Regularity of Global Attractor for the Reaction-Diffusion Equation

Abstract

By using an iteration procedure, regularity estimates for the linear semigroups, and a classical existence theorem of global attractor, we prove that the reaction-diffusion equation possesses a global attractor in Sobolev space 𝐻^𝑘 for all 𝑘>0, which attracts any bounded subset of 𝐻^𝑘(Ω) in the 𝐻^𝑘-norm.

1. Introduction

This paper is concerned with the following initial-boundary problem of reaction-diffusion systems involving an unknown function 𝑢=𝑢(𝑥,𝑡): 𝜕𝑢(𝑢)(),(),𝑢()𝜕𝑡=𝑎Δ𝑢−𝑔inΩ×0,∞𝑢=0,in𝜕Ω×0,∞𝑥,0=𝜑,inΩ, (1.1) where 𝑎>0 is a given constant. Δ is the Laplace operator. Ω denotes an open bounded set of 𝑅^𝑛(𝑛=1,2,3) with smooth boundary 𝜕Ω. 𝑔(𝑠) is a polynomial on 𝑠∈𝑅¹, which is given by 𝑔(𝑠)=^𝑝_𝑘=1𝑎_𝑘𝑠^𝑘, (1.2) where 𝑝 should be an odd number, that is, 𝑝=2𝑚+1(𝑚≥1), and 𝑎_𝑝>0. (1.3)

The reaction-diffusion systems (1.1) have been extensively studied during the last decades, one of the motivations being that such systems could account for phenomena occurring in living organisms. In 1952, Turing [1] proposed that a combination of chemical reaction and diffusion produces spatial patterns of chemical concentration, under certain conditions. Such patterns are of interest because they give a possible explanation for the development of pattern and form in developmental biology [2–4] and experimental chemical systems [5]. Schneider et al. [6–9] have studied existence of periodic travelling wave solutions and positive periodic solution of reaction-diffusion systems. In [10–17], asymptotic behaviour of the nonlinear reaction-diffusion equation, such as global attractors, inertial manifolds, and approximate inertial manifolds, has been studied.

The global asymptotical behaviors of solutions and existence of global attractors are important for the study of the dynamical properties of general nonlinear dissipative dynamical systems. So, many authors are interested in the existence of global attractors such as [12–19]. As for the reaction-diffusion equation (1.1), the existence of global solutions and global attractors in 𝐿²(Ω) has been proved by Temam [16], Marion [18, 19], and Zhong et al. [17]. For convenience, we introduce the main results as follows.

Lemma 1.1. Under the conditions (1.2) and (1.3) for 𝜑∈𝐻, the following three claims hold.(1)Equation (1.1) has a unique global weak solution 𝑢∈𝐶((0,∞);𝐻)⋂𝐿²((0,𝑇);𝐻_1/2), for 𝑇>0;(2)Equation (1.1) has a unique strong solution 𝑢∈𝐶([0,𝑇);𝐻_1/2)⋂𝐿²((0,𝑇);𝐻₁) for any 𝑇>0;(3)Equation (1.1) has a global attractor 𝒜⊂𝐻, which attracts any bounded set of 𝐻 in the 𝐻-norm.
Here the spaces 𝐻,𝐻_1/2 and 𝐻₁ are defined as follows: 𝐻=𝐿²(Ω),𝐻_1/2=𝐻¹₀(Ω),𝐻₁=𝐻¹₀(Ω)𝐻²(Ω). (1.4)

In this paper, we shall use the regularity estimates for the linear semigroups, combining with the classical existence theorem of global attractors, to prove that the reaction-diffusion equation possesses, in any 𝑘th differentiable function spaces 𝐻^𝑘(Ω), a global attractor, which attracts any bounded set of 𝐻^𝑘(Ω) in 𝐻^𝑘-norm. The basic idea is an iteration procedure, which is from recent books and papers [20–24].

2. Preliminaries

Let 𝑋 and 𝑋₁ be two Banach spaces and 𝑋₁⊂𝑋 a compact and dense inclusion. Consider the abstract nonlinear evolution equation defined on 𝑋, given by𝑑𝑢(𝑢),𝑢()𝑑𝑡=𝐿𝑢+𝐺𝑥,0=𝑢₀, (2.1) where 𝑢(𝑡) is an unknown function, 𝐿∶𝑋₁→𝑋 a linear operator, and 𝐺∶𝑋₁→𝑋 a nonlinear operator.

