Global Attractor for a Chemotaxis Model with Reaction Term

Abstract

A chemotaxis model with reproduction term in a bounded domain Ω⊂𝑅^𝑛 is discussed in this paper, where 𝑛=2,3. First, the existence of a global-in-time solution is given, and then a global attractor for this model is obtained.

1. Introduction

This paper deals with global attractor of a quasilinear parabolic system introduced in [1] by Horstmann and Winkler to model chemotaxis. Chemotaxis phenomenon is quite common in biosystem. The survival of many organisms, from microscopic bacteria to the largest mammals, depends on their ability to navigate in a complex environment through the detection, integration, and processing of a variety of internal and external signals. This movement is crucial for many aspects of behavior, including the location of food sources, avoidance of predators, and attracting mates. The ability to migrate in response to external signals is shared by many cell populations. The directed movement of cells and organisms in response to chemical gradients is called chemotaxis.

The classical chemotaxis model has been extensively studied in the last few years (see [1–5]); Payne and Straughan [4] tackle precise nonlinear decay for classical model. Decay for a nonlinear chemotaxis system modeling glia cell movement in the brain is treated by Quinlan and Straughan [5].

In this paper, we are concerned with the following chemotaxis model: 𝜕𝑢()𝜕𝑡=▵𝑢−∇⋅𝑢+1^𝛼∇𝑔(𝑤)+𝑓(𝑢,𝑤),𝑥∈Ω,𝑡>0,𝜕𝑤𝜕𝑡=▵𝑤+𝑢−𝑤,𝑥∈Ω,𝑡>0,𝜕𝑢=𝜕𝑛𝜕𝑤𝑢()𝜕𝑛=0,𝑥∈𝜕Ω,𝑥,0=𝑢₀(𝑥),𝑤(𝑥,0)=𝑤₀(𝑥),𝑥∈Ω, (1) where 𝑢(𝑥,𝑡) represents the density of a biological species, which could be a cell, a germ, or an insect, while 𝑤(𝑥,𝑡) denotes the concentration of a chemical substance on position 𝑥∈Ω and at time 𝑡∈[0,∞). 𝛼 is a positive constant. Ω is an open bounded subset of 𝑅^𝑛 with the boundary of class 𝐶³. 𝑔(𝑠), 𝑓(𝑠) are two real smooth functions for 𝑠∈[0,+∞) satisfying the following conditions: (i)𝑓(0,𝑤)=0;(ii)there exists 𝑀>0 such that 𝑓(𝑢,𝑤)≤0 as 𝑢>𝑀; (iii)there exists 𝜆>0 such that |𝑔^′(𝑠)|<𝜆 for all 𝑠∈[0,+∞).𝑔^′(𝑤)(𝑢+1)^𝛼 denotes a chemotactic sensitivity function. (𝑢+1)^𝛼 can be replaced by (𝑢+𝜀)^𝛼, where 𝜀 is an arbitrary positive constant. For convenience, we choose 𝜀=1. 𝑓(𝑢,𝑤) describes the intrinsic rate of growth of cells population. For 𝛼=1,𝜀=0,and𝑔(𝑤)=𝑤, system (1) equals the most common formulation of the Keller-Segel model.

To our knowledge it has never been analyzed whether the global attractor of system (1) exists if the positive power 𝛼≠1; this system is more motivated from the mathematical point of view than from the biological one, but it will help to get more insights in the understanding of the behavior of the problem. We will look at this aspect of the critical exponent 𝛼 which decides whether global attractor can exist or not.

The existence of the global attractor for semilinear reaction diffusion equations in bounded and unbounded domains has been studied extensively [6, 7]. However, chemotaxis model is always a strongly coupled quasilinear parabolic system. There are a few articles to discuss the global attractor of such a system. We construct a local solution to (1) by the semigroup method and then discuss its regularity by a priori estimate method we set up for a strongly coupled quasilinear parabolic system.

For readers' convenience, the following standard result on attractor is first presented here (see [3, 6–9]).

