The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations

Abstract

We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domain Ω. Assuming 𝑓=𝑓(𝑥,𝑡)∈𝐿²_loc, we establish the existence of the uniform attractor in 𝐿²(Ω) and 𝐷(𝐴^1/2). The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

1. Introduction

In this paper, we study the behavior of solutions of the nonautonomous 2D g-Navier-Stokes equations. These equations are a variation of the standard Navier-Stokes equations, and they assume the form,𝜕𝑢1𝜕𝑡−𝜈Δ𝑢+(𝑢⋅∇)𝑢+∇𝑝=𝑓inΩ,𝑔(∇⋅𝑔𝑢)=∇𝑔𝑔⋅𝑢+∇⋅𝑢=0inΩ, (1.1)where 𝑔=𝑔(𝑥₁,𝑥₂) is a suitable smooth real-valued function defined on (𝑥₁,𝑥₂)∈Ω and Ω is a suitable bounded domain in ℝ². Notice that if 𝑔(𝑥₁,𝑥₂)=1, then (1.1) reduce to the standard Navier-Stokes equations.

In addition, we assume that the function 𝑓(⋅,𝑡)=∶𝑓(𝑡)∈𝐿²_loc(ℝ;𝐸) is translation bounded, where 𝐸=𝐿²(Ω)or𝐻⁻¹(Ω). This property implies that‖𝑓‖²_𝐿2𝑏=‖𝑓‖²_{𝐿2𝑏(ℝ;𝐸)}=sup_{𝑡∈ℝ𝑡}^𝑡+1‖𝑓(𝑠)‖²_𝐸𝑑𝑠<∞. (1.2)

We consider this equation in an appropriate Hilbert space and show that there is an attractor 𝔄 which all solutions approach as 𝑡→∞. The basic idea of our construction, which is motivated by the works of [1, 2].

In [1, 2] the author established the global regularity of solutions of the g-Navier-Stokes equations. For the boundary conditions, we will consider the periodic boundary conditions, while same results can be got for the Dirichlet boundary conditions on the smooth bounded domain. For many years, the Navier-Stokes equations were investigated by many authors and the existence of the attractors for 2D Navier-Stokes equations was first proved by Ladyzhenskaya [3, 4] and independently by Foias and Temam [5]. The finite-dimensional property of the global attractor for general dissipative equations was first proved by Mallet-Paret [6]. For the analysis on the Navier-Stokes equations, one can refer to [7] and specially [8] for the periodic boundary conditions.

The book in [9] considers some special classes of such systems and studies systematically the notion of uniform attractor parallelling to that of global attractor for autonomous systems. Later on, [10] presents a general approach that is well suited to study equations arising in mathematical physics. In this approach, to construct the uniform (or trajectory) attractors, instead of the associated process {𝑈_𝜎(𝑡,𝜏),𝑡≥𝜏,𝜏∈ℝ} one should consider a family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ in some Banach space 𝐸, where the functional parameter 𝜎₀(𝑠),𝑠∈ℝ is called the symbol and Σ is the symbol space including 𝜎₀(𝑠). The approach preserves the leading concept of invariance which implies the structure of uniform attractor described by the representation as a union of sections of all kernels of the family of processes. The kernel is the set of all complete trajectories of a process.

In the paper, we study the existence of compact uniform attractor for the nonautonomous the two dimensional g-Navier-Stokes equations in the periodic boundary conditions Ω. We apply measure of noncompactness method to nonautonomous g-Navier-Stokes equations equation with external forces 𝑓(𝑥,𝑡) in 𝐿²_loc(ℝ;𝐸) which is normal function (see Definition 4.2). Last, the fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

2. Functional Setting

Let Ω=(0,1)×(0,1) and we assume that the function 𝑔(𝑥)=𝑔(𝑥₁,𝑥₂) satisfies the following properties:

(1) 𝑔(𝑥)∈𝐶^∞_per(Ω) and

(2) there exist constants 𝑚₀=𝑚₀(𝑔) and 𝑀₀=𝑀₀(𝑔) such that, for all 𝑥∈Ω,0<𝑚₀≤𝑔(𝑥)≤𝑀₀. Note that the constant function 𝑔≡1 satisfies these conditions.

We denote by 𝐿²(Ω,𝑔) the space with the scalar product and the norm given by(𝑢,𝑣)_𝑔=_Ω(𝑢⋅𝑣)𝑔𝑑𝑥,|𝑢|²_𝑔=(𝑢,𝑢)_𝑔, (2.1)as well as 𝐻¹(Ω,𝑔) with the norm‖𝑢‖_𝐻¹_(Ω,𝑔)=[(𝑢,𝑢)_𝑔+²_𝑖=1(𝐷_𝑖𝑢,𝐷_𝑖𝑢)_𝑔]^1/2, (2.2)where 𝜕𝑢/𝜕𝑥_𝑖=𝐷_𝑖𝑢.

Then for the functional setting of (1.1), we use the following functional spaces𝐻_𝑔=𝐶𝑙_𝐿²_per(Ω,𝑔){𝑢∈𝐶^∞_per(Ω)∶∇⋅𝑔𝑢=0,∫_Ω𝑉𝑢𝑑𝑥=0},_𝑔={𝑢∈𝐻¹_per(Ω,𝑔)∶∇⋅𝑔𝑢=0,∫_Ω𝑢𝑑𝑥=0}, (2.3)where 𝐻_𝑔 is endowed with the scalar product and the norm in 𝐿²(Ω,𝑔), and 𝑉_𝑔 is the spaces with the scalar product and the norm given by((𝑢,𝑣))_𝑔=_Ω(∇𝑢⋅∇𝑣)𝑔𝑑𝑥,‖𝑢‖_𝑔=((𝑢,𝑢))_𝑔. (2.4)Also, we define the orthogonal projection 𝑃_𝑔 as𝑃_𝑔∶𝐿²_per(Ω,𝑔)→𝐻_𝑔 (2.5)and we have that 𝑄⊆𝐻^⟂_𝑔, where𝑄=𝐶𝑙_𝐿²_per(Ω,𝑔){∇𝜙∶𝜙∈𝐶¹(Ω,ℝ)}. (2.6)Then, we define the 𝑔-Laplacian operator−Δ_𝑔1𝑢≡𝑔1(∇⋅𝑔∇)𝑢=−Δ𝑢−𝑔(∇𝑔⋅∇)𝑢 (2.7)to have the linear operator𝐴_𝑔𝑢=𝑃_𝑔1[−𝑔(∇⋅(𝑔∇𝑢))]. (2.8)For the linear operator 𝐴_𝑔, the following hold (see [1, 2]):

(1) 𝐴_𝑔 is a positive, self-adjoint operator with compact inverse, where the domain of 𝐴_𝑔,𝐷(𝐴_𝑔)=𝑉_𝑔∩𝐻²(Ω,𝑔).