A family of operators 𝑆(𝑡)∶𝑋→𝑋(𝑡≥0) is called a semigroup generated by (2.1) provided that 𝑆(𝑡) satisfies the properties:(1)𝑆(𝑡)∶𝑋→𝑋 is a continuous mapping for any 𝑡≥0;(2)𝑆(0)=id∶𝑋→𝑋 the identity; (3)𝑆(𝑡+𝑠)=𝑆(𝑡)⋅𝑆(𝑠)forall𝑡,𝑠≥0, and the solution of (2.1) can be expressed as 𝑢𝑡,𝑢₀=𝑆(𝑡)𝑢₀. (2.2)

Next, we introduce the concepts and definitions of invariant sets, global attractors, 𝜔-limit sets for the semigroup 𝑆(𝑡).

Definition 2.1. Let 𝑆(𝑡) be a semigroup defined on 𝑋. A set Σ⊂𝑋 is called an invariant set of 𝑆(𝑡) if 𝑆(𝑡)Σ=Σforall𝑡≥0. An invariant set Σ is an attractor of 𝑆(𝑡) if Σ is compact, and there exists a neighborhood 𝑈⊂𝑋 of Σ such that for any 𝑢₀∈𝑈: inf_𝑣∈Σ𝑆(𝑡)𝑢₀−𝑣_𝑋⟶0,as𝑡⟶∞. (2.3)
In this case, we say that Σ attracts 𝑈. Especially, if Σ attracts any bounded set of 𝑋, Σ is called a global attractor of 𝑆(𝑡).
For a set 𝐷⊂𝑋, we define the 𝜔-limit set of 𝐷 as follows: 𝜔(𝐷)=_{𝑠≥0𝑡≥𝑠}𝑆(𝑡)𝐷, (2.4) where the closure is taken in the 𝑋-norm. The following Lemma 2.2 is the classical existence theorem of global attractor by Temam [16].

Lemma 2.2. Let 𝑆(𝑡)∶𝑋→𝑋 be the semigroup generated by (2.1). Assume that the following conditions hold:(1)𝑆(𝑡) has a bounded absorbing set 𝐵⊂𝑋, that is, for any bounded set 𝐴⊂𝑋 there exists a time 𝑡_𝐴≥0 such that 𝑆(𝑡)𝑢₀∈𝐵 for  all 𝑢₀⊂𝐴 and 𝑡>𝑡_𝐴;(2)𝑆(𝑡) is uniformly compact, that is, for any bounded set 𝑈⊂𝑋 and some 𝑇>0 sufficiently large, the set ⋃_𝑡≥𝑇𝑆(𝑡)𝑈 is compact in 𝑋.
Then, that the 𝜔-limit set 𝒜=𝜔(𝐵) of 𝐵 is a global attractor of (2.1), and 𝒜 is connected providing B is connected.
Note that we used to assume that the linear operator 𝐿 in (2.1) is a sectorial operator which generates an analytic semigroup 𝑒^𝑡𝐿. It is known that there exists a constant 𝜆≥0 such that 𝐿−𝜆𝐼 generates the fractional power operators ℒ^𝛼 and fractional order spaces 𝑋_𝛼 for 𝛼∈𝑅¹, where ℒ=−(𝐿−𝜆𝐼). Without loss of generality, we assume that 𝐿 generates the fractional power operators ℒ^𝛼 and fractional order spaces 𝑋_𝛼 as follows: ℒ^𝛼=(−𝐿)^𝛼∶𝑋_𝛼⟶𝑋,𝛼∈𝑅¹, (2.5) where 𝑋_𝛼=𝐷(ℒ^𝛼) is the domain of ℒ^𝛼. By the semigroup theory of linear operators (Pazy [25]), we know that 𝑋_𝛽⊂𝑋_𝛼 is a compact inclusion for any 𝛽>𝛼.
Thus, Lemma 2.2 can be equivalently expressed in the following Lemma 2.3 [24].

Lemma 2.3. Let 𝑢(𝑡,𝑢₀)=𝑆(𝑡)𝑢₀(𝑢₀∈𝑋,𝑡≥0) be a solution of (2.1) and 𝑆(𝑡) the semigroup generated by (2.1). Let 𝑋_𝛼 be the fractional order space generated by 𝐿. Assume(1)for some 𝛼≥0, there is a bounded set 𝐵⊂𝑋_𝛼; for any 𝑢₀∈𝑋_𝛼, there exists 𝑡_𝑢0>0 such that 𝑢𝑡,𝑢₀∈𝐵,∀𝑡>𝑡_𝑢0; (2.6)(2)there is a 𝛽>𝛼, for any bounded set 𝑈⊂𝑋_𝛽 there are 𝑇>0 and 𝐶>0 such that 𝑢𝑡,𝑢_0𝑋𝛽≤𝐶,∀𝑡>𝑇,𝑢₀∈𝑈. (2.7)
Then (2.1) has a global attractor 𝒜⊂𝑋_𝛼, which attracts any bounded set of 𝑋_𝛼 in the 𝑋_𝛼-norm.
For sectorial operators, we also have the following properties, which can be found in [25].