Proposition 1. Suppose that 𝑋 is a metric space and 𝑆(𝑡)_𝑡≥0 is a semigroup of continuous operators in 𝑋. If 𝑆(𝑡)_𝑡≥0 has a bounded absorbing set and is asymptotically compact, then 𝑆(𝑡)_𝑡≥0 possesses a global attractor which is a compact invariant set and attracts every bounded set in 𝑋.

Definition 2. The semigroup 𝑆(𝑡) is asymptotically compact; that is, if 𝑢_𝑛 is bounded in 𝑋 and 𝑡_𝑛→∞, then 𝑆(𝑡_𝑛)𝑢_𝑛 is precompact in 𝑋.
We first show that there is a unique global solution to (1) for any nonnegative initial functions 𝑢₀(𝑥)∈𝐿²(Ω), 𝑤₀(𝑥)∈𝑊^1,2(Ω). And then we define a continuous semigroup 𝑆(𝑡)_𝑡≥0 in 𝐿²(Ω)×𝑊^1,2(Ω). Next, we find a bounded absorbing set and prove that the semigroup 𝑆(𝑡) is asymptotically compact. Thus, from the standard result, it is shown that there is a global attractor to (1) in 𝐿²(Ω)×𝑊^1,2(Ω).

2. Preliminary

Some well-known inequalities and embedding results that will be used in the sequel are presented.

Lemma 3 (see [10]). If 𝑝,𝑞≥1 and 𝑝(𝑛−𝑞)<𝑛𝑞, then, for 𝑟∈(0,𝑝), ‖𝑢‖_𝐿^𝑝_(Ω)≤𝑐‖𝑢‖^𝑎_𝑊1,𝑞⋅‖𝑢‖_𝐿^(1−𝑎)_𝑟(Ω)∀𝑢∈𝑊^1,𝑞(Ω), (2) where 𝑎=(𝑛/𝑟−𝑛/𝑝)/(1−𝑛/𝑞+𝑛/𝑟)∈(0,1).

Lemma 4 (see [10]). Let 1≤𝑞≤𝑝≤∞ and 𝑓∈𝐿^𝑞(Ω). Then 𝑒^𝑡▵𝑓_𝑝≤(4𝜋𝑡)^{(−𝑛/2)(1/𝑞−1/𝑝)}𝑓_𝑞,∇𝑒^𝑡▵𝑓_𝑝≤𝑐𝑡^{(−𝑛/2)(1/𝑞−1/𝑝)−1/2}𝑓_𝑞,(−▵+1)^𝛽𝑒^𝑡▵𝑓_𝑝≤𝑐𝑡^{−𝛽(−𝑛/2)(1/𝑞−1/𝑝)}𝑒^{(1−𝜇)𝑡}𝑓_𝑞𝑝≠𝑞, (3) where 𝛽>0, 𝜇>0, and 𝑐 is a positive constant depending only on 𝑝,𝑞,Ω.
(𝑒^𝑡▵𝑓)(𝑥)=∫_Ω𝐺(𝑥−𝑦,𝑡)𝑓(𝑦)𝑑𝑦 and 𝐺(𝑥,𝑡) is the Green function of the heat equation 𝜕𝑢/𝜕𝑡−▵𝑢=0,𝑥∈Ω,𝑡>0 with the homogeneous Neumann boundary condition 𝜕𝑢/𝜕𝑛=0.

Lemma 5 (see [11]). Let 𝐴_𝑝=−▵ and 𝐷(𝐴_𝑝)={𝜑∈𝑊^2,𝑝_(Ω)|𝜕𝜑/𝜕𝑁|_𝜕Ω=0}. Then 𝐷𝐴_𝑝+1^𝛽↪𝑊^1,𝑝(Ω)1,𝑖𝑓𝛽>2,𝐷𝐴_𝑝+1^𝛽↪𝐶^𝛿(Ω)𝑛,𝑖𝑓2𝛽−𝑝)>𝛿≥0,(𝐴+1^𝛽𝑒^{−𝑡(𝐴+1)}𝑢_𝐿𝑝(Ω)≤𝑐𝑡^−𝛽‖𝑢‖_𝐿^𝑝_(Ω). (4)

Lemma 6. Let 𝑟 and 𝑠 be nonnegative real numbers satisfying 𝑟+𝑠<2. Then for any 𝜀>0 there exists a constant 𝑐_𝜀>0 such that 𝑎^𝑟𝑏^𝑠≤𝜀𝑎²+𝑏²+𝑐_𝜀∀𝑎,𝑏>0. (5)

