Global Attractor for Doubly Nonlinear Parabolic Equation

Abstract

Our aim in this paper is to study the long-time behavior for a class of doubly nonlinear parabolic equations. First we show that the problem has a unique solution. Then we prove that the semigroup corresponding to the problem is norm-to-weak continuous in 𝐿^𝑞 and 𝐻¹⁰. Finally we establish the existence of global attractor of the problem in 𝐿^𝑞 and 𝐻¹⁰.

1. Introduction

We study the long-time behavior (in terms of attractors) of the solution of the following problem: 𝜕𝛼(𝑢)𝜕𝑡−Δ𝑢𝑢+(𝑥𝑓,(𝑡𝑢))|=_𝜕Ω𝑔(𝑥),𝑥∈Ω,𝑢(𝑥,0)=𝑢₀=0,(𝑥).(1.1) Here Ω is a bounded smooth domain in ℝ^𝑁(𝑁≥1) and 𝑔(𝑥) is a given function in 𝐿²(Ω).

Such equations appear, for example, in the study of gas filtration (so-called porous medium equation [1]). It has been extensively studied when 𝛼 is linear [1–5], and the existence of attractors has been proved in [6–11] (for 𝛼=𝐼).

Our aim in this paper is to extend the result of [1–4] to the more general equation (1.1). We make the following assumptions: 𝛼𝛼(𝑠)∈𝒞¹(ℝ),𝛼(0)=0,𝛼^(𝑠)≥𝛼₀,𝛼₀>0,𝑠∈ℝ,(1.2)₁|𝑠|^𝑟+2−𝛼𝛼₃≤𝛼(𝑠)𝑠≤𝛼₂|𝑠|^𝑟+2+𝛼₃,𝛼₁,𝛼₂>0,𝛼₃≥0,𝑟≥0,(1.3)_𝑓(𝑠)∈𝒞(ℝ),𝛾₁(𝑠)≤𝛼₄|𝑠|^𝑟+𝛼₅,𝛼₄>0,𝛼₅≥0,𝑠∈ℝ,(1.4)|𝑠|^𝑞−𝛾₃≤𝑓(𝑠)𝑠≤𝛾₂|𝑠|^𝑞+𝛾₃,𝑠∈ℝ,𝛾₁,𝛾₂>0,𝛾₃≥0,𝑞≥𝑟+2.(1.5) There exists a constant 𝐶₀≥0, such that 𝐶₀𝛼(𝑠)+𝑓(𝑠)isincreasing.(1.6)

By hypotheses (1.2)–(1.5), 𝛼 and 𝑓 are nonlinear functions with polynomial growth of arbitrary order. Here 𝛼 is more general than in [1–4](where 𝛼 is linear growth), which is an essential difficulty in proving the existence of global attractor. To problem (1.1), the key points are to obtain the continuous and compactness of semigroup. By using Legendre transform and the asymptotic a priori estimate method introduced in [10, 11], we show the existence of global attractor.

This paper is organized as follows. In Section 2, we recall some basic concepts about the global attractor. In Section 3, we show the uniqueness of solution and norm-to-weak continuous semigroup for (1.1). In Section 4, we verify the asymptotic compactness of the semigroup 𝑆(𝑡) in 𝐿^𝑞(Ω) and prove the existence of the (𝐿^𝑟+2(Ω),𝐿^𝑞(Ω))-global attractor under the hypotheses (1.2)–(1.6). Finally, in Section 5, we prove the existence of the (𝐿^𝑟+2(Ω),𝐻¹⁰(Ω))-global attractor for 𝑁≤2.

Throughout this paper we use the following notation: 𝐻=𝐿²(Ω), and the norms in 𝐻¹⁰(Ω) and 𝐿^𝑝(Ω)(1≤𝑝≤∞) are denoted by ‖𝑢‖²=∫_Ω|∇𝑢|²𝑑𝑥 and |𝑢|^𝑝𝑝=∫_Ω|𝑢|^𝑝𝑑𝑥, respectively; Ω(𝑢≥𝑀)={𝑥∈Ω∶𝑢(𝑥)≥𝑀} and Ω(𝑢≤−𝑀)={𝑥∈Ω∶𝑢(𝑥)≤−𝑀}; 𝑚(Ω) or |Ω| denotes Lebesgue measure of Ω; sometimes for special differentiation, we denote the different positive constants by 𝑐₁, 𝑐₂,….

2. Preliminaries

In this section, we recall some basic concepts about the global attractors.

Definition 2.1 (see [6–8]). Let {𝑆(𝑡)}_𝑡≥0 be a semigroup on a Banach space 𝑋. A subset 𝒜⊂𝑋 is called a global attractor for the semigroup if 𝒜 is compact in 𝑋 and enjoys the following properties:(1)𝒜 is an invariant set, that is, 𝑆(𝑡)𝒜=𝒜 for any 𝑡≥0;(2)𝒜 attracts all bounded sets of 𝑋, that is, for any bounded subset 𝐵 of 𝑋dist(𝑆(𝑡)𝐵,𝒜)⟶0,as𝑡⟶∞,(2.1)where dist(𝐴,𝐵) is the Hausdorff semidistance of the two sets 𝐴 and 𝐵: dist(𝐴,𝐵)=sup_𝑥∈𝐴inf_𝑦∈𝐵‖𝑥−𝑦‖_𝑋.(2.2)
And a subset 𝐵₀ of 𝑋 is called a bounded absorbing set of the semigroup {𝑆(𝑡)}_𝑡≥0 in 𝑋, if for any bounded 𝐵 of 𝑋, there exists some 𝑇_𝐵≥0, such that 𝑆(𝑡)𝐵⊂𝐵₀ for any 𝑡≥𝑡₀.