Lemma 2.4. Let 𝐿∶𝑋₁→𝑋 be a sectorial operator which generates an analytic semigroup 𝑇(𝑡)=𝑒^𝑡𝐿. If all eigenvalues 𝜆 of 𝐿 satisfy Re𝜆<−𝜆₀ for some real number 𝜆₀>0, then for ℒ^𝛼(ℒ=−𝐿) we have (1)𝑇(𝑡)∶𝑋→𝑋_𝛼 is bounded for all 𝛼∈𝑅¹ and 𝑡>0;(2)𝑇(𝑡)ℒ^𝛼𝑥=ℒ^𝛼𝑇(𝑡)𝑥  for  all  𝑥∈𝑋_𝛼;(3) for each 𝑡>0,ℒ^𝛼𝑇(𝑡)∶𝑋→𝑋 is bounded, and ℒ^𝛼𝑇(𝑡)≤𝐶_𝛼𝑡^−𝛼𝑒^−𝛿𝑡, (2.8) where some 𝛿>0,𝐶_𝛼>0 is a constant only depending on 𝛼;(4) the 𝑋_𝛼-norm can be defined by ‖𝑥‖_𝑋𝛼=ℒ^𝛼𝑥_𝑋; (2.9)(5) if ℒ is symmetric, for any 𝛼,𝛽∈𝑅¹, we have ℒ^𝛼𝑢,𝑣_𝐻=ℒ^𝛼−𝛽𝑢,ℒ^𝛽𝑣_𝐻. (2.10)

3. Main Results

Let 𝐻 and 𝐻₁ be the spaces defined as in (1.4). We define the operators 𝐿∶𝐻₁→𝐻 and 𝐺∶𝐻₁→𝐻 by (𝑢),𝐿𝑢=𝑎Δ𝑢,𝐺𝑢=−𝑔 (3.1) where 𝑔(𝑢) is the same as one of (1.2). Thus, the reaction-diffusion equation (1.1) can be written in the abstract form (2.1). It is well known that the linear operator 𝐿∶𝐻₁→𝐻 given by (3.1) is a sectorial operator and ℒ=−𝐿. The space 𝐻₁ is the same as (1.4), 𝐻_1/2 is given by 𝐻_1/2= closure of 𝐻₁ in 𝐻¹(Ω), and 𝐻_𝑘=𝐻^2𝑘(Ω)∩𝐻₁ for 𝑘≥1.

The main result in this paper is given by the following theorem, which provides the existence of global attractors of the reaction-diffusion equation (1.1) in any 𝑘th order space 𝐻_𝑘.

Theorem 3.1. Let the function 𝑔 be a polynomial of order 𝑝𝑔(𝑢)=^𝑝_𝑘=1𝑎_𝑘𝑢^𝑘,𝑝=2𝑚+1(𝑚≥1,𝑚∈𝑁), (3.2) with leading coefficient 𝑎_𝑝>0. (3.3)
Assume 𝑝=3 for 𝑛=3. Then, for any 𝛼≥0 (1.1) has a global attractor 𝒜 in 𝐻_𝛼, and 𝒜 attracts any bounded set of 𝐻_𝛼 in the 𝐻_𝛼-norm.