Lemma 7 (see [2]). Suppose that 𝐻^𝑠 is an interpolation space of 𝐻^𝑠0 and 𝐻^𝑠1, where 0≤𝑠₀<𝑠₁<∞, 0<𝜃<1, 𝑠=(1−𝜃)𝑠₀+𝜃𝑠₁, and then ‖⋅‖_𝐻^𝑠≤𝑐‖⋅‖_𝐻^1−𝜃𝑠₀‖⋅‖^𝜃_𝐻^𝑠₁. (6) Here 𝐻^𝑠(Ω), 𝑠>0, denotes the fractional Sobolev space in Ω.

Lemma 8 (see [1]). Let 𝛽>0, 𝑝∈(1,+∞), for all 𝜀>0; there exists 𝑐(𝜀)>0 for any 𝑤∈𝐿^𝑝(Ω), such that (𝐴+1)^𝛽𝑒^−𝑡𝐴∇⋅𝑤_𝐿𝑝(Ω)≤𝑐(𝜀)𝑡^{−𝛽−(1/2)−𝜀}‖𝑤‖_𝐿^𝑝_(Ω). (7)

3. Local Existence and Uniqueness

The local existence of a solution to system (1) is discussed in this section. First, an estimate to 𝑤 in (1) is shown.

Lemma 9. Assume that there exist 1≤𝑞≤𝑝≤∞, (1/𝑞)−(1/𝑝)<1/𝑛, and 𝑢∈𝐿^∞([0,∞);𝐿^𝑞(Ω)). Then for any 𝜏>0‖𝑤(𝑡)‖_𝑊^1,𝑝≤𝜏^−𝛼𝑤_0𝐿1+𝑐Γ𝛾sup_{𝜏<𝑠<𝑡}‖𝑢(𝑠)‖_𝑞∀𝑡≥𝜏, (8) where 𝛼>0, 𝛾=1−𝛽−(𝑛/2)((1/𝑞)−(1/𝑝)), Γ(⋅) is the Gamma function, and 𝑤(𝑥,𝑡)=𝑒^{𝑡(▵−1)}𝑤₀+^𝑡₀𝑒^{(𝑡−𝑠)(▵−1)}𝑢(𝑠)𝑑𝑠. (9)
The proof of the lemma can be found in another paper [3].

Theorem 10. Suppose 0≤𝑢₀(𝑥)∈𝐿²(Ω), 0≤𝑤₀(𝑥)∈𝑊^1,2(Ω), and 𝛼<1/𝑛. Then there is a 𝑇≤∞ (depending on ‖𝑢₀‖_𝐿2(Ω),‖𝑤₀‖_𝑊1,2(Ω)) such that there exists a unique nonnegative solution (𝑢(𝑥,𝑡),𝑤(𝑥,𝑡)) to (1) in [0,𝑇] and 𝑢∈𝐶⁰[0,𝑇];𝐿²(Ω)𝐶^2,1(Ω;(0,𝑇]),𝑤∈𝐶⁰[0,𝑇];𝑊^1,2(Ω)𝐶^2,1((𝐶Ω;0,𝑇])₀(0,𝑇];𝐶³(Ω). (10)