Definition 2.2 (see [11]). Let 𝑋 be a Banach space and let {𝑆(𝑡)}_𝑡≥0 be a family of operator in 𝑋. We say that {𝑆(𝑡)}_𝑡≥0 is norm-to-weak continuous semigroup in 𝑋, if {𝑆(𝑡)}_𝑡≥0 satisfies(1)𝑆(0)=𝐼 (the identity);(2)𝑆(𝑡)𝑆(𝑠)=𝑆(𝑡+𝑠),∀𝑡,𝑠≥0;(3)𝑆(𝑡_𝑛)𝑥_𝑛⇀𝑆(𝑥)𝑥 if 𝑡_𝑛→𝑡 and 𝑥_𝑛→𝑥 in 𝑋.

Definition 2.3 (see [11]). A set 𝒜⊂𝑋, which is invariant, closed in 𝑋, compact in 𝑍 and attracts the bounded subsets of 𝑋 in the topology of 𝑍, is called an (𝑋,𝑍)-global attractor.

Definition 2.4 (see [11]). Let {𝑆(𝑡)}_𝑡≥0 be a semigroup on Banach space 𝑋. {𝑆(𝑡)}_𝑡≥0 is called (𝑋,𝑍)-asymptotically compact, if for any bounded (in𝑋) sequence{𝑥_𝑛}^∞𝑛=1⊂𝑋 and 𝑡_𝑛≥0, 𝑡_𝑛→∞ as 𝑛→∞, {𝑆(𝑡_𝑛)𝑥_𝑛}^∞𝑛=1 has a convergence subsequence with respect to the topology of 𝑍.

Definition 2.5 (see [11]). Let {𝑆(𝑡)}_𝑡≥0 be a semigroup on Banach space 𝑋. A bounded subset 𝐵₀ of 𝑍 is called an (𝑋,𝑍)-bounded absorbing set, if for any bounded set 𝐵 of 𝑋, there exists some 𝑡_𝐵≥0, such that 𝑆(𝑡)𝐵⊂𝐵₀ for any 𝑡≥𝑡_𝐵.

Theorem 2.6 (see [11]). Let 𝑋 be a Banach space and let {𝑆(𝑡)}_𝑡≥0 be a norm-to-weak continuous semigroup on 𝑋. Then {𝑆(𝑡)}_𝑡≥0 has a global attractor in 𝑋 provided that the following conditions hold:(1){𝑆(𝑡)}_𝑡≥0 has a bounded absorbing set 𝐵₀ in 𝑋;(2){𝑆(𝑡)}_𝑡≥0 is asymptotically compact in 𝑋.

Theorem 2.7 (see [11]). Let {𝑆(𝑡)}_𝑡≥0 be a norm-to-weak continuous semigroup on 𝑋. One assumes that 𝑆(𝑡)𝑋⊂𝐿^𝑝(Ω). Then {𝑆(𝑡)}_𝑡≥0 has an (𝑋,𝐿^𝑝(Ω))-global attractor provided that the following conditions hold:(1){𝑆(𝑡)}_𝑡≥0 has an (𝑋,𝐿^𝑝(Ω))-bounded absorbing set 𝐵₀⊂𝐿^𝑝(Ω);(2)there is a 𝑞(1≤𝑞≤𝑝) such that {𝑆(𝑡)}_𝑡≥0 is (𝑋,𝐿^𝑞(Ω))-asymptotically compact;(3)for any 𝜀>0 and for any bounded 𝐵⊂𝑋, there exist constants 𝑀=𝑀(𝜀,𝐵) and 𝑇=𝑇(𝜀,𝐵), such that _{Ω(|𝑆(𝑡)𝑢0|≥𝑀)}||𝑆(𝑡)𝑢₀||^𝑝𝑑𝑥<𝜀forany𝑢₀∈𝐵and𝑡≥𝑇.(2.3)

Lemma 2.8 (see [11]). Let {𝑆(𝑡)}_𝑡≥0 be a semigroup on 𝐿^𝑝(Ω)(𝑝≥1) and suppose that {𝑆(𝑡)}_𝑡≥0 has a bounded absorbing set in 𝐿^𝑝(Ω). Then for any 𝜀>0 and any bounded subset 𝐵⊂𝐿^𝑝(Ω), there exist positive constants 𝑇=𝑇(𝐵) and 𝑀=𝑀(𝜀) such that 𝑚Ω||𝑆(𝑡)𝑢₀||≥𝑀≤𝐶𝜀forany𝑢₀∈𝐵and𝑡≥𝑇,(2.4) where the positive constant 𝐶 is independent of 𝐵, 𝑇, and 𝜀.

3. Uniqueness of Solution and Norm-to-Weak Continuous Semigroup

The existence of weak solution for (1.1) can be obtained by the standard Faedo-Galerkin approximation method. Here we only state the result.

Lemma 3.1. Assume that 𝑔(𝑥)∈𝐿²(Ω), 𝛼 and 𝑓 satisfying (1.3)–(1.5), 𝑢₀(𝑥)∈𝐿^𝑟+2(Ω). Then for any initial data 𝑢₀(𝑥)∈𝐿^𝑟+2(Ω), there exists solution 𝑢(𝑥,𝑡) for (1.1) which satisfies 𝑢(𝑥,𝑡)∈𝐶0,𝑇;𝐿²(Ω)∩𝐿²0,𝑇;𝐻¹⁰(Ω)∩𝐿^𝑞(0,𝑇;𝐿^𝑞(Ω)).(3.1)

We now show that the solution is uniqueness and continuous dependent on initial conditions.