Proof. From Lemma 1.1, we know that the solution of system (1.1) is a weak solution for any 𝜑∈𝐻. Hence, the solution 𝑢(𝑡,𝜑) of system (1.1) can be written as 𝑢𝑡,𝜑=𝑒^𝑡𝐿𝜑+^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝐺(𝑢)𝑑𝜏. (3.4)
By (3.1), we rewrite (3.4) as 𝑢𝑡,𝜑=𝑒^𝑡𝐿𝜑−^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)𝑑𝜏. (3.5)
Next, according to Lemma 2.3, we prove Theorem 3.1 in the following six steps.
Step 1. We prove that for any bounded set 𝑈⊂𝐻_1/2 there is a constant 𝐶>0 such that the solution 𝑢(𝑡,𝜑) of system (1.1) is uniformly bounded by the constant 𝐶 for any 𝜑∈𝑈 and 𝑡≥0. To do that, we firstly check that system (1.1) has a global Lyapunov function as follows: 𝐹(𝑢)=_Ω12|∇𝑢|²+𝑓(𝑢)𝑑𝑥, (3.6) where𝑓(𝑧)=^𝑧₀𝑔(𝑧)𝑑𝑧=^𝑝_𝑘=11𝑎𝑘+1^𝑘𝑧^𝑘+1. (3.7)
In fact, if 𝑢(𝑡,⋅) is a weak solution of system (1.1), we have 𝑑𝐹𝑢=(𝑢),𝑑𝑡𝑡,𝜑𝐷𝐹𝑑𝑢𝑑𝑡_𝐻. (3.8)
By (3.1) and (3.6), we get 𝑑𝑢(𝑢)(𝑢).𝑑𝑡=𝐿𝑢+𝐺=−𝐷𝐹 (3.9)
Hence, it follows from (3.8) and (3.9) that 𝑑𝐹(𝑢)=⟨(𝑢)(𝑢𝑑𝑡𝐷𝐹,−𝐷𝐹)⟩_𝐻=−‖𝐷𝐹(𝑢)‖²_𝐻, (3.10) which implies that (3.6) is a Lyapunov function.
Integrating (3.10) from 0 to 𝑡 gives 𝐹𝑢𝑡,𝜑=−^𝑡₀‖𝐷𝐹(𝑢)‖²_𝐻𝑑𝑡+𝐹𝜑. (3.11)
Using (1.2) and (3.6), we have 𝐹(𝑢)=_Ω12|∇𝑢|²+𝑓(𝑢)𝑑𝑥=_Ω12|∇𝑢|²+1𝑎𝑝+1^𝑝𝑢^𝑝+1+^𝑝−1_𝑘=11𝑎𝑘+1^𝑘𝑢^𝑘+1𝑑𝑥≥_Ω12|∇𝑢|²+1𝑎𝑝+1^𝑝𝑢^𝑝+1−^𝑝−1_𝑘=11𝑎𝑘+1^𝑘|𝑢|^𝑘+1≥𝑑𝑥_Ω12|∇𝑢|²+1𝑎𝑝+1^𝑝𝑢^𝑝+1−^𝑝−1_𝑘=11𝑎𝑘+1^𝑘𝜀|𝑢|^𝑝+1+𝜀^{−(𝑘+1)/(𝑝−𝑘)}=𝑑𝑥_Ω12|∇𝑢|²+1𝑎𝑝+1^𝑝𝑢^𝑝+1−𝜀^𝑝−1_𝑘=11𝑎𝑘+1^𝑘|𝑢|^𝑝+1−^𝑝−1_𝑘=11𝑎𝑘+1^𝑘𝜀^{−(𝑘+1)/(𝑝−𝑘)}𝑑𝑥. (3.12) Choosing 𝜀 such that 𝜀(∑^𝑝−1_𝑘=1(1/(𝑘+1))|𝑎_𝑘|)=(1/2(𝑝+1))𝑎_𝑝, and noting that 𝑝 is an odd number, that is, 𝑝=2𝑚+1(𝑚≥1), we get 𝐹(𝑢)≥_Ω12|∇𝑢|²+1𝑎𝑝+1^𝑝𝑢^𝑝+1−12𝑎𝑝+1^𝑝|𝑢|^𝑝+1−^𝑝−1_𝑘=11𝑎𝑘+1^𝑘𝜀^{−(𝑘+1)/(𝑝−𝑘)}𝑑𝑥=_Ω12|∇𝑢|²+12𝑎𝑝+1^𝑝|𝑢|^𝑝+1−^𝑝−1_𝑘=11𝑎𝑘+1^𝑘𝜀^{−(𝑘+1)/(𝑝−𝑘)}𝑑𝑥≥𝐶_1Ω|∇𝑢|²+|𝑢|^𝑝+1𝑑𝑥−𝐶₂. (3.13) Combining with (3.11) yields 𝐶_1Ω|∇𝑢|²+|𝑢|^𝑝+1𝑑𝑥−𝐶₂≤−^𝑡₀‖𝐷𝐹(𝑢)‖²_𝐻𝜑,𝐶𝑑𝑡+𝐹_1Ω|∇𝑢|²+|𝑢|^𝑝+1𝑑𝑥+^𝑡₀‖𝐷𝐹(𝑢)‖²_𝐻𝑑𝑡≤𝐹𝜑+𝐶₂,_Ω|∇𝑢|²+|𝑢|^𝑝+1𝑑𝑥≤𝐶,∀𝑡≥0,𝜑∈𝑈, (3.14) which implies 𝑢𝑡,𝜑_𝐻1/2≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_1/2. (3.15) where 𝐶₁,𝐶₂, and 𝐶 are positive constants. 𝐶 only depends on 𝜑.