Proof. Choose 𝑇∈(0,1) and 𝑅>0 to be fixed. In Banach space 𝑋=𝐶⁰([0,𝑇];𝐿²(Ω))×𝐶⁰([0,𝑇];𝑊^1,2(Ω)), we define a bounded closed set 𝑆∶=(𝑢,𝑤)∈𝑋∣‖𝑢,𝑤‖_𝑋≤𝑅. (11) Let 𝜓(𝑡)=𝜓𝑢,𝑤)(₁(𝑡)𝜓𝑢,𝑤)(₂(𝑡)=𝑒𝑢,𝑤)(_𝑡▵𝑢₀−∫^𝑡₀𝑒^{(𝑡−𝑠)▵}∇𝑢^𝛼(𝑤)(𝑢𝑒∇𝑔+𝑓)𝑑𝑠_{𝑡(▵−1)}𝑤₀+∫^𝑡₀𝑒^{(𝑡−𝑠)(▵−1)}𝑢(𝑠),𝑑𝑠 (12) where 𝑓(𝑢)=𝑢(𝑎−𝑏𝑢).
Next, we prove that 𝜓 is a contractive mapping from 𝑆 into itself for 𝑇 small enough and 𝑅 sufficiently large. By Lemma 4, then 𝜓₁(𝑢,𝑤)_𝐿2≤𝑒^−𝑡𝐴𝑢₀(𝑥)_𝐿2+𝑐^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}∇𝑢^𝛼∇𝑔(𝑤)_𝐿2+𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑢(𝑠))_𝐿2≤𝑢𝑑𝑠₀(𝑥)_𝐿2+𝑐^𝑡₀(𝑡−𝑠)^{(−𝑛/2)((1/𝑞)−(1/2))−(1/2)}𝑢^𝛼∇𝑔(𝑤)_𝐿𝑞+𝑑𝑠_𝑡0(𝑡−𝑠)^{(−𝑛/2)(1−(1/2))}𝑓(𝑢(𝑠))_𝐿1𝑑𝑠, (13) where 𝐴=−▵. Since 𝛼<1/𝑛, there is a 𝑞∈[2𝑛/(2+𝑛),2), such that 2𝑞𝛼/(2−𝑞)≤2 and 1/2−(𝑛/2)((1/𝑞)−(1/2))>0. By Hölder’s inequality 𝑢^𝛼∇𝑔(𝑤)_𝐿𝑞≤𝜆𝑢^𝛼∇𝑤_𝐿𝑞≤𝑐(𝜆)‖𝑢‖^𝛼_{𝐿2𝑞𝛼/(2−𝑞)}‖∇𝑤‖_𝐿²≤𝑐(𝜆)‖𝑢‖^𝛼_𝐿2‖∇𝑤‖_𝐿²≤𝑐(𝜆)𝑅², (14) then 𝑐^𝑡₀(𝑡−𝑠)^{(−𝑛/2)((1/𝑞)−(1/2))−(1/2)}𝑢^𝛼∇𝑔(𝑤)_𝐿𝑞(𝜆)𝑅𝑑𝑠≤𝑐²𝑇^{(1/2)−(𝑛/2)((1/𝑞)−(1/2))}. (15)
Similarly, for 𝑛≤3, there is ∫^𝑡₀(𝑡−𝑠)^{(−𝑛/2)(1−(1/2))}‖𝑓(𝑢(𝑠))‖_𝐿1𝑑𝑠≤𝑅𝑇^1−(𝑛/4). Hence 𝜓₁(𝑢,𝑤)_𝐿2≤𝑢₀(𝑥)_𝐿2+𝑐𝑅²𝑇^{1/2−(𝑛/2)((1/𝑞)−(1/2))}+𝑅𝑇^1−(𝑛/4). (16)
By Lemma 5, for any 𝑡∈[0,𝑇), there is a 𝛾∈(1/2,1) such that 𝜓₂(𝑢,𝑤)_𝑊1,2≤𝑒^{−𝑡(𝐴+1)}𝑤₀(𝑥)_𝑊1,2+^𝑡₀𝑒^{−(𝑡−𝑠)(𝐴+1)}𝑢(𝑠)_𝑊1,2≤𝑤𝑑𝑠₀(𝑥)_𝑊1,2+𝑐^𝑡₀(𝐴+1)^𝛾𝑒^{−(𝑡−𝑠)(𝐴+1)}𝑢(𝑠)_𝐿2≤𝑤𝑑𝑠₀(𝑥)_𝑊1,2+𝑐^𝑡₀(𝑡−𝑠)^−𝛾‖𝑢(𝑠)‖_𝐿²≤𝑤𝑑𝑠₀(𝑥)_𝑊1,2+𝑐𝑅𝑇^1−𝛾. (17)