Theorem 3.2. Assume that 𝑔(𝑥)∈𝐿²(Ω),𝑢₀(𝑥)∈𝐿^𝑟+2(Ω), 𝛼 and 𝑓 satisfying (1.3)–(1.6). Then there exists a unique solution of (1.1).

Proof. Suppose that 𝑢(𝑡),𝑣(𝑡) be two solutions of (1.1) with initial conditions 𝑢₀(𝑥), 𝑣₀(𝑥); then 𝜕(𝛼(𝑢)−𝛼(𝑣))𝜕𝑡−Δ(𝑢−𝑣)+𝑓(𝑢)−𝑓(𝑣)=0,(3.2) that is 𝜕(𝛼(𝑢)−𝛼(𝑣))𝜕𝑡−Δ(𝑢−𝑣)+𝐶⁰𝛼(𝑢)+𝑓(𝑢)−𝐶₀𝛼(𝑣)+𝑓(𝑣)=𝐶₀(𝛼(𝑢)−𝛼(𝑣)).(3.3) We define the sign function by sign(𝜏)=⎧⎪⎨⎪⎩10−1iiifff𝜏𝜏𝜏>=<000,,.(3.4) Multiplying (3.3) by sign(𝑢−𝑣) and integrating in Ω, we obtain 𝑑𝑑𝑡_Ω+||𝛼(𝑢)−𝛼(𝑣)||𝑑𝑥−_ΩΔ(𝑢−𝑣)sign(𝑢−𝑣)_Ω=𝐶₀𝐶₀𝛼(𝑢)+𝑓(𝑢)−𝐶₀𝛼(𝑣)+𝑓(𝑣)sign(𝛼(𝑢)−𝛼(𝑣))𝑑𝑥_Ω||𝛼(𝑢)−𝛼(𝑣)||𝑑𝑥.(3.5) Using  (1.6), we get _Ω𝐶₀𝛼(𝑢)+𝑓(𝑢)−𝐶₀𝛼(𝑣)+𝑓(𝑣)sign(𝛼(𝑢)−𝛼(𝑣))𝑑𝑥≥0.(3.6) Since sign(𝑢−𝑣)=lim_𝜀→0⁺(𝑢−𝑣)/(𝜀+|𝑢−𝑣|), by dominated convergence theorem, we have −_ΩΔ(𝑢−𝑣)sign(𝑢−𝑣)𝑑𝑥=−lim_𝜀→0+_ΩΔ(𝑢−𝑣)𝑢−𝑣=lim_𝜀→0+𝜀+|𝑢−𝑣|𝑑𝑥_Ω∇(𝑢−𝑣)∇𝑢−𝑣𝜀+|𝑢−𝑣|𝑑𝑥≥0.(3.7) So 𝑑𝑑𝑡_Ω||𝛼(𝑢)−𝛼(𝑣)||𝑑𝑥≤𝐶₀_Ω||𝛼(𝑢)−𝛼(𝑣)||𝑑𝑥.(3.8) By Gronwall inequality, we get _Ω||𝛼(𝑢(𝑡))−𝛼(𝑣(𝑡))||𝑑𝑥≤_Ω||𝛼𝑢₀−𝛼𝑣₀||𝑑𝑥.(3.9) From (1.2), we have _Ω||𝑢(𝑡)−𝑣(𝑡)||𝑑𝑥≤1𝛼₀_Ω||𝛼𝑢₀−𝛼𝑣₀||𝑑𝑥,(3.10) which gives continuous dependence on initial conditions and uniqueness of solution in 𝐿¹(Ω).
By Theorem 3.2, we can define the operator semigroup {𝑆(𝑡)}_𝑡≥0 in 𝐿¹(Ω) as the following: 𝑆(𝑡)𝑢₀∶ℝ⁺×𝐿^𝑟+2(Ω)⟶𝐿¹(Ω),(3.11) which is continuous in 𝐿¹(Ω).
Since 𝛼 is a continuous increasing function with 𝛼(0)=0, we define for 𝑡∈ℝ, 𝜓(𝑡)=^𝑡0𝛼(𝜏)𝑑𝜏.(3.12) Then the Legendre transform 𝜓^∗ is defined by 𝜓^∗(𝜏)=sup_𝑠∈ℝ{𝜏𝑠−𝜓(𝑠)}.(3.13) Note that 𝜓^∗(𝜏)≥0,𝜓^∗(𝛼(𝜏))+𝜓(𝜏)=𝜏𝛼(𝜏),𝜓^∗(𝛼(𝜏))≤𝜏𝛼(𝜏).(3.14)

Theorem 3.3. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). Then the semigroup {𝑆(𝑡)}_𝑡≥0 is norm-to-weak continuous in 𝐿^𝑞(Ω) and 𝐻¹⁰(Ω).