Step 2. We prove that for any bounded set 𝑈⊂𝐻_𝛼(1/2≤𝛼<1) there exists 𝐶>0 such that 𝑢𝑡,𝜑_𝐻𝛼≤𝐶,∀𝑡≥0,𝜑∈𝑈,𝛼<1. (3.16) By 𝐻_1/2↪𝐿^2𝑝(Ω), we have 𝑔(𝑢)²_𝐻=_Ω𝑔(𝑢)²𝑑𝑥=_Ω^𝑝_𝑘=1𝑎_𝑘𝑢^𝑘2𝑑𝑥≤_Ω𝑎_𝑝|𝑢|^𝑝+^𝑝−1_𝑘=1𝑎_𝑘𝜀|𝑢|^𝑝+𝜀^{−𝑘/(𝑝−𝑘)2}𝑑𝑥≤𝐶_Ω|𝑢|^2𝑝𝑑𝑥+1≤𝐶‖𝑢‖_𝐻^2𝑝_1/2+1, (3.17) which implies  that 𝑔∶𝐻_1/2→𝐻 is bounded.
Hence, it follows from (2.9) and (3.5) that 𝑢𝑡,𝜑_𝐻𝛼=𝑒^𝑡𝐿𝜑+^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)𝑑𝜏_𝐻𝛼≤𝜑_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+𝐶^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}‖𝑢‖_𝐻^2𝑝_1/2≤𝜑+1𝑑𝜏_𝐻𝛼+𝐶^𝑡₀𝜏^𝛽𝑒^−𝛿𝑡𝑑𝜏≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼, (3.18) where 𝛽=𝛼(0<𝛽<1). Hence, (3.16) holds.
Step 3. We prove that for any bounded set 𝑈⊂𝐻_𝛼(1≤𝛼<3/2) there exists 𝐶>0 such that 𝑢𝑡,𝜑_𝐻𝛼≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼3,𝛼<2. (3.19) In fact, by the embedding theorems of fractional order spaces [25]: 𝐻_𝛼↪𝐶⁰(Ω)∩𝐻¹(Ω)3,𝛼≥4, (3.20) we have 𝑔(𝑢)²_𝐻1/2=_Ω∇𝑔(𝑢)²𝑑𝑥=_Ω∇^𝑝_𝑘=1𝑎_𝑘𝑢^𝑘2𝑑𝑥≤_Ω^𝑝_𝑘=1𝑘𝑎_𝑘𝑢^(𝑘−1)∇𝑢²≤𝑑𝑥_Ω𝑝𝑎_𝑝|𝑢|^𝑝−1+^𝑝−1_𝑘=1𝑘𝑎_𝑘𝜀|𝑢|^𝑝−1+𝜀^{−(𝑘−1)/(𝑝−𝑘)2}|∇𝑢|²𝑑𝑥≤𝐶_Ω|𝑢|^2𝑝−2+1|∇𝑢|²𝑑𝑥≤𝐶_Ωsup_𝑥∈Ω|𝑢|^2𝑝−2+1|∇𝑢|²𝑢‖𝑑𝑥≤𝐶‖_𝐻^2𝑝−2_𝛼+1‖𝑢‖²_𝐻1/2≤𝐶‖𝑢‖_𝐻^2𝑝−2_𝛼+1‖𝑢‖²_𝐻𝛼, (3.21) which implies𝑔∶𝐻_𝛼⟶𝐻_1/23isboundedfor𝛼≥4. (3.22) Therefore, it follows from (3.16) and (3.22) that 𝑔(𝑢)_𝐻1/2<𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼,34≤𝛼<1. (3.23)
Then, by using same method as that in Step 2, we get from (3.23) that 𝑢𝑡,𝜑_𝐻𝛼=𝑒^𝑡𝐿𝜑+^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)𝑑𝜏_𝐻𝛼≤𝜑_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+𝐶^𝑡₀ℒ^𝛼−(1/2)𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻1/2≤𝜑𝑑𝜏_𝐻𝛼+𝐶^𝑡₀𝜏^−𝛽𝑒^−𝛿𝑡𝑑𝜏≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼, (3.24) where 𝛽=𝛼−1/2(0<𝛽<1). Hence, (3.19) holds.