Equations (16) and (17) imply that 𝜓𝑆⊂𝑆 for any fixed positive 𝑅 large enough and 𝑇 small enough. Now we show that 𝜓 is a contractive operator from 𝑆 to 𝑆. For for all (𝑢,𝑤),(𝑢,𝑤)∈𝑆, 𝜓₁(𝑢,𝑤)−𝜓₁(𝑢,𝑤)_𝐿2≤𝑐^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}∇𝑢^𝛼∇𝑔(𝑤)−∇𝑢^𝛼∇𝑔(𝑤)_𝐿2+𝑑𝑠_𝑡0𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑢)−𝑓(𝑢)_𝐿2𝑑𝑠≤𝑐^𝑡₀(𝑡−𝑠)^{(−𝑛/2)((1/𝑞)−(1/2))−(1/2)}𝑢^𝛼∇𝑤−𝑢^𝛼∇𝑤_𝐿𝑞+𝑑𝑠_𝑡0(𝑡−𝑠)^−𝑛/4𝑓(𝑢)−𝑓(𝑢)_𝐿1𝑑𝑠≤𝑐𝑅²𝑇^{1/2−(𝑛/2)((1/𝑞)−(1/2))}+𝑇^1−𝑛/4‖(𝑢,𝑤)−(𝑢,𝑤)‖_𝑋[),𝜓∀𝑡∈0,𝑇₂(𝑢,𝑤)−𝜓₂(𝑢,𝑤)_𝑊1,2≤𝑐^𝑡₀(𝐴+1)^𝛾𝑒^{−(𝑡−𝑠)(𝐴+1)}(𝑢(𝑠)−𝑢(𝑠))_𝐿2𝑑𝑠≤𝑐^𝑡₀(𝑡−𝑠)^−𝛾‖𝑢(𝑠)−𝑢(𝑠)‖_𝐿²𝑑𝑠≤𝑐𝑇^1−𝛾‖(𝑢,𝑤)−(𝑢,𝑤)‖_𝑋∀𝑡∈[0,𝑇). (18)
Equation (18) implies that 𝜓 is a contractive mapping if 𝑇 is sufficiently small. By Banach's fixed point theorem, there exists a unique fixed point (𝑢,𝑤)∈𝑋 which is just a local solution to (1) in 𝑋.
Since 𝑢₀(𝑥)≥0, 𝑤₀(𝑥)≥0, then, for any given smooth function 𝑤, 𝑢=0 is the subsolution of the following problem: 𝜕𝑢()𝜕𝑡−▵𝑢+∇⋅𝑢+1^𝛼∇𝑔(𝑤)−𝑢(𝑎−𝑏𝑢)=0,𝑥∈Ω,𝑡>0,𝜕𝑢𝑢()𝜕𝑛=0,𝑥∈𝜕Ω,𝑥,0=𝑢₀(𝑥),𝑥∈Ω (19) and, for any given smooth function 𝑢, 𝑤=0 is the subsolution of the following problem: 𝜕𝑤𝜕𝑡−▵𝑤+𝑤=𝑢,𝑥∈Ω,𝑡>0,𝜕𝑤𝑤()𝜕𝑛=0,𝑥∈𝜕Ω,𝑥,0=𝑤₀(𝑥),𝑥∈Ω. (20) By the comparison principle, for any 𝑡∈[0,𝑇_max) and 𝑥∈Ω, 𝑢(𝑥,𝑡)≥0; 𝑤(𝑥,𝑡)≥0.
Now, we discuss the regularity of the solution to (1). From the previous analysis, 𝑢(𝑡,𝑥) is bounded in 𝐿²(Ω) for any 𝑡∈[0,𝑇]. Then by Lemma 9, for any 𝜏∈(0,𝑇], we deduce that 𝑤(𝑡)∈𝑊^1,𝑝(Ω), for all 𝑡∈[𝜏/2,𝑇], where 𝑛<𝑝<2𝑛/(𝑛−2) and 𝑛=2,3. From the semigroup representation of the solution to (1) and Lemmas 3 and 4, for any 𝑞∈(2,𝑛/(𝑛−2)), 𝑡∈[𝜏,𝑇], ‖𝑢(𝑥,𝑡)‖_𝐿^𝑞≤𝑒^{−(𝑡−𝜏/2)𝐴}𝑢_𝜏/2(𝑥)_𝐿𝑞+𝑐^𝑡_𝜏/2𝑒^{−(𝑡−𝑠)𝐴}∇(𝑢+1)^𝛼∇𝑔(𝑤)_𝐿𝑞+𝑑𝑠_𝑡𝜏/2𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑢(𝑠))_𝐿𝑞≤𝜏𝑑𝑠𝑡−2^{(−𝑛/2)((1/2)−(1/𝑞))}𝑢_𝜏/2(𝑥)_𝐿2+𝑐𝜆^𝑡_𝜏/2(𝑡−𝑠)^{(−𝑛/2)((1/2)−(1/𝑞))}(𝑢+1)^𝛼∇𝑤_𝐿2+𝑑𝑠_𝑡𝜏/2(𝑡−𝑠)^{(−𝑛/2)(1−(1/𝑞))}𝑓(𝑢(𝑠))_𝐿1𝑑𝑠. (21) Since 