(2) There exist countable eigenvalues of 𝐴_𝑔 satisfying0<𝜆_𝑔≤𝜆₁≤𝜆₂≤𝜆₃≤⋯, (2.9)where 𝜆_𝑔=4𝜋²𝑚/𝑀 and 𝜆₁ is the smallest eigenvalue of 𝐴_𝑔. In addition, there exists the corresponding collection of eigenfunctions {𝑒₁,𝑒₂,𝑒₃,…} which forms an orthonormal basis for 𝐻_𝑔.

Next, we denote the bilinear operator 𝐵_𝑔(𝑢,𝑣)=𝑃_𝑔(𝑢⋅∇)𝑣 and the trilinear form𝑏_𝑔(𝑢,𝑣,𝑤)=^𝑛_{𝑖,𝑗=1Ω}𝑢_𝑖(𝐷_𝑖𝑣_𝑗)𝑤_𝑗𝑔𝑑𝑥=(𝑃_𝑔(𝑢⋅∇)𝑣,𝑤)_𝑔, (2.10)where 𝑢,𝑣,𝑤 lie in appropriate subspaces of 𝐿²(Ω,𝑔). Then, the form 𝑏_𝑔 satisfies𝑏_𝑔(𝑢,𝑣,𝑤)=−𝑏_𝑔(𝑢,𝑤,𝑣)for𝑢,𝑣,𝑤∈𝐻_𝑔. (2.11)

We denote a linear operator 𝑅 on 𝑉_𝑔 by𝑅𝑢=𝑃_𝑔[1𝑔(∇𝑔⋅∇)𝑢]for𝑢∈𝑉_𝑔, (2.12)and have 𝑅 as a continuous linear operator from 𝑉_𝑔 into 𝐻_𝑔 such that|(𝑅𝑢,𝑢)|≤|∇𝑔|_∞𝑚₀‖𝑢‖_𝑔|𝑢|_𝑔≤|∇𝑔|_∞𝑚₀𝜆_𝑔^1/2‖𝑢‖_𝑔for𝑢∈𝑉_𝑔. (2.13)

We now rewrite (1.1) as abstract evolution equations,𝑑𝑢𝑑𝑡+𝜈𝐴_𝑔𝑢+𝐵_𝑔𝑢+𝜈𝑅𝑢=𝑃_𝑔𝑓,𝑢(𝜏)=𝑢_𝜏. (2.14)

Hereafter 𝑐 will denote a generic scale invariant positive constant, which is independent of the physical parameters in the equation and may be different from line to line and even in the same line.

3. Abstract Results

Let 𝐸 be a Banach space, and let a two-parameter family of mappings {𝑈(𝑡,𝜏)}={𝑈(𝑡,𝜏)∣𝑡≥𝜏,𝜏∈ℝ} act on 𝐸:𝑈(𝑡,𝜏)∶𝐸→𝐸,𝑡≥𝜏,𝜏∈ℝ. (3.1)Definition 3.1. A two-parameter family of mappings {𝑈(𝑡,𝜏)} is said to be a process in 𝐸 if𝑈(𝑡,𝑠)𝑈(𝑠,𝜏)=𝑈(𝑡,𝜏),∀𝑡≥𝑠≥𝜏,𝜏∈ℝ,𝑈(𝜏,𝜏)=𝐼𝑑,𝜏∈ℝ. (3.2)

By ℬ(𝐸) we denote the collection of the bounded sets of 𝐸. We consider a family of processes {𝑈_𝜎(𝑡,𝜏)} depending on a parameter 𝜎∈Σ. The parameter 𝜎 is said to be the symbol of the process {𝑈_𝜎(𝑡,𝜏)} and the set Σ is said to be the symbol space. In the sequel Σ is assumed to be a complete metric space.

A family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ is said to be uniformly (with respect to (𝑤.𝑟.𝑡.)𝜎∈Σ) bounded if for any 𝐵∈ℬ(𝐸) the set_{𝜎∈Σ𝜏∈𝑅𝑡≥𝜏}𝑈_𝜎(𝑡,𝜏)𝐵∈ℬ(𝐸). (3.3)

A set 𝐵₀⊂𝐸 is said to be uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) absorbing for the family of processes {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈Σ if for any 𝜏∈𝑅 and every 𝐵∈ℬ(𝐸) there exists 𝑡₀=𝑡₀(𝜏,𝐵)≥𝜏 such that ⋃_𝜎∈Σ𝑈_𝜎(𝑡,𝜏)𝐵⊆𝐵₀ for all 𝑡≥𝑡₀.

A set 𝑃⊂𝐸 is said to be uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) attracting for the family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ if for an arbitrary fixed 𝜏∈𝑅,lim_𝑡→+∞(sup_𝜎∈Σdist_𝐸(𝑈_𝜎(𝑡,𝜏)𝐵,𝑃))=0. (3.4)

A family of processes possessing a compact uniformly absorbing set is called uniformly compact and a family of processes possessing a compact uniformly attracting set is called uniformly asymptotically compact.Definition 3.2. A closed set 𝒜_Σ⊂𝐸 is said to be the uniform (𝑤.𝑟.𝑡.𝜎∈Σ) attractor of the family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ if it is uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) attracting and it is contained in any closed uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) attracting set 𝒜^′ of the family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ∶𝒜_Σ⊆𝒜^′.

A family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ acting in 𝐸 is said to be (𝐸×Σ,𝐸)-continuous, if for all fixed 𝑡 and 𝜏,𝑡≥𝜏,𝜏∈ℝ the mapping (𝑢,𝜎)↦𝑈_𝜎(𝑡,𝜏)𝑢 is continuous from 𝐸×Σ into 𝐸.