Proof. Let 𝑢_𝑚0(𝑥)→𝑢₀(𝑥) in 𝐿^𝑟+2(Ω), 𝑢_𝑚(𝑡),𝑢(𝑡) be the solutions of (1.1) corresponding to initial date 𝑢_𝑚0(𝑥), 𝑢₀(𝑥). In (1.1), replace 𝑢(𝑡) by 𝑢_𝑚(𝑡). Multiplying (1.1) by 𝑢_𝑚(𝑡) and integrating in Ω, we get 𝑑𝑑𝑡_Ω𝜓^∗𝛼𝑢_𝑚(𝑡)𝑑𝑥+||∇𝑢_𝑚||²²+𝑓𝑢_𝑚,𝑢_𝑚=𝑔,𝑢_𝑚.(3.15) Applying Young inequality, we have ||||_Ω𝑔(𝑥)𝑢_𝑚𝑑𝑥||||≤𝛾¹2||𝑢^𝑚||^𝑞𝑞+𝑐_Ω||𝑔(𝑥)||^𝑞′𝑑𝑥.1𝑞+𝑞1^=1.(3.16) So 𝑑𝑑𝑡_Ω𝜓^∗𝛼𝑢_𝑚(𝑡)𝑑𝑥+||∇𝑢_𝑚||²²+𝛾¹2||𝑢^𝑚||^𝑞𝑞≤𝛾₃||Ω||+𝑐_Ω||𝑔||^𝑞′𝑑𝑥.(3.17) Integrating from 0 to 𝑇, we obtain _Ω𝜓^∗𝛼𝑢_𝑚(𝑇)𝑑𝑥+^𝑇0||∇𝑢_𝑚||²²𝑑𝑡+𝛾¹2^𝑇0≤||𝑢_𝑚||^𝑞𝑞𝑑𝑡_Ω≤𝜓^∗𝛼𝑢_𝑚0𝑑𝑥+𝑐𝑇_Ω≤𝛼₂𝑢_𝑚0𝛼𝑢_𝑚0𝑑𝑥+𝛼₃||Ω||+𝑐𝑇||𝑢_𝑚0||^𝑟+2𝑟+2+𝑐𝑇+𝛼₃||Ω||.(3.18)𝑢_𝑚0→𝑢₀ in 𝐿^𝑟+2(Ω), so there existence 𝑀>0, such that |𝑢_𝑚0|^𝑟+2𝑟+2≤𝑀. {𝑢_𝑚(𝑡)} Is bounded in 𝐿²(0,𝑇;𝐻¹⁰(Ω)) and 𝐿^𝑞(0,𝑇;𝐿^𝑞(Ω)). Therefore, there exists weak convergent subsequence {𝑢_𝑚𝑘(𝑡)} of {𝑢_𝑚(𝑡)} in 𝐿²(0,𝑇;𝐻¹⁰(Ω)) and 𝐿^𝑞(0,𝑇;𝐿^𝑞(Ω). Let 𝑢_𝑚(𝑡)⇀𝑣(𝑡) obviously, 𝑣(𝑡) be a solution of (1.1) satisfying the initial value condition 𝑣(0)=𝑢₀(𝑥). By the unique of solution for (1.1), we have 𝑢(𝑡)=𝑣(𝑡), that is, 𝑢_𝑚𝑘⇀𝑢(𝑡) in 𝐿²(0,𝑇;𝐻¹⁰(Ω)) and 𝐿^𝑞(0,𝑇;𝐿^𝑞(Ω)). By Definition 2.2, Theorem 3.3 holds.

Remark 3.4. The semigroup {𝑆(𝑡)}_𝑡≥0 is norm-to-weak continuous in 𝐿²(Ω).

4. (𝐿^𝑟+2(Ω),𝐿^𝑞(Ω))-Global Attractor

By Theorem 3.3, we can define operator semigroup {𝑆(𝑡)}_𝑡≥0 as the following: 𝑆𝑆((𝑡𝑡))𝑢𝑢₀₀∶ℝ⁺×𝐿^𝑟+2(Ω)⟶𝐿^𝑞(Ω),∶ℝ⁺×𝐿^𝑟+2(Ω)⟶𝐻¹⁰(Ω).(4.1)

Theorem 4.1. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). Then the semigroup {𝑆(𝑡)}_𝑡≥0 exists bounded absorbing sets in 𝐿^𝑞(Ω) and 𝐻¹⁰(Ω); that is, for arbitrary bounded set 𝐵⊂𝐿^𝑟+2(Ω), there exist 𝑀>0, 𝑇(𝐵)>0, ∀𝑡≥𝑇, ∀𝑢₀(𝑥)∈𝐵. We have ||𝑢(𝑡)||^𝑞𝑞+||∇𝑢||²²≤𝑀.(4.2)