𝛼<1/𝑛, then there exists 𝑝₁∈(𝑛/2,1/𝛼] such that 2𝛼𝑝₁≤2, 2𝑝^′₁<2𝑛/(𝑛−2) and (𝑢+1)^𝛼∇𝑤_𝐿2≤𝑐‖𝑢‖^𝛼_{𝐿12𝛼𝑝}‖∇𝑤‖_{𝐿′12𝑝}≤𝑐‖𝑢‖^𝛼_𝐿2‖∇𝑤‖_{𝐿′12𝑝}. (22) Here 1/𝑝₁+1/𝑝^′₁=1. Then ‖𝑢(𝑥,𝑡)‖_𝐿^𝑞≤𝑐𝜏^{(−𝑛/2)((1/2)−(1/𝑞))}𝑢₀(𝑥)_𝐿2+𝑐sup_{𝜏/2<𝑠<𝑡}‖𝑢‖_𝐿²‖∇𝑤‖_{𝐿′12𝑝}^𝑡_𝜏/2(𝑡−𝑠)^{(−𝑛/2)((1/2)−(1/𝑞))−(1/2)}𝑑𝑠+𝑐^𝑡_𝜏/2(𝑡−𝑠)^{(−𝑛/2)(1−1/𝑞)}𝑑𝑠𝑡∈[𝜏,𝑇] (23) which implies that, for any 𝑞∈(2,𝑛/(𝑛−2)), ‖𝑢(𝑥,𝑡)‖_𝐿^𝑞 is bounded for all 𝑡∈[𝜏,𝑇]. By using Lemma 9 again and repeating the above process, it can be proved that 𝑢∈𝐶⁰(𝜏,𝑇];𝐿^∞(Ω),𝑤∈𝐶⁰(𝜏,𝑇];𝐶⁰(Ω). (24) There exist 𝑝, 𝛽, and 𝜀 satisfying 𝑝>𝑛, 1/2>𝛽>𝑛/2𝑝, and 𝜀<1/2−𝛽 such that, for any small enough constant 𝜂>0, ‖𝑢(𝑥,𝑡)‖_𝐶⁰≤𝑒^{−(𝑡−𝜏)𝐴}𝑢_𝜏(𝑥)_𝐶0+𝑐^𝑡_𝜏𝑒^{−(𝑡−𝑠)𝐴}∇(𝑢+1)^𝛼∇𝑔(𝑤)_𝐶0+𝑑𝑠_𝑡𝜏𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑢(𝑠))_𝐶0≤)𝑑𝑠(𝐴+1^𝛽𝑒^{−(𝑡−𝜏)𝐴}𝑢_𝜏(𝑥)_𝐿𝑝+𝑐𝜆^𝑡_𝜏(𝐴+1)^𝛽𝑒^{−(𝑡−𝑠)𝐴}∇(𝑢+1)^𝛼∇𝑤_𝐿𝑝+𝑑𝑠_𝑡𝜏(𝐴+1)^𝛽𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑢(𝑠))_𝐿𝑝𝑑𝑠≤𝜂^{(−𝑛/2)((1/2)−(1/𝑝))}𝑒^{(1−𝜇)𝑡}𝑢_𝜏(𝑥)_𝐿2+𝑐sup_{𝜏<𝑠<𝑡}(𝑢+1)^𝛼∇𝑤_𝐿𝑝^𝑡_𝜏(𝑡−𝑠)^{−𝛽−1/2−𝜀}𝑑𝑠+𝑐^𝑡_𝜏(𝑡−𝑠)^{−𝛽−𝜀1}𝑑𝑠𝑡∈𝜏+𝜂,𝑇, (25) where 𝜀₁=(𝑛/2)((1/𝑞)−(1/𝑝))≤1−𝛽, which implies that 𝑢∈𝐶⁰((0,𝑇];𝐶⁰(Ω)). In conclusion, we see that 𝑢(𝑥,𝑡)∈𝐶[0,𝑇];𝐿²(Ω)𝐶^2,1((,𝑤()[]Ω;0,𝑇])𝑥,𝑡∈𝐶0,𝑇;𝑊^1,2(Ω)𝐶^2,1((𝐶(]Ω;0,𝑇])0,𝑇;𝐶³(Ω). (26)
By semigroup techniques and Schauder estimates (Theorem IV. 5.1–5.3 in [7] and Lemmas 3.2–3.3 in [10]), we have that 𝑢∈𝐶⁰[0,𝑇];𝐿²(Ω)𝐶⁰(0,𝑇];𝐻²_𝑁(Ω),𝑤∈𝐶⁰[0,𝑇];𝑊^1,2(Ω)𝐶⁰(0,𝑇];𝐻³_𝑁(Ω). (27) Here we denote that 𝐻^𝑠_𝑁={𝑢∈𝐻^𝑠(Ω);𝜕𝑢/𝜕𝑛=0,on𝜕Ω}. The proof of Theorem 10 is completed.

