
(4.83)∫-∞te-βλk(t-s)e-βλ1s(∫-∞seαr(‖g(r)‖L2(Ω)2+‖g′(r)‖L2(Ω)2)dr)ds⟶0ask⟶+∞.
Combining (4.81), (4.83) and taking T0=t-τ=1, we get k is sufficient large then from (4.79) we deduce
(4.84)‖w2(t)‖L2(Ω)2≤δ‖w(τ)‖L2(Ω)2,here0<δ<1.
From (4.73) and (4.31), we have
(4.85)‖w1(t)‖L2(Ω)2≤l0‖w(τ)‖L2(Ω)2,‖w2(t)‖L2(Ω)2≤δ‖w(τ)‖L2(Ω)2,∀t∈R.
Here, l0=e2C3; 0<δ<1; T0=1. Thus, the process U(t,τ) associated to (1.2) satisfies all conditions of Theorem 2.10. This completes the proof.



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*Source: 178057-2011-04-14.xml*