Proof. Let 𝑢(𝑡) be the solution of (1.1) with initial date 𝑢₀(𝑥); taking scalar product with 𝑢(𝑡) in (1.1), we deduce that 𝑑𝑑𝑡_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥+||∇𝑢||²²+(𝑓(𝑢),𝑢)=(𝑔,𝑢).(4.3) By (1.5), we obtain 𝑑𝑑𝑡_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥+||∇𝑢||²²+𝛾₁|𝑢|^𝑞𝑞−𝛾₃||Ω||≤(𝑔,𝑢).(4.4) Using Young inequality, we get ||(𝑔,𝑢)||≤𝛾¹2_Ω|𝑢|^𝑞𝑑𝑥+𝑐₁_Ω||𝑔||^𝑞𝑑𝑥.(4.5) By (4.4), we have 𝑑𝑑𝑡_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥+||∇𝑢||²²+𝛾¹2|𝑢|^𝑞𝑞≤𝛾₃||Ω||+𝑐₁_Ω||𝑔||^𝑞𝑑𝑥.(4.6) Since 𝑞≥𝑟+2, by (1.3), there exist 𝑐₂,𝑐₃>0 such that 𝛾₁4_Ω|𝑢|^𝑞𝑑𝑥≥𝑐₃_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥−𝑐₂.(4.7) Hence, 𝑑𝑑𝑡_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥+||∇𝑢||²²+𝛾¹4|𝑢|^𝑞𝑞+𝑐₃_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥≤𝑐₄.(4.8) We get 𝑑𝑑𝑡_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥+𝑐₃_Ω𝜓^∗(𝛼(𝑢))𝑑𝑥≤𝑐₄.(4.9) By the Gronwall lemma, we have _Ω𝜓^∗(𝛼(𝑢(𝑡)))≤𝑒^−𝑐3𝑡_Ω𝜓^∗𝛼𝑢₀𝑑𝑥+𝑐⁴𝑐₃.(4.10) Therefore, _Ω𝜓^∗𝛼𝑢₀𝑑𝑥≤_Ω𝑢₀𝛼𝑢₀𝑑𝑥.(4.11) We obtain _Ω𝜓^∗(𝛼(𝑢(𝑡)))≤𝛼₂||𝑢₀||^𝑟+2𝑟+2𝑒^−𝑐3𝑡+𝑐⁴𝑐₃+𝛼₃||Ω||≤𝑐₅.(4.12) It follows from (4.8) that there exists 𝑇₁>0,∀𝑡≥𝑇₁; we get ^𝑡𝑡+1||∇𝑢||²²𝑑𝑠+^𝑡𝑡+1|𝑢|^𝑞𝑞𝑑𝑠≤𝑐₆.(4.13) Multiplying (1.1) by 𝑢_𝑡 and integrating over Ω, we get _Ω𝛼^(𝑢)𝑢^2𝑡𝑑𝑥+𝑑𝑑𝑡_Ω||∇𝑢||²+𝐹(𝑢)𝑑𝑥=𝑔,𝑢_𝑡≤12_Ω𝛼^(𝑢)𝑢^2𝑡𝑑𝑥+12𝛼₀||𝑔(𝑥)||²²,(4.14) where 𝐹(𝑢)=∫^𝑢0𝑓(𝑠)𝑑𝑠. Now by (1.2) we get 𝑑𝑑𝑡_Ω||∇𝑢||²+𝐹(𝑢)𝑑𝑥≤12𝛼₀||𝑔(𝑥)||²².(4.15) It follows from (1.5) that there exist 𝛾^1,𝛾^2>0,𝛾^3≥0 such that 𝛾^1|𝑠|^𝑞−𝛾^3≤𝐹(𝑠)≤𝛾^2|𝑠|^𝑞+𝛾^3.(4.16) Using the uniform Gronwall Lemma that there exists 𝑇₂>0,∀𝑡≥𝑇₂, we have _Ω||∇𝑢||²+𝐹(𝑢)𝑑𝑥≤𝑐₇.(4.17) By (4.16), ∀𝑡≥𝑇=max{𝑇₁,𝑇₂}, we obtain ||∇𝑢||²²+|𝑢|^𝑞𝑞≤𝑐₈.(4.18) Therefor, the semigroup {𝑆(𝑡)} exists bounded absorbing set in 𝐻¹⁰(Ω) and 𝐿^𝑞(Ω); it follows from Theorem 2.6 that we have the following.

Theorem 4.2. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). Then the semigroup {𝑆(𝑡)}_𝑡≥0 has a (𝐿^𝑟+2(Ω),𝐿²(Ω))-global attractor, which is nonempty, compact, invariant in 𝐿²(Ω) and attracts every bounded subset of 𝐿^𝑟+2(Ω) with respect to 𝐿²(Ω) norm.

In the following, we will give the asymptotic a priori estimate of {𝑆(𝑡)}_𝑡≥0 with respect to 𝐿^𝑞(Ω)-norm, which plays a crucial role in the proof of the (𝐿^𝑟+2(Ω),𝐿^𝑞(Ω))-global attractor.

Lemma 4.3. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)∈𝐿²(Ω), 𝑞≥max{2𝑟,𝑟+2}, ∀𝐵⊂𝐿^𝑟+2(Ω). For any 𝜀>0, there exist positive constants 𝑀(=𝑀(𝜀)) and 𝑇(=𝑇(𝜀)) such that _{Ω(|𝑢(𝑡)|≥𝑀)}||𝑢(𝑡)||^𝑞𝑑𝑥<𝑐𝜀,forany𝑡≥𝑇and𝑢₀(𝑥)∈𝐵.(4.19)