Step 4. We prove that for any bounded set 𝑈⊂𝐻_𝛼(3/2≤𝛼<2) there exists 𝐶>0 such that 𝑢𝑡,𝜑_𝐻𝛼≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼,𝛼<2. (3.25)
In fact, by the embedding theorems of fractional order spaces [25]: 𝐻²↪𝑊^1,6↪𝑊^1,4,𝐻_𝛼↪𝐶⁰(Ω)∩𝐻²(Ω),𝛼≥1, (3.26)
we have 𝑔(𝑢)²_𝐻1=_ΩΔ𝑔(𝑢)²𝑑𝑥=_ΩΔ^𝑝_𝑘=1𝑎_𝑘𝑢^𝑘2=𝑑𝑥_Ω𝑝𝑘=2𝑘(𝑘−1)𝑎_𝑘𝑢^(𝑘−2)(∇𝑢)²+𝑘𝑎_𝑘𝑢^𝑘−1Δ𝑢+𝑎₁Δ𝑢²≤𝑑𝑥_Ω𝑝𝑝−1𝑎_𝑝|𝑢|^𝑝−2(∇𝑢)²+𝑝𝑎_𝑝|𝑢|^𝑝−1+Δ𝑢_{𝑝−1𝑘=2}𝑘(𝑘−1)𝑎_𝑘𝜀|𝑢|^𝑝−2+𝜀^{−(𝑘−2)/(𝑝−𝑘)}(∇𝑢)²+𝑘𝑎_𝑘𝜀|𝑢|^𝑝−1+𝜀^{−(𝑘−1)/(𝑝−𝑘)}Δ𝑢+𝑎₁|Δ𝑢|²𝑑𝑥≤𝐶_Ω|𝑢|^𝑝−2|∇𝑢|²+|∇𝑢|²+|𝑢|^𝑝−1|Δ𝑢|+|Δ𝑢|²𝑑𝑥≤𝐶_Ω|𝑢|^2𝑝−4|∇𝑢|⁴+|∇𝑢|⁴+|𝑢|^2𝑝−2|Δ𝑢|²+|Δ𝑢|²𝑑𝑥≤𝐶_Ωsup_𝑥∈Ω|𝑢|^2𝑝−4|∇𝑢|⁴+|∇𝑢|⁴+sup_𝑥∈Ω|𝑢|^2𝑝−2|Δ𝑢|²+|Δ𝑢|²|𝑢|𝑑𝑥≤𝐶_𝐻^2𝑝−4_𝛼‖𝑢‖⁴_𝑊1,4+‖𝑢‖⁴_𝑊1,4+|𝑢|_𝐻^2𝑝−2_𝛼‖𝑢‖²_𝐻2+‖𝑢‖²_𝐻2≤𝐶|𝑢|_𝐻^2𝑝−4_𝛼‖𝑢‖⁴_𝐻2+‖𝑢‖⁴_𝐻2+|𝑢|_𝐻^2𝑝−2_𝛼‖𝑢‖²_𝐻𝛼+‖𝑢‖²_𝐻𝛼≤𝐶|𝑢|_𝐻^2𝑝_𝛼+‖𝑢‖⁴_𝐻𝛼+‖𝑢‖²_𝐻𝛼, (3.27) which implies that 𝑔∶𝐻_𝛼⟶𝐻₁isboundedfor𝛼≥1. (3.28)
Therefore, it follows from (3.19) and (3.28) that 𝑔(𝑢)_𝐻1<𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼3,1≤𝛼<2. (3.29)
Then, we get from (3.29) that 𝑢𝑡,𝜑_𝐻𝛼=𝑒^𝑡𝐿𝜑+^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)𝑑𝜏_𝐻𝛼≤𝜑_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+^𝑡₀ℒ^𝛼−1𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻1≤𝜑𝑑𝜏_𝐻𝛼+𝐶^𝑡₀𝜏^−𝛽𝑒^−𝛿𝑡𝑑𝜏≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼, (3.30) where 𝛽=𝛼−1(0<𝛽<1). Hence, (3.25) holds.
Step 5. We prove that for any bounded set 𝑈⊂𝐻_𝛼(𝛼≤0) there exists 𝐶>0 such that 𝑢𝑡,𝜑_𝐻𝛼≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼,𝛼≥0. (3.31) In fact, by the embedding theorems of fractional order spaces [25]: 𝐻³↪𝑊^2,6↪𝑊^1,6,𝐻³↪𝑊^2,6↪𝑊^2,4,𝐻_𝛼↪𝐶⁰(Ω)∩𝐻³(Ω)3,𝛼≥2, (3.32) we have 𝑔(𝑢)²_𝐻3/2=_Ω∇Δ𝑔(𝑢)²𝑑𝑥=_Ω∇Δ^𝑝_𝑘=1𝑎_𝑘𝑢^𝑘2=𝑑𝑥_Ω𝑝𝑘=2𝑘(𝑘−1)(𝑘−2)𝑎_𝑘|𝑢|^𝑘−3|∇𝑢|³+3𝑘(𝑘−1)𝑎_𝑘|𝑢|^𝑘−2|∇𝑢||Δ𝑢|+𝑘𝑎_𝑘|𝑢|^𝑘−1|∇𝑢|+𝑎₁|∇𝑢||Δ𝑢|²≤𝑑𝑥_Ω𝑝𝑝−1𝑝−2𝑎_𝑝|𝑢|^𝑝−3|∇𝑢|³+3𝑝𝑝−1𝑎_𝑝|𝑢|^𝑝−2|∇𝑢||Δ𝑢|+𝑝𝑎_𝑝|𝑢|^𝑝−1||+∇𝑢_{𝑝−1𝑘=3}𝑘(𝑘−1)(𝑘−2)𝑎_𝑘𝜀|𝑢|^𝑝−3+𝜀^{−(𝑘−3)/(𝑝−𝑘)}|∇𝑢|³+3𝑘(𝑘−1)𝑎_𝑘𝜀|𝑢|^𝑝−2+𝜀^{−(𝑘−2)/(𝑝−𝑘)}|∇𝑢||Δ𝑢|+𝑘𝑎_𝑘𝜀|𝑢|^𝑝−1+𝜀^{