4. Global Solution and Some A Priori Estimates

In this section, the global-in-time existence of a solution to system (1) is proved. The following a priori estimates will play a crucial role in the proof of our result.

Lemma 11. Suppose that 0≤𝑢₀∈𝐿²(Ω), (𝑢,𝑣) is a solution to (1), and then there exists a positive constant 𝑐₀ such that ‖𝑢(𝑡)‖₁≤𝑐₀,∀𝑡>0. (28)

Proof. Integrating the first equation of (1) on Ω, we obtain 𝑑‖𝑢(𝑡)‖₁=𝑑𝑡_Ω𝑎𝑢−𝑏𝑢²𝑑𝑥=𝑎‖𝑢‖₁−𝑏_Ω𝑢²‖𝑢‖𝑑𝑥≤𝑎₁−𝑏|Ω|‖𝑢‖²₁. (29) From the above equation, it is easy to know that 𝑑‖𝑢(𝑡)‖₁/𝑑𝑡<0 if ‖𝑢(𝑡)‖₁≥𝑎|Ω|/𝑏. Then there exists a constant 𝑐₀ (depending on 𝑎, 𝑏, and Ω) such that ‖𝑢(𝑡)‖₁≤𝑐₀.

Lemma 12. Suppose that 0≤𝑢₀∈𝐿²(Ω), 0≤𝑤₀∈𝑊^1,2(Ω), and 𝛼<1/𝑛; (𝑢,𝑤) is a local solution to (1) in [0,𝑇] satisfying 𝑢∈𝐶⁰[0,𝑇];𝐿²(Ω)𝐶⁰(0,𝑇];𝐻¹(Ω)𝐶⁰(0,𝑇];𝐻²_𝑁(Ω),𝑤∈𝐶⁰[0,𝑇];𝑊^1,2(Ω)𝐶⁰(0,𝑇];𝐻²_𝑁(Ω)𝐶⁰(0,𝑇];𝐻³_𝑁(Ω). (30) Then for any 𝜏>0, ‖𝑢(𝑡)‖²_𝐿2+‖𝑤(𝑡)‖²_𝑊1,2≤𝑐1+𝑒^−𝜐𝑡𝑢₀²_𝐿2+𝑤₀²_𝑊1,2,‖𝑢(𝑡0≤𝑡≤𝑇,)‖_𝐻¹+‖𝑤(𝑡)‖_𝐻²≤𝐶,𝜏<𝑡≤𝑇, (31) where 𝐶 depends only on Ω, 𝑢_𝜏, 𝑤_𝜏, and 𝜐>0.