Proof. By (1.3) and (1.5), we find that there exists 𝑀₁>0,∀|𝑢|≥𝑀₁ such that 𝑓(𝑢)𝑢≥𝛾¹2|𝑢|^𝑞,𝛼¹2|𝑢|^𝑟+1≤||𝛼(𝑢)||≤2𝛼₂|𝑢|^𝑟+1.(4.20)
Letting 𝑀₂=max{1,𝛼₁/(2)|𝑀₁|^𝑟+1}, when |𝛼(𝑢)|≥𝑀₂, then |𝑢|≥𝑀₁. Multiplying (1.1) with |(𝛼(𝑢)−𝑀₂)₊|^{(𝑞/(𝑟+1))−2}(𝛼(𝑢)−𝑀₂)₊, we get 𝑟+1𝑞𝑑𝑑𝑡_Ω||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥+_Ω∇𝑢∇||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−2}𝛼(𝑢)−𝑀₂₊+𝑑𝑥_Ω=𝑓(𝑢)||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−2}𝛼(𝑢)−𝑀₂₊𝑑𝑥_Ω𝑔(𝑥)||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−2}𝛼(𝑢)−𝑀₂₊𝑑𝑥,(4.21) where (𝛼(𝑢)−𝑀₂)₊ denote the positive part of (𝛼(𝑢)−𝑀₂), that is: 𝛼(𝑢)−𝑀₂₊=𝛼(𝑢)−𝑀²0,,𝛼𝛼((𝑢𝑢))≥<𝑀𝑀₂₂,.(4.22) Thus we have _Ω=∇𝑢∇||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−2}𝛼(𝑢)−𝑀₂₊𝑑𝑥_{Ω(𝛼(𝑢)≥𝑀)}=𝑞∇𝑢∇||𝛼(𝑢)−𝑀₂||^{𝑞/(𝑟+1)−1}𝑑𝑥𝑟+1−1_{Ω(𝛼(𝑢)≥𝑀)}𝛼^(𝑢)||𝛼(𝑢)−𝑀₂||^{𝑞/(𝑟+1)−2}||∇𝑢||²𝑑𝑥≥0,_Ω≥𝛾𝑓(𝑢)||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−2}𝛼(𝑢)−𝑀₂₊𝑑𝑥₁2_Ω≥𝑐_Ω|𝑢|^𝑞−1||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−1}𝑑𝑥≥𝑐_Ω||𝛼(𝑢)||^{(𝑞−1)/(𝑟+1)}||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−1}𝑑𝑥||||||𝛼(𝑢)−𝑀₂₊||^{(2𝑞−𝑟−2)/(𝑟+1)}𝑑𝑥,_Ω≤𝑔(𝑥)||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−2}𝛼(𝑢)−𝑀₂₊𝑑𝑥_Ω||||≤𝑐||𝑔(𝑥)||||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)−1}𝑑𝑥2_Ω||𝛼(𝑢)−𝑀₂₊||^{(2𝑞−𝑟−2)/(𝑟+1)}𝑑𝑥+𝑐_{Ω(𝛼(𝑢)≥𝑀2)}||𝑔(𝑥)||^{(2𝑞−𝑟−2)/(𝑞−1)}𝑑𝑥.(4.23) We obtain 𝑑𝑑𝑡_Ω≤𝑐||𝛼(𝑢)−_{Ω(𝛼(𝑢)≥𝑀2)}𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥+𝑐_Ω||𝛼(𝑢)−𝑀₂₊||^{(2𝑞−𝑟−2)/(𝑟+1)}𝑑𝑥||𝑔(𝑥)||^{(2𝑞−𝑟−2)/(𝑞−1)}𝑑𝑥.(4.24) Since 𝑞≥𝑟+2, we have 2𝑞−𝑟−2𝑟+1≥𝑟+𝑞1,1<2𝑞𝑞−−𝑟1−2≤2.(4.25) So 𝑑𝑑𝑡_Ω≤𝑐||𝛼(𝑢)−_{Ω(𝛼(𝑢)≥𝑀2)}𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥+𝑐_Ω||𝛼(𝑢)−𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥||𝑔(𝑥)||^{(2𝑞−𝑟−2)/(𝑞−1)}𝑑𝑥.(4.26) By Gronwall inequality, there exists 𝑇(𝐵)>0,∀𝑡≥𝑇(𝐵) such that _Ω||𝛼(𝑢(𝑡))−𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥≤𝑒^{−𝑐(𝑡−𝑇)}_Ω+𝑐||𝛼_{Ω(𝛼(𝑢)≥𝑀2)}(𝑢(𝑇))−𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥||𝑔(𝑥)||^{(2𝑞−𝑟−2)/(𝑞−1)}𝑑𝑥.(4.27)
For any 𝜀>0 there exists 𝛿>0; for any 𝑒⊂Ω, 𝑚(𝑒)<𝛿, we have _𝑒||𝑔(𝑥)||^{(2𝑞−𝑟−2)/(𝑞−1)}𝑑𝑥<𝜀.(4.28)
Leting 𝑇(𝑡)=𝛼(𝑢(𝑡)), obviously {𝑇(𝑡)}_𝑡≥0 is a semigroup in 𝐿¹(Ω). By (1.5) and (4.18), {𝑇(𝑡)}_𝑡≥0 has a bounded absorbing set in 𝐿¹(Ω). Combining Lemma 2.8, for any 𝜀>0, there exist 𝑀≥𝑀₂,𝑇₁≥𝑇(𝐵), for any 𝑡>𝑇₁, we have 𝑚(Ω(𝛼(𝑢)≥𝑀))<min{𝛿,𝜀}.(4.29) Hence _Ω||𝛼(𝑢(𝑡))−𝑀₂₊||^{𝑞/(𝑟+1)}𝑑𝑥<𝑐𝜀,(4.30) that is _{Ω(𝛼(𝑢)≥𝑀)}||𝛼(𝑢(𝑡))||−𝑀^{𝑞/(𝑟+1)}𝑑𝑥<𝑐𝜀.(4.31)
By repeating the same step above and multiplying (1.1) with |(𝛼(𝑢)+𝑀₂)₋|^{𝑞/(𝑟+1)−2}(𝛼(𝑢)+𝑀₂)₋, we get _{Ω(𝛼(𝑢)≤−𝑀)}||𝛼(𝑢(𝑡))||−𝑀^{𝑞/(𝑟+1)}𝑑𝑥<𝑐𝜀,(4.32) where 𝛼(𝑢)+𝑀₂₋=𝛼(𝑢)+𝑀²0,,𝛼𝛼((𝑢𝑢))≤>−−𝑀𝑀₂₂,.(4.33)
Combining (4.31) and (4.32), we have _{Ω(|𝛼(𝑢)|≥2𝑀)}=||𝛼(𝑢(𝑡))||^{𝑞/(𝑟+1)}𝑑𝑥_{Ω(|𝛼(𝑢)|≥2𝑀)}≤𝑐||𝛼_{Ω(|𝛼(𝑢)|≥2𝑀)}(𝑢(𝑡))||−𝑀+𝑀^{𝑞/(𝑟+1)}𝑑𝑥≤𝑐_{Ω(|𝛼(𝑢)|≥𝑀)}||𝛼(𝑢(𝑡))||−𝑀^{𝑞/(𝑟+1)}𝑑𝑥+_{Ω(|𝛼(𝑢)|≥2𝑀)}𝑀^{𝑞/(𝑟+1)}𝑑𝑥||𝛼(𝑢(𝑡))||−𝑀^{𝑞/(𝑟+1)}𝑑𝑥+_{Ω(|𝛼(𝑢)|≥𝑀)}||𝛼(𝑢(𝑡))||−𝑀^{𝑞/(𝑟+1)}𝑑𝑥≤𝑐𝜀.(4.34)
Thanks to (4.20), we have _{Ω(|𝑢(𝑡)|≥𝑀)}||𝑢(𝑡)||^𝑞𝑑𝑥<𝑐𝜀.(4.35)
From Theorem 4.2, we know that the semigroup {𝑆(𝑡)}_𝑡≥0 is asymptotically compact in 𝐿²(Ω); using Theorem 4.1, Lemma 4.3, and Theorem 2.7, we get the following.