−(𝑘−1)/(𝑝−𝑘)}|∇Δ𝑢|+6𝑎₂|∇𝑢||Δ𝑢|+2𝑎₂|𝑢||∇𝑢|+𝑎₁|∇Δ𝑢|²𝑑𝑥≤𝐶_Ω|𝑢|^𝑝−3|∇𝑢|³+|∇𝑢|³+|𝑢|^𝑝−2||+||+|𝑢|∇𝑢||Δ𝑢∇𝑢||Δ𝑢_𝑝−1|∇Δ𝑢|+|𝑢||∇𝑢|+|∇Δ𝑢|²𝑑𝑥≤𝐶_Ω|𝑢|^2𝑝−6|∇𝑢|⁶+|∇𝑢|⁶+|𝑢|^2𝑝−4|∇𝑢|²|Δ𝑢|²+|∇𝑢|²|Δ𝑢|²+|𝑢|^2𝑝−2|∇Δ𝑢|²+|𝑢|²|∇𝑢|²+|∇Δ𝑢|²𝑑𝑥≤𝐶_Ωsup_𝑥∈Ω|𝑢|^2𝑝−6|∇𝑢|⁶+|∇𝑢|⁶+sup_𝑥∈Ω|𝑢|^2𝑝−4|∇𝑢|²|Δ𝑢|²+|∇𝑢|²|Δ𝑢|²+sup_𝑥∈Ω|𝑢|^2𝑝−2|∇Δ𝑢|²+|𝑢|²|∇𝑢|²+|∇Δ𝑢|²‖𝑢‖𝑑𝑥≤𝐶_𝐻^2𝑝−6_𝛼‖𝑢‖⁶_𝑊1,6+‖𝑢‖⁶_𝑊1,6+‖𝑢‖_𝐻^2𝑝−4_𝛼‖𝑢‖²_𝑊1,2‖𝑢‖²_𝑊2,2+‖𝑢‖⁴_𝑊2,4+‖𝑢‖_𝐻^2𝑝−2_𝛼‖𝑢‖²_𝑊3,2+‖𝑢‖²‖𝑢‖²_𝑊1,2+‖𝑢‖²_𝑊3,2≤𝐶‖𝑢‖_𝐻^2𝑝−6_𝛼‖𝑢‖⁶_𝐻𝛼+‖𝑢‖_𝐻^2𝑝−4_𝛼‖𝑢‖⁴_𝑊2,4+‖𝑢‖_𝐻^2𝑝−2_𝛼|𝑢|²_𝐻3+‖𝑢‖⁶_𝐻3+‖𝑢‖⁴_𝑊2,4+‖𝑢‖²_𝐻3≤𝐶‖𝑢‖_𝐻^2𝑝_𝛼+‖𝑢‖_𝐻^2𝑝−4_𝛼‖𝑢‖⁴_𝐻𝛼+‖𝑢‖_𝐻^2𝑝−2_𝛼‖𝑢‖²_𝐻𝛼+‖𝑢‖⁶_𝐻𝛼+‖𝑢‖⁴_𝐻𝛼+‖𝑢‖²_𝐻𝛼≤𝐶‖𝑢‖_𝐻^2𝑝_𝛼+‖𝑢‖⁶_𝐻𝛼+‖𝑢‖⁴_𝐻𝛼+‖𝑢‖²_𝐻𝛼, (3.33) which implies  that 𝑔∶𝐻_𝛼⟶𝐻_3/23isboundedfor𝛼≥2. (3.34)
Therefore, it follows from (3.25) and (3.34) that 𝑔(𝑢)_𝐻3/2<𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼,32≤𝛼<2. (3.35)
Then, we get from (3.35) that 𝑢𝑡,𝜑_𝐻𝛼=𝑒^𝑡𝐿𝜑+^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)𝑑𝜏_𝐻𝛼≤𝜑_𝐻𝛼+^𝑡₀ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻≤𝜑𝑑𝜏_𝐻𝛼+𝐶^𝑡₀ℒ^𝛼−(3/2)𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)_𝐻3/2≤𝜑𝑑𝜏_𝐻𝛼+𝐶^𝑡₀𝜏^−𝛽𝑒^−𝛿𝑡𝑑𝜏≤𝐶,∀𝑡≥0,𝜑∈𝑈⊂𝐻_𝛼, (3.36) where 𝛽=𝛼−3/2(0<𝛽<1). Hence, (3.31) holds for 2≤𝛼<5/2.
By doing the same procedures as Steps 1–4, we can prove that (3.31) holds for all 𝛼≥0.
Step 6. We show that for any 𝛼≥0, system (1.1) has a bounded absorbing set in 𝐻_𝛼. We first consider the case of 𝛼=1/2.
It is well known that the reaction-diffusion equation possesses a global attractor in 𝐻 space, and the global attractor of this equation consists of equilibria with their stable and unstable manifolds. Thus, each trajectory has to converge to a critical point. From (3.31) and (3.10), we deduce that for any 𝜑∈𝐻_1/2 the solution 𝑢(𝑡,𝜑) of system (1.1) converges to a critical point of 𝐹. Hence, we only need to prove the following two properties: (1)𝐹(𝑢)→∞⇔‖𝑢‖_𝐻1/2→∞; (2)the set 𝑆={𝑢∈𝐻_1/2∣𝐷𝐹(𝑢)=0} is bounded.