A curve 𝑢(𝑠),𝑠∈ℝ is said to be a complete trajectory of the process {𝑈(𝑡,𝜏)} if𝑈(𝑡,𝜏)𝑢(𝜏)=𝑢(𝑡),∀𝑡≥𝜏,𝜏∈ℝ. (3.5)

The kernel𝒦 of the process {𝑈(𝑡,𝜏)} consists of all bounded complete trajectories of the process {𝑈(𝑡,𝜏)}:𝒦={𝑢(⋅)∣𝑢(⋅)satisfies(3.6),‖𝑢(𝑠)‖_𝐸≤𝑀_𝑢for𝑠∈ℝ}. (3.6)

The set𝒦(𝑠)={𝑢(𝑠)∣𝑢(⋅)∈𝒦}⊆𝐸 (3.7)is said to be the kernel section at time 𝑡=𝑠,𝑠∈ℝ.

For convenience, let 𝐵_𝑡=⋃_𝜎∈Σ⋃_𝑠≥𝑡𝑈_𝜎(𝑠,𝑡)𝐵, the closure 𝐵 of the set 𝐵 and ℝ_𝜏={𝑡∈ℝ∣𝑡≥𝜏}. Define the uniform (𝑤.𝑟.𝑡.𝜎∈Σ) 𝜔-limit set 𝜔_𝜏,Σ(𝐵) of 𝐵 by 𝜔_𝜏,Σ(𝐵)=⋂_𝑡≥𝜏𝐵_𝑡 which can be characterized, analogously to that for semigroups, the following:𝑦∈𝜔_𝜏,Σ(𝐵)⟺therearesequences{𝑥_𝑛}⊂𝐵,{𝜎_𝑛}⊂Σ,{𝑡_𝑛}⊂ℝ_𝜏suchthat𝑡_𝑛→+∞,𝑈_𝜎𝑛(𝑡_𝑛,𝜏)𝑥_𝑛→𝑦(𝑛→∞). (3.8)

We recall characterize the existence of the uniform attractor for a family of processes satisfying (3.8) in term of the concept of measure of noncompactness that was put forward first by Kuratowski (see [11, 12]).

Let 𝐵∈ℬ(𝐸). Its Kuratowski measure of noncompactness 𝜅(𝐵) is defined by𝜅(𝐵)=inf{𝛿>0∣𝐵admitsafinitecoveringbysetsofdiameter≤𝛿}. (3.9)Definition 3.3. A family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ is said to be uniformly (𝑤.𝑟.𝑡.𝜎∈Σ)𝜔-limit compact if for any 𝜏∈ℝ and 𝐵∈ℬ(𝐸) the set 𝐵_𝑡 is bounded for every 𝑡 and lim_𝑡→∞𝜅(𝐵_𝑡)=0.

We present now a method to verify the uniform (𝑤.𝑟.𝑡.𝜎∈Σ)𝜔-limit compactness (see [13, 14]).Definition 3.4. A family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ is said to satisfy uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) Condition (C) if for any fixed 𝜏∈ℝ,𝐵∈ℬ(𝐸) and 𝜀>0, there exist 𝑡₀=𝑡(𝜏,𝐵,𝜀)≥𝜏 and a finite-dimensional subspace 𝐸₁ of 𝐸 such that
(i)𝑃(⋃_𝜎∈Σ⋃_𝑡≥𝑡0𝑈_𝜎(𝑡,𝜏)𝐵) is bounded; and(ii)‖(𝐼−𝑃)(⋃_𝜎∈Σ⋃_𝑡≥𝑡0𝑈_𝜎(𝑡,𝜏)𝑥)‖≤𝜀,∀𝑥∈𝐵, where 𝑃∶𝐸→𝐸₁ is a bounded projector.

Therefore we have the following results.Theorem 3.5. Let Σ be a metric space and let {𝑇(𝑡)} be a continuous invariant semigroup 𝑇(𝑡)Σ=Σ on Σ. A family of processes {𝑈_𝜎(𝑡,𝜏)},𝜎∈Σ acting in 𝐸 is (𝐸×Σ,𝐸)-(weakly) continuous and possesses the compact uniform (𝑤.𝑟.𝑡.𝜎∈Σ) attractor 𝐴_Σ satisfying𝒜_Σ=𝜔_0,Σ(𝐵₀)=𝜔_𝜏,Σ(𝐵₀)=_𝜎∈Σ𝒦_𝜎(0),∀𝜏∈ℝ, (3.10)if it
(i)has a bounded uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) absorbing set 𝐵₀, and(ii)satisfies uniformly (𝑤.𝑟.𝑡.𝜎∈Σ) Condition (C)
Moreover, if 𝐸 is a uniformly convex Banach space then the converse is true.

4. Uniform Attractor of Nonautonomous g-Navier-Stokes Equations

This section deals with the existence of the attractor for the two-dimensional nonautonomous g-Navier-Stokes equations with periodic boundary condition (see [1, 2]).

It is similar to autonomous case that we can establist the existence of solution of (2.14) by the standard Faedo-Galerkin method.

In [1, 2], the authors have shown that the semigroup 𝑆(𝑡)∶𝐻_𝑔→𝐻_𝑔(𝑡≥0) associated with the autonomous systems (2.14) possesses a global attractor. The main objective of this section is to prove that the nonautonomous system (2.14) has uniform attractors in 𝐻_𝑔 and 𝑉_𝑔.

To this end, we first state some the following results of existence and uniqueness of solutions of (2.14).Proposition 4.1. Let 𝑓∈𝑉^′ be given. Then for every 𝑢_𝜏∈𝐻_𝑔 there exists a unique solution 𝑢=𝑢(𝑡) on [0,∞) of (2.14), satisfying 𝑢(𝜏)=𝑢_𝜏. Moreover,one has𝑢(𝑡)∈𝐶[0,𝑇;𝐻_𝑔)∩𝐿²(0,𝑇;𝑉_𝑔),∀𝑇>0. (4.1)Finally, if 𝑢_𝜏∈𝑉_𝑔, then𝑢(𝑡)∈𝐶[0,𝑇;𝑉_𝑔)∩𝐿²(0,𝑇;𝐷(𝐴_𝑔)),∀𝑇>0. (4.2)

Proof. The Proof of Proposition 4.1 is similar to autonomous in [1, 15].

Now we will write (2.14) in the operator form𝜕_𝑡𝑢=𝐴_𝜎(𝑡)(𝑢),𝑢|_𝑡=𝜏=𝑢_𝜏, (4.3)where 𝜎(𝑠)=𝑓(𝑥,𝑠) is the symbol of (4.3). Thus, if 𝑢_𝜏∈𝐻_𝑔, then problem (4.3) has a unique solution 𝑢(𝑡)∈𝐶([0,𝑇];𝐻_𝑔)∩𝐿²([0,𝑇];𝑉_𝑔). This implies that the process {𝑈_𝜎(𝑡,𝜏)} given by the formula 𝑈_𝜎(𝑡,𝜏)𝑢_𝜏=𝑢(𝑡) is defined in 𝐻_𝑔.

Now recall the following facts that can be found in [13].Definition 4.2. A function 𝜑∈𝐿²_loc(ℝ;𝐸) is said to be normal if for any 𝜀>0, there exists 𝜂>0 such thatsup_{𝑡∈ℝ𝑡}^𝑡+𝜂‖𝜑(𝑠)‖²_𝐸𝑑𝑠≤𝜀. (4.4)Remark 4.3. Obviously, 𝐿²_𝑛(ℝ;𝐸)⊂𝐿²_𝑏(ℝ;𝐸). Denote by 𝐿²_𝑐(ℝ;𝐸) the class of translation compact functions 𝑓(𝑠),𝑠∈ℝ, whose family of ℋ(𝑓) is precompact in 𝐿²_loc(ℝ;𝐸). It is proved in [13] that 𝐿²_𝑛(ℝ;𝐸) and 𝐿²_𝑐(ℝ;𝐸) are closed subspaces of 𝐿²_𝑏(ℝ;𝐸), but the latter is a proper subset of the former (for further details see [13]).

We now define the symbol spaceℋ(𝜎₀) for (4.3). Let a fixed symbol 𝜎₀(𝑠)=𝑓₀(𝑠)=𝑓₀(⋅,𝑠) be normal functions in 𝐿²_loc(ℝ;𝐸); that is, the family of translation {𝑓₀(𝑠+ℎ),ℎ∈ℝ} forms a normal function set in 𝐿²_loc([𝑇₁,𝑇₂];𝐸), where [𝑇₁,𝑇₂] is an arbitrary interval of the time axis ℝ. Thereforeℋ(𝜎₀)=ℋ(𝑓₀)=[𝑓₀(𝑥,𝑠+ℎ)∣ℎ∈ℝ]_𝐿²_{loc(ℝ;𝐸)}. (4.5)

Now, for any 𝑓(𝑥,𝑡)∈ℋ(𝑓₀), the problem (4.3) with 𝑓 instead of 𝑓₀ possesses a corresponding process {𝑈_𝑓(𝑡,𝜏)} acting on 𝑉_𝑔. As is proved in [10], the family {𝑈_𝑓(𝑡,𝜏)∣𝑓∈ℋ(𝑓₀)} of processes is (𝑉_𝑔×ℋ(𝑓₀);𝑉_𝑔)-continuous.

Let𝒦_𝑓={𝑢_𝑓(𝑥,𝑡)for𝑡∈ℝ∣𝑢_𝑓(𝑥,𝑡)issolutionof(4.3)satisfying‖𝑢_𝑓(⋅,𝑡)‖_𝐻≤𝑀_𝑓∀𝑡∈ℝ} (4.6)be the so-called kernel of the process {𝑈_𝑓(𝑡,𝜏)}.Proposition 4.4. The process {𝑈_𝑓(𝑡,𝜏)}∶𝐻_𝑔→𝐻_𝑔(𝑉_𝑔) associated with the (4.3) possesses absorbing setsℬ₀={𝑢∈𝐻_𝑔∣|𝑢|_𝑔≤𝜌₀},ℬ₁={𝑢∈𝑉_𝑔∣‖𝑢‖_𝑔≤𝜌₁} (4.7)which absorb all bounded sets of 𝐻_𝑔. Moreover ℬ₀ and ℬ₁ absorb all bounded sets of 𝐻_𝑔 and 𝑉_𝑔 in the norms of 𝐻_𝑔 and 𝑉_𝑔, respectively.

Proof. The proof of Proposition 4.4 is similar to autonomous g-Navier-Stokes equation. We can obtain absorbing sets in 𝐻_𝑔 and 𝑉_𝑔 the following from [1] and the proof of the main results as follow.

The main results in this section are as follows.Theorem 4.5. If 𝑓₀(𝑥,𝑠) is normal function in 𝐿²_loc(ℝ;𝑉^′_𝑔), then the processes {𝑈_𝑓0(𝑡,𝜏)} corresponding to problem (2.14) possess compact uniform (𝑤.𝑟.𝑡.𝜏∈ℝ) attractor 𝔄₀ in 𝐻_𝑔 which coincides with the uniform (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓₀)) attractor 𝔄_ℋ(𝑓0) of the family of processes {𝑈_𝑓(𝑡,𝜏)},𝑓∈ℋ(𝑓₀):𝔄₀=𝔄_ℋ(𝑓0)=𝜔_0,ℋ(𝑓0)(ℬ₀)=_{𝑓∈ℋ(𝑓0)}𝒦_𝑓(0), (4.8)where ℬ₀ is the uniformly (𝑤.𝑟.𝑡.𝑓∈ℋ_𝑔(𝑓₀)) absorbing set in 𝐻_𝑔 and 𝒦_𝑓 is the kernel of the process {𝑈_𝑓(𝑡,𝜏)}. Furthermore, the kernel 𝒦_𝑓 is nonempty for all 𝑓∈ℋ(𝑓₀).

Proof. As in the previous section, for fixed 𝑁, let 𝐻₁ be the subspace spanned by 𝑤₁;…;𝑤_𝑁, and 𝐻₂ the orthogonal complement of 𝐻₁ in 𝐻_𝑔. We write𝑢=𝑢₁+𝑢₂;𝑢₁∈𝐻₁,𝑢₂∈𝐻₂forany𝑢∈𝐻_𝑔. (4.9)
Now, we only have to verify Condition (C). Namely, we need to estimate |𝑢₂(𝑡)|₂, where 𝑢(𝑡)=𝑢₁(𝑡)+𝑢₂(𝑡) is a solution of (2.14) given in Proposition 4.1.
Multiplying (2.14) by 𝑢₂, we have(𝑑𝑢𝑑𝑡,𝑢₂)_𝑔+(𝜈𝐴_𝑔𝑢,𝑢₂)_𝑔+(𝐵(𝑢,𝑢),𝑢₂)_𝑔=(𝑓,𝑢₂)_𝑔−(𝑅𝑢,𝑢₂)_𝑔. (4.10)It follows that12𝑑𝑑𝑡|𝑢₂|²_𝑔+𝜈‖𝑢_𝑔‖²_𝑔≤|(𝐵(𝑢,𝑢),𝑢₂)_𝑔|+|(𝑓,𝑢₂)_𝑔|+(𝑅𝑢,𝑢₂)_𝑔. (4.11)Since 𝑏_𝑔 satisfies the following inequality (see [15]):|𝑏_𝑔(𝑢,𝑣,𝑤)|≤𝑐|𝑢|_𝑔^1/2‖𝑢‖_𝑔^1/2‖𝑣‖_𝑔|𝑤|_𝑔^1/2‖𝑤‖_𝑔^1/2,∀𝑢,𝑣,𝑤∈𝑉_𝑔, (4.12)thus,|(𝐵(𝑢,𝑢),𝑢₂)_𝑔|≤𝑐|𝑢|_𝑔^1/2‖𝑢‖_𝑔^3/2|𝑢₂|_𝑔^1/2‖𝑢₂‖_𝑔^1/2≤𝑐𝜆_𝑚+1|𝑢|_𝑔^1/2‖𝑢‖_𝑔^3/2‖𝑢₂‖_𝑔≤𝜈6‖𝑢₂‖²_𝑔+𝑐𝜌₀𝜌³₁. (4.13)Next, the Cauchy inequality,|(𝑅𝑢,𝑢₂)_𝑔𝜈|=|(𝑔(∇𝑔⋅∇)𝑢,𝑢₂)_𝑔|≤𝜈𝑚₀|∇𝑔|_∞‖𝑢‖_𝑔|𝑢₂|_𝑔≤𝜈6‖𝑢₂‖²_𝑔+3𝜈𝜌²₁|∇𝑔|²_∞2𝑚²₀𝜆_𝑔𝜆_𝑚+1. (4.14)Finally, we have|(𝑓,𝑢₂)_𝑔|≤|𝑓|_𝑉′𝑔‖𝑢₂𝜈‖≤6‖𝑢₂‖²_𝑔+32𝜈|𝑓|²_𝑉′𝑔. (4.15)Putting (4.13)–(4.15) together, there exist constant 𝑀₁=𝑀₁(𝑚₀,|∇𝑔|_∞,𝜌₀,𝜌₁) such that12𝑑𝑑𝑡|𝑢₂|²_𝑔+12𝜈‖𝑢₂‖²_𝑔≤3|𝑓|_𝑉′𝑔2𝜈+𝑀₁. (4.16)Therefore, we deduce that𝑑𝑑𝑡|𝑢₂|²_𝑔+𝜈𝜆_𝑚+1|𝑢₂|²₂≤2𝑀₁+3𝜈|𝑓|²_𝑉′𝑔. (4.17)Here 𝑀₁ depends on 𝜆_𝑚+1, is not increasing as 𝜆_𝑚+1 increasing.
By the Gronwall inequality, the above inequality implies|𝑢₂(𝑡)|²_𝑔≤|𝑢₂(𝑡₀+1)|²₂𝑒^{−𝜈𝜆𝑚+1(𝑡−(𝑡0+1))}+2𝑀₁𝜈𝜆_𝑚+1+3𝜈∫^𝑡_𝑡0+1𝑒^{−𝜈𝜆𝑚+1(𝑡−𝑠)}|𝑓|²_𝑉′𝑑𝑠. (4.18)Applying (4.4) for any 𝜀3𝜈^𝑡_𝑡0+1𝑒^{−𝜈𝜆𝑚+1(𝑡−𝑠)}|𝑓|²_𝑉′𝑔𝜀𝑑𝑠<3. (4.19)Using (2.9) and let 𝑡₁=𝑡₀+1+(1/𝜈𝜆_𝑚+1)ln(3𝜌²₀/𝜀), then 𝑡≥𝑡₁ implies2𝑀𝜈𝜆_𝑚+1<𝜀3;|𝑢₂(𝑡₀+1)|²₂𝑒^{−𝜈𝜆𝑚+1(𝑡−(𝑡0+1))}≤𝜌²₀𝑒^{−𝜈𝜆𝑚+1(𝑡−(𝑡0+1))/2}<𝜀3. (4.20)Therefore, we deduce from (4.18) that|𝑢₂|²₂≤𝜀,∀𝑡≥𝑡₁,𝑓∈ℋ(𝑓₀), (4.21)which indicates {𝑈_𝑓(𝑡,𝜏)},𝑓∈ℋ(𝑓₀) satisfying uniform (𝑤.𝑡.𝑟.𝑓∈ℋ(𝑓₀)) Condition (C) in 𝐻_𝑔. Applying Theorem 3.5 the proof is complete.

Theorem 4.6. If 𝑓₀(𝑥,𝑠) is normal function in 𝐿²_𝑙𝑜𝑐(ℝ;𝐻_𝑔), then the processes {𝑈_𝑓0(𝑡,𝜏)} corresponding to problem (2.14) possesses compact uniform (𝑤.𝑟.𝑡.𝜏∈ℝ) attractor 𝔄₁ in 𝑉_𝑔 which coincides with the uniform (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓₀)) attractor 𝔄_ℋ(𝑓0) of the family of processes {𝑈_𝑓(𝑡,𝜏)},𝑓∈ℋ(𝑓₀):𝔄₁=𝔄_ℋ(𝑓0)=𝜔_0,ℋ(𝑓0)(ℬ₁)=_{𝑓∈ℋ(𝑓0)}𝒦_𝑓(0), (4.22)where ℬ₁ is the uniformly (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓₀)) absorbing set in 𝑉_𝑔 and 𝒦_𝑓 is the kernel of the process {𝑈_𝑓(𝑡,𝜏)}. Furthermore, the kernel 𝒦_𝑓 is nonempty for all 𝑓∈ℋ(𝑓₀).

Proof. Using Proposition 4.4, we have the family of processes {𝑈_𝑓(𝑡,𝜏)}, 𝑓∈ℋ(𝑓₀) corresponding to (4.3) possesses the uniformly (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓₀)) absorbing set in 𝑉_𝑔.
Now we prove the existence of compact uniform (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓₀)) attractor in 𝑉_𝑔 by applying the method established in Section 3, that is, we testify that the family of processes {𝑈_𝑓(𝑡,𝜏)},𝑓∈ℋ(𝑓₀) corresponding to (4.3) satisfies uniform (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓₀)) Condition (C).
Multiplying (2.14) by 𝐴_𝑔𝑢₂(𝑡), we have(𝑑𝑣𝑑𝑡,𝐴_𝑔𝑢₂)+(𝜈𝐴_𝑔𝑢,𝐴_𝑔𝑢₂)+(𝐵_𝑔(𝑢,𝑢),𝐴_𝑔𝑢₂)_𝑔=(𝑓,𝐴_𝑔𝑢₂)−(𝑅𝑢,𝐴_𝑔𝑢₂)_𝑔. (4.23)It follows that12𝑑𝑑𝑡‖𝑢₂‖²_𝑔+𝜈|𝐴_𝑔𝑢₂|²_𝑔≤|(𝐵_𝑔(𝑢,𝑢),𝐴_𝑔𝑢₂)_𝑔|+|(𝑓,𝐴_𝑔𝑢₂)_𝑔|+|(𝑅𝑢,𝐴_𝑔𝑢₂)_𝑔|. (4.24)To estimate (𝐵_𝑔(𝑢,𝑢),𝐴𝑢₂)_𝑔, we recall some inequalities [16]: for every 𝑢,𝑣∈𝐷(𝐴_𝑔):|𝐵_𝑔(𝑢,𝑣)|≤𝑐|𝑢|_𝑔^1/2‖𝑢‖_𝑔^1/2‖𝑣‖_𝑔^1/2|𝐴_𝑔𝑣|_𝑔^1/2,|𝑢|_𝑔^1/2|𝐴_𝑔𝑢|_𝑔^1/2‖𝑣‖_𝑔 (4.25)(see [16])|𝑤|_𝐿∞(Ω)²≤𝑐‖𝑤‖_𝑔(1+log|𝐴_𝑔𝑤|𝜆_𝑔‖𝑤‖²_𝑔)^1/2 (4.26)from which we deduce that|𝐵_𝑔(𝑢,𝑣)|≤𝑐|𝑢|_𝐿^∞_(Ω)|∇𝑣|_𝑔|𝑢|_𝑔|∇𝑣|_𝐿^∞(Ω), (4.27)and using, (4.26)|𝐵_𝑔(𝑢,𝑣)|≤𝑐‖𝑢‖_𝑔‖𝑣‖_𝑔(1+log|𝐴_𝑔𝑢|²𝜆_𝑔‖𝑤‖²_𝑔)^1/2,|𝑢|_𝑔|𝐴_𝑔𝑣|_𝑔(1+log|𝐴_𝑔^3/2𝑣|²𝜆_𝑔‖𝐴_𝑔𝑣‖²_𝑔)^1/2. (4.28)Expanding and using Young's inequality, together with the first one of (4.28) and the second one of (4.25), we have|(𝐵_𝑔(𝑢,𝑢),𝐴_𝑔𝑢₂)|≤|(𝐵_𝑔(𝑢₁,𝑢₁+𝑢₂),𝐴_𝑔𝑢₂)|+|(𝐵_𝑔(𝑢₂,𝑢₁+𝑢₂),𝐴_𝑔𝑢₂)|≤𝑐𝐿^1/2‖𝑢₁‖_𝑔|𝐴_𝑔𝑢₂|_𝑔(‖𝑢₁‖_𝑔+‖𝑢₂‖_𝑔)+𝑐|𝑢₂|_𝑔^1/2|𝐴_𝑔𝑢₂|_𝑔^3/2≤𝜈6|𝐴_𝑔𝑢₂|²_𝑔+𝑐𝜈𝜌⁴₁𝑐𝐿+𝜈³𝜌²₀𝜌⁴₁,𝑡≥𝑡₀+1, (4.29)where we use|𝐴_𝑔𝑢₁|²_𝑔≤𝜆_𝑚‖𝑢₁‖²_𝑔 (4.30)and set𝜆𝐿=1+log^𝑚+1𝜆_𝑔. (4.31)Next, using the Cauchy inequality,|(𝑅𝑢,𝐴_𝑔𝑢₂)_𝑔𝜈|=|(𝑔(∇𝑔⋅∇)𝑢,𝐴_𝑔𝑢₂)_𝑔|≤𝜈𝑚₀|∇𝑔|_∞‖𝑢‖_𝑔|𝐴_𝑔𝑢₂|_𝑔≤𝜈6|𝐴_𝑔𝑢₂|²_𝑔+3𝜈2|∇𝑔|²_∞𝜌²₁. (4.32)Finally, we estimate |(𝑓,𝐴_𝑔𝑢₂)| by|(𝑓,𝐴_𝑔𝑢₂)|≤|𝑓|_𝑔|𝐴_𝑔𝑢₂|₂≤𝜈6|𝐴_𝑔𝑢₂|²_𝑔+32𝜈|𝑓|²_𝑔. (4.33)Putting (4.29)–(4.33) together, there exists a constant 𝑀₂ such that𝑑𝑑𝑡‖𝑢₂‖²_𝑔+𝜈𝜆_𝑚+1‖𝑢₂‖²_𝑔≤3𝜈|𝑓|_𝑔+𝑀₂. (4.34)Here 𝑀₂=𝑀₂(𝜌₀,𝜌₁,𝐿,𝜈,|∇𝑔|) depends on 𝜆_𝑚+1, is not increasing as 𝜆_𝑚+1 increasing. Therefore, by the Gronwall inequality, the above inequality implies‖𝑢₂‖²_𝑔≤‖𝑢₂(𝑡₀+1)‖²_𝑔𝑒^{−𝜈𝜆𝑚+1(𝑡−(𝑡0+1))}+2𝑀₂𝜈𝜆_𝑚+1+3𝜈^𝑡_𝑡0+1𝑒^{−𝜈𝜆𝑚+1(𝑡−𝑠)}|𝑓|²_𝑔𝑑𝑠. (4.35) Applying (4.4) for any 𝜀3𝜈^𝑡_𝑡0+1𝑒^{−𝜈𝜆𝑚+1(𝑡−𝑠)}|𝑓|²_𝑔𝜀𝑑𝑠<3. (4.36) Using (2.9) and let 𝑡₁=𝑡₀+1+(1/𝜈𝜆_𝑚+1)ln(3𝜌²₁/𝜀), then 𝑡≥𝑡₁ implies2𝑀₂𝜈𝜆_𝑚+1<𝜀3;‖𝑢₂(𝑡₀+1)‖²_𝑔𝑒^{−𝜈𝜆𝑚+1(𝑡−(𝑡0+1))}≤𝜌²₁𝑒^{−𝜈𝜆𝑚+1(𝑡−(𝑡0+1))}<𝜀3. (4.37)Therefore, we deduce from (4.35) that‖𝑢₂‖²_𝑔≤𝜀,∀𝑡≥𝑡₁,𝑓∈ℋ(𝑓₀), (4.38)which indicates {𝑈_𝑓(𝑡,𝜏)},𝑓∈ℋ(𝑓₀) satisfying uniform (𝑤.𝑡.𝑟.𝑓∈ℋ(𝑓₀)) Condition (C) in 𝑉_𝑔.

5. Dimension of the Uniform Attractor

In this section we estimate the fractal dimension (for definition see, e.g., [2, 10, 15]) of the kernel sections of the uniform attractors obtained in Section 4 by applying the methods in [17].

Process {𝑈(𝑡,𝜏)} is said to be uniformly quasidifferentiable on {𝒦(𝑠)}_𝜏∈ℝ, if there is a family of bounded linear operators {𝐿(𝑡,𝜏;𝑢)∣𝑢∈𝒦(𝑠),𝑡≥𝜏,𝜏∈ℝ},𝐿(𝑡,𝜏;𝑢)∶𝐸→𝐸 such thatlimsup_{𝜀→0𝜏∈ℝ}sup_{𝑢,𝑣∈𝒦(𝑠)0<|𝑢−𝑣|≤𝜀}|𝑈(𝑡,𝜏)𝑣−𝑈(𝑡,𝜏)𝑢−𝐿(𝑡,𝜏;𝑢)(𝑣−𝑢)||𝑣−𝑢|=0. (5.1)

We want to estimate the fractal dimension of the kernel sections 𝒦(𝑠) of the process {𝑈(𝑡,𝜏)} generated by the abstract evolutionary (2.14). Assume that {𝐿(𝑡,𝜏;𝑢)} is generated by the variational equation corresponding to (2.14)𝜕_𝑡𝑤=𝐹^′(𝑢,𝑡)𝑤,𝑤|_𝑡=𝜏=𝑤_𝜏∈𝐸,𝑡≥𝜏,𝜏∈ℝ, (5.2)that is, 𝐿(𝑡,𝜏;𝑢_𝜏)𝑤_𝜏=𝑤(𝑡) is the solution of (5.2), and 𝑢(𝑡)=𝑈(𝑡,𝜏)𝑢_𝜏 is the solution of (2.14) with initial value 𝑢_𝜏∈𝒦(𝜏). For natural number 𝑗∈𝑁, we set^0𝑥0𝑞^02𝑑𝑐_𝑗=lim_𝑇→+∞sup_𝜏∈𝑅sup_{𝑢𝜏∈𝒦(𝜏)}(1𝑇_𝜏^𝜏+𝑇Tr(𝐹^′(𝑢(𝑠),𝑠))𝑑𝑠), (5.3)where Tr is trace of the operator.

We will need the following Theorem VIII.3.1 in [10,] and [2]. Theorem 5.1. Under the assumptions above, let us suppose that 𝑈_𝜏∈ℝ𝒦(𝜏) is relatively compact in 𝐸, and there exists 𝑞_𝑗,𝑗=1,2,…, such that^0𝑥0𝑞^02𝑑𝑐_𝑗≤𝑞_𝑗𝑞,forany𝑗≥1,_𝑛0≥0,𝑞_𝑛0+1<0,forsome𝑛₀𝑞≥1,_𝑗≤𝑞_𝑛0+(𝑞_𝑛0−𝑞_𝑛0+1)(𝑛₀−𝑗),∀𝑗=1,2,…. (5.4)Then,𝑑_𝐹(𝒦(𝜏))≤𝑑₀∶=𝑛₀+𝑞_𝑛0𝑞_𝑛0−𝑞_𝑛0+1,∀𝜏∈ℝ. (5.5)

We now consider (2.14) with 𝑓∈𝐿²_𝑛(ℝ;𝑉^′_𝑔). The equations possess a compact uniform (𝑤.𝑟.𝑡.𝑓∈ℋ(𝑓)) attractor 𝔄_ℋ(𝑓) and ⋃_𝜏∈ℝ𝒦_𝑓(𝜏)⊂𝔄_ℋ(𝑓). By [2, 10, 15], we know that the associated process {𝑈_𝑓(𝑡,𝜏)} is uniformly quasidifferentiable on {𝒦_𝑓(𝜏)}_𝜏∈ℝ and the quasidifferential is Hölder-continuous with respect to 𝑢_𝜏∈𝒦_𝑓(𝜏). The corresponding variational equation is𝜕_𝑡𝑤=−𝜈𝐴_𝑔𝑢−𝐵_𝑔𝑢−𝜈𝑅𝑢+𝑃_𝑔𝑓≡𝐹^′(𝑢(𝑡),𝑡)𝑤,𝑤|_𝑡=𝜏=𝑤_𝜏∈𝐸,𝜏∈ℝ. (5.6)

We have the main results in this section.Theorem 5.2. Suppose that 𝑓(𝑡) satisfies the assumptions of Theorem 4.5. Then, if 𝛾=1−(2|∇𝑔|_∞)/(𝑚₀𝜆_𝑔^1/2)>0, the Uniform attractor 𝔄₀ defined by (4.8) satisfies𝑑_𝐹(𝔄₀)≤𝛽𝛼, (5.7)where𝑐𝛼=²𝜈𝑚₀𝜆^′₁𝛾2𝑀₀,𝑐𝛽=¹𝑑₁2𝜈³𝑚₀𝛾sup_{𝜑𝑗∈𝐻𝑔|𝜑𝑗|≤1𝑗=1,2,…,𝑚}1𝑇_𝜏^𝜏+𝑇‖𝑓(𝑠)‖²_𝑉′𝑔𝑑𝑠, (5.8)the constant 𝑐₁,𝑐₂ of (3.29) and (3.32) of Chapter 𝑉𝐼 in [15] and [2], 𝜆^′₁ is the first eigenvalue of the Stokes operator and 𝑑₁=|∇𝑔|²_∞/4𝑚₀+|∇𝑔|_∞+𝑀₀.

Proof. With Theorem 4.5 at our disposal we may apply the abstract framework in [2, 10, 15, 17].
For 𝜉₁,𝜉₂,…,𝜉_𝑚∈𝐻_𝑔, let 𝑣_𝑗(𝑡)=𝐿(𝑡,𝑢_𝜏)⋅𝜉_𝑗, where 𝑢_𝜏∈𝐻_𝑔. Let {𝜑_𝑗(𝑠);𝑗=1,2,…,𝑚} be an orthonormal basis for span {𝑣_𝑗;𝑗=1,2,…,𝑚}. Since 𝑣_𝑗∈𝑉_𝑔 almost everywhere 𝑠≥𝜏, we can also assume that 𝜑_𝑗(𝑠)∈𝑉_𝑔 almost everywhere 𝑠≥𝜏. Then, similar to the Proof process of Theorems 4.5 and 4.6, we may obtain^𝑚_𝑖=1(𝐹^′(𝑈(𝑠,𝜏)𝑢_𝜏,𝑠)𝜑_𝑖,𝜑_𝑖)_𝑔=−𝜈^𝑚_𝑖=1‖𝜑_𝑗‖²_𝑔−^𝑚_𝑖=1𝑏_𝑔(𝜑_𝑗,𝑈(𝑠,𝜏)𝑢_𝜏,𝜑_𝑗)−^𝑚_𝑖=1(𝜈𝑔(∇𝑔⋅∇)𝜑_𝑗,𝜑_𝑗)_𝑔, (5.9)almost everywhere 𝑠≥𝜏. From this equality, and in particular using the Schwarz and Lieb-Thirring inequality (see [2, 10, 15, 17]), one obtains^𝑚_𝑖=1‖𝜑‖²_𝑔≥𝜆₁+⋯+𝜆_𝑚≥𝑚₀𝑀₀(𝜆^′₁+⋯+𝜆^′_𝑚𝑚)≥⁰𝑀₀𝑐₂𝜆^′₁𝑚²,Tr_𝑗(𝐹^′(𝑈(𝑠,𝜏)𝑢_𝜏,𝑠)_𝑔≤−𝜈(1−|∇𝑔|_∞𝑚₀𝜆₁^1/2)^𝑚_𝑖=1‖𝜑_𝑗‖²_𝑔+‖𝑈(𝑠,𝜏)𝑢_𝜏‖_𝑔(𝑐₁𝑑₁𝑚₀^𝑚_𝑖=1‖𝜑_𝑗‖²_𝑔)^1/2𝜈≤−2(1−2|∇𝑔|_∞𝑚₀𝜆₁^1/2)^𝑚_𝑖=1‖𝜑_𝑗‖²_𝑔+𝑐₁𝑑₁2𝜈𝑚₀‖𝑈(𝑠,𝜏)𝑢_𝜏‖²_𝑔≤−𝜈𝑚₀2𝑀₀(1−2|∇𝑔|_∞𝑚₀𝜆₁^1/2)𝑐₂𝜆^′₁𝑚²+𝑐₁𝑑₁2𝜈𝑚₀‖𝑈(𝑠,𝜏)𝑢_𝜏‖²_𝑔, (5.10)on the other hand, we can deduce (2.14) that𝑑𝑑𝑡|𝑈(𝑠,𝜏)𝑢_𝜏|²_𝑔+𝜈‖𝑈(𝑠,𝜏)𝑢_𝜏‖²_𝑔≤‖𝑓‖²_𝑉′𝑔𝜈+2𝜈𝑚₀𝜆_𝑔^1/2|∇𝑔|_∞‖𝑈(𝑠,𝜏)𝑢_𝜏‖²_𝑔 (5.11)for 𝜆_𝑔=4𝜋²𝑚₀/𝑀₀, and then^𝑡_𝜏‖𝑈(𝑠,𝜏)𝑢_𝜏‖²_𝑔1𝑑𝑠≤(𝜈^2𝑡_𝜏‖𝑓(𝑠)‖²_𝑉′𝑔𝑑𝑠+|𝑢_𝜏|²𝜈)(1−2|∇𝑔|_∞𝑚₀𝜆_𝑔^1/2)⁻¹,𝑡≥𝜏. (5.12)
Now we define𝑞_𝑚=sup_{𝜑𝑗∈𝐻𝑔|𝜑𝑗|≤1𝑗=1,2,…,𝑚}(1𝑇_𝜏^𝜏+𝑇Tr_𝑗(𝐹^′(𝑈(𝑠,𝜏)𝑢_𝜏,𝑠)𝑑𝑠))_𝑔, (5.13)Using Theorem 5.1, we have^0𝑥0𝑞^02𝑑𝑐_𝑚≤−𝜈𝑚₀2𝑀₀(1−2|∇𝑔|_∞𝑚₀𝜆₁^1/2)𝑐₂𝜆^′₁𝑚²+𝑐₁𝑑₁2𝜈𝑚₀(sup_{𝜑𝑗∈𝐻𝑔|𝜑𝑗|≤1𝑗=1,2,…,𝑚}(1𝑇_𝜏^𝜏+𝑇‖𝑈(𝑠,𝜏)𝑢_𝜏‖²_𝑔𝑑𝑠))≤−𝜈𝑚₀2𝑀₀(1−2|∇𝑔|_∞𝑚₀𝜆₁^1/2)𝑐₂𝜆^′₁𝑚²+𝑐₁𝑑₁2𝜈𝑚₀(1𝜈²sup_{𝜑𝑗∈𝐻𝑔|𝜑𝑗|≤1𝑗=1,2,…,𝑚}(1𝑇_𝜏^𝜏+𝑇‖𝑓(𝑠)‖²_𝑉′𝑔𝑑𝑠+|𝑢_𝜏|²𝜈𝑇)(1−2|∇𝑔|_∞𝑚₀𝜆_𝑔^1/2)⁻¹𝑞),_𝑚=limsup𝑞_𝑇→∞^{0𝑥002𝑑𝑐}_𝑚≤−𝛼𝑚²+𝛽, (5.14)Hencedim_𝐹𝒜₀(𝜏)≤𝛽𝛼. (5.15)

Acknowledgment

The author would like to thank the reviewers and the editor for their valuable suggestions and comments.