Theorem 4.4. Assume that the conditions (1.2)–(1.6) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). Then the semigroup {𝑆(𝑡)}_𝑡≥0 has a (𝐿^𝑟+2(Ω),𝐿^𝑞(Ω))-global attractor, which is nonempty, compact, invariant in 𝐿^𝑞(Ω) and attracts every bounded subset of 𝐿^𝑟+2(Ω) with respect to 𝐿^𝑞(Ω) norm.

5. (𝐿^𝑟+2(Ω),𝐻¹⁰(Ω))-Global Attractor

In this section, we want to prove the (𝐿^𝑟+2(Ω),𝐻¹⁰(Ω))-global attractor. However, for general N-dimension space, our methods can not work, so we make the following assumption: 𝑁≤2,𝛼(𝑠)∈𝐶²(ℝ),𝑓(𝑠)∈𝐶¹(Ω),𝑠∈ℝ.(5.1)

Lemma 5.1. Assume that the conditions (1.2), (1.3), (1.5), (1.6), and (5.1) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). For any bounded set 𝐵⊂𝐿^𝑟+2(Ω), there exists 𝑡^∗(𝐵)>0; for any 𝑡≥𝑡^∗, we have _Ω𝛼^(𝑢)||𝑢_𝑡||²𝑑𝑥≤𝑐.(5.2)

Proof. Multiplying (1.1) by 𝑢_𝑡 and integrating over Ω, we have _Ω𝛼′(𝑢)||𝑢_𝑡||²𝑑𝑥+𝑑𝑑𝑡_Ω||∇𝑢||²+𝐹(𝑢)𝑑𝑥=𝑔(𝑥),𝑢_𝑡≤_Ω𝛼^(𝑢)2||𝑢^𝑡||²𝑑𝑥+||𝑔(𝑥)||²²2𝛼₀,(5.3) and then _Ω𝛼′(𝑢)||𝑢_𝑡||²+𝑐𝑑𝑑𝑡_Ω||∇𝑢||²+𝐹(𝑢)𝑑𝑥≤𝑐.(5.4) By (4.16) and (4.18), there exists 𝑇(𝐵)>0 for any 𝑡≥𝑇(𝐵); we get _Ω||∇𝑢||²+𝐹(𝑢)𝑑𝑥≤𝑐.(5.5) Integrating (5.4) from 𝑡 to 𝑡+1, we obtain ^𝑡𝑡+1_Ω𝛼^(𝑢)||𝑢_𝑡||²𝑑𝑥𝑑𝑠≤𝑐,𝑡≥𝑇(𝐵).(5.6) By differentiating (1.1) in time 𝑡, we have 𝛼^(𝑢)𝑢^2𝑡+𝛼^(𝑢)𝑢_𝑡𝑡−Δ𝑢_𝑡+𝑓^(𝑢)𝑢_𝑡=0.(5.7) Letting 𝑟(𝑡)=𝛼′(𝑢)𝑢^2𝑡, we get 𝑑𝑑𝑡_Ω𝑟(𝑡)𝑑𝑥=_Ω𝛼^(𝑢)𝑢^3𝑡𝑑𝑥+2_Ω𝛼^(𝑢)𝑢_𝑡𝑢_𝑡𝑡𝑑𝑥,(5.8) that is 12𝑑𝑑𝑡_Ω𝑟(𝑡)𝑑𝑥=12_Ω𝛼^(𝑢)𝑢^3𝑡𝑑𝑥+_Ω𝛼^(𝑢)𝑢_𝑡𝑢_𝑡𝑡𝑑𝑥.(5.9) Multiplying (5.7) by 𝑢_𝑡 and integrating over Ω, we get _Ω𝛼^(𝑢)𝑢^3𝑡𝑑𝑥+_Ω𝛼^(𝑢)𝑢_𝑡𝑡𝑢_𝑡𝑑𝑥=−_Ω||∇𝑢_𝑡||²𝑑𝑥−_Ω𝑓^(𝑢)𝑢^2𝑡𝑑𝑥.(5.10) Hence from (5.9), we obtain 12𝑑𝑑𝑡_Ω𝑟(𝑡)𝑑𝑥=−12_Ω𝛼^(𝑢)𝑢^3𝑡𝑑𝑥−_Ω||∇𝑢_𝑡||²𝑑𝑥−_Ω𝑓^(𝑢)𝑢^2𝑡𝑑𝑥.(5.11)
Since 𝑁≤2, from Theorem 4.1 the semigroup {𝑆(𝑡)}_𝑡≥0 has a bounded absorbing set in 𝐻¹⁰(Ω). Applying Sobolev embedding theorem, the semigroup {𝑆(𝑡)}_𝑡≥0 is bounded in 𝐿^∞(Ω); hence, 12𝑑𝑑𝑡_Ω𝑟(𝑡)𝑑𝑥+_Ω||∇𝑢_𝑡||²𝑑𝑥≤𝑀_Ω||𝑢_𝑡||³𝑑𝑥+c_Ω||𝑢_𝑡||²𝑑𝑥.(5.12)
Since 𝑁≤2, the imbedding 𝐻^1/3(Ω)↪𝐿³(Ω) is compact; hence ||𝑢_𝑡||³³≤𝑐||𝑢_𝑡||^3𝐻_1/3(Ω).(5.13) Using interpolation inequality for Sobolev space, we get ||𝑢_𝑡||^3𝐻_1/3(Ω)≤𝑐||∇𝑢_𝑡||₂||𝑢_𝑡||²²≤12||∇𝑢^𝑡||²²+𝑐||𝑢_𝑡||⁴².(5.14) Therefore, from (5.12), we have 12𝑑𝑑𝑡_Ω𝑟(𝑡)𝑑𝑥+_Ω||∇𝑢_𝑡||²𝑑𝑥≤𝑐||𝑢_𝑡||²²+𝑐||𝑢_𝑡||⁴²≤≤𝑐𝑐||𝑢_𝑡||⁴²_Ω+𝑐=𝑐_Ω||𝑢_𝑡||²𝑑𝑥²+𝑐𝛼^(𝑢)||𝑢_𝑡||²𝑑𝑥²+𝑐≤𝑐_Ω𝑟(𝑡)𝑑𝑥²+𝑐.(5.15) Thanks to uniform Gronwall inequality, there exists 𝑇(𝐵)>0 for any 𝑡≥𝑇(𝐵); we have _Ω𝑟(𝑡)𝑑𝑥≤𝑐.(5.16)
Now we prove the semigroup {𝑆(𝑡)}_𝑡≥0 is asymptotically compact in 𝐻¹⁰(Ω).