Property (1) is obviously true, we now prove property (2) in the following. It is easy to check if 𝐷𝐹(𝑢)=0, 𝑢 is a solution of the following equation: (𝑢)𝑢|𝑎Δ𝑢−𝑔=0,_𝜕Ω=0, (3.37) where 𝑔(𝑢) is given by (1.2). Taking the scalar product of (3.37) with 𝑢, then we derive that _Ω|∇𝑢|²+^2𝑚+1_𝑘=1𝑎_𝑘𝑢^𝑘+1𝑑𝑥=0. (3.38)
By (1.3) and (3.38), we have _Ω|∇𝑢|²+𝑎_2𝑚+1𝑢^2(𝑚+1)−^2𝑚_𝑘=1𝑎_𝑘|𝑢|^𝑘+1𝑑𝑥≤0. (3.39)
Using Hölder inequality and the above inequality, we have _Ω|∇𝑢|²+𝑢^2(𝑚+1)𝑑𝑥≤𝐶, (3.40) where 𝐶>0 is a constant. Thus, property (2) is proved.
Now, we show that system (1.1) has a bounded absorbing set in 𝐻_𝛼 for any 𝛼≥1/2, that is, for any bounded set 𝑈⊂𝐻_𝛼 there are 𝑇>0 and a constant 𝐶>0 independent of 𝜑 such that 𝑢𝑡,𝜑_𝐻𝛼≤𝐶,∀𝑡≥𝑇,𝜑∈𝑈. (3.41)
From the above discussion, we know that (3.41) holds as 𝛼=1/2. By (3.5), we have 𝑢𝑡,𝜑=𝑒^{(𝑡−𝑇)𝐿}𝑢𝑇,𝜑−^𝑡₀𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢)𝑑𝜏. (3.42)
Let 𝐵⊂𝐻_1/2 be the bounded absorbing set of system (1.1), and 𝑇₀>0 such that 𝑢𝑡,𝜑∈𝐵,∀𝑡≥𝑇₀,𝜑∈𝑈⊂𝐻_𝛼1𝛼≥2. (3.43)
It is well known that 𝑒^𝑡𝐿≤𝐶𝑒^−𝑡𝜆1, (3.44) where 𝜆₁>0 is the first eigenvalue of the equation: (𝑢)𝑢|𝑎Δ𝑢−𝑔=0,_𝜕Ω=0. (3.45) Hence, for any given 𝑇>0 and 𝜑∈𝑈⊂𝐻_𝛼(𝛼≥1/2), we have 𝑒^{(𝑡−𝜏)𝐿}𝑢𝑡,𝜑_𝐻𝛼=ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑢𝑡,𝜑_𝐻⟶0,as𝑡⟶∞. (3.46) From (3.42)-(3.43) and Lemma 2.4 for any 1/2≤𝛼<1, we get that 𝑢𝑡,𝜑_𝐻𝛼≤𝑒^{(𝑡−𝑇0)𝐿}𝑢𝑇₀,𝜑_𝐻𝛼+^𝑡_𝑇0ℒ^𝛼𝑒^{(𝑡−𝜏)𝐿}𝑔(𝑢≤𝑒)𝑑𝜏_{(𝑡−𝑇0)𝐿}𝑢𝑇₀,𝜑_𝐻𝛼+𝐶^𝑡−𝑇0₀𝜏^−𝛼𝑒^−𝜆1𝜏𝑑𝜏, (3.47) where 𝐶>0 is a constant independent of 𝜑.
Then, we infer from (3.46) and (3.47) that (3.41) holds for all 1/2≤𝛼<1. By the iteration method, we have that (3.41) holds for all 𝛼≥1/2.
Finally, this theorem follows from (3.31)–(3.41) and Lemma 2.3. The proof is completed.

Acknowledgment

This work is funded by the National Natural Science Foundation of China (no. 11071177) and the NSF of Sichuan Education Department of China (no. 11ZA1102).