Lemma 5.2. Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). Then the semigroup {𝑆(𝑡)}_𝑡≥0 is asymptotically compact in 𝐻¹⁰(Ω).

Proof. Let 𝐵 is a bounded absorbing set in 𝐻¹⁰(Ω), 𝑥_𝑛∈𝐵, 𝑡_𝑛→∞ as 𝑛→∞, 𝑢_𝑛(𝑡)=𝑆(𝑡_𝑛)𝑥_𝑛. Now we will prove that there exists Cauchy sequence of {𝑢_𝑛(𝑡)} in 𝐻¹⁰(Ω). By Theorem 4.2, we know that {𝑆(𝑡)}_𝑡≥0 exists global attractor in 𝐿²(Ω), therefore there exists Cauchy sequence of {𝑢_𝑛(𝑡)} in 𝐿²(Ω), we denote by {𝑢_𝑛(𝑡)}. Hence, ∀0<𝜀<1,∃𝑁,∀𝑛,𝑚>𝑁: ||𝑢_𝑛−𝑢_𝑚||²²<𝜀².(5.17) In view of (1.1), we get 𝛼^𝑢_𝑚𝑢_𝑚𝑡−𝛼^𝑢_𝑛𝑢_𝑛𝑡−Δ𝑢_𝑚−𝑢_𝑛+𝑓𝑢_𝑚−𝑓𝑢_𝑛=0.(5.18) Multiplying the above equality by 𝑢_𝑚−𝑢_𝑛 and integrating over Ω, we obtain ||∇𝑢_𝑚−∇𝑢_𝑛||²²≤≤𝑐−_ΩΩ𝛼^𝑢_𝑚𝑢_𝑚𝑡−𝛼^𝑢_𝑛𝑢_𝑛𝑡𝑢_𝑚−𝑢_𝑛𝑑𝑥−_Ω𝑓𝑢_𝑚−𝑓𝑢_𝑛𝑢_𝑚−𝑢_𝑛𝑑𝑥≤𝑐||𝑢_𝑚𝑡||𝑢_𝑚𝑡||||𝑢_𝑚−𝑢_𝑛||𝑑𝑥+_Ω||𝑢_𝑛𝑡||||𝑢_𝑚−𝑢_𝑛||𝑑𝑥−_Ω𝑓^(𝜉)||𝑢_𝑚−𝑢_𝑛||²𝑑𝑥||₂||𝑢_𝑚−𝑢_𝑛||₂+||𝑢_𝑛𝑡||₂||𝑢_𝑚−𝑢_𝑛||₂+𝑐||𝑢_𝑚−𝑢_𝑛||²².(5.19) Hence, for any 𝑛,𝑚>𝑁, we get ||∇𝑢_𝑚−∇𝑢_𝑛||²²<𝑐𝜀.(5.20) That is to say, the semigroup {𝑆(𝑡)}_𝑡≥0 is asymptotically compact in 𝐻¹⁰(Ω).

Thus from Theorem 4.1 and Lemma 5.2, we have the following.

Theorem 5.3. Assume that the conditions (1.2), (1.3), (1.5), (1.6) and (5.1) are satisfied, 𝑔(𝑥)∈𝐿²(Ω). Then the semigroup {𝑆(𝑡)}_𝑡≥0 has a (𝐿^𝑟+2(Ω),𝐻¹⁰(Ω))-global attractor, which is nonempty, compact, invariant in 𝐻¹⁰(Ω) and attracts every bounded subset of 𝐿^𝑟+2(Ω) with respect to 𝐻¹⁰(Ω) norm.

Acknowledgment

This work is supported in part by the NSFC Grant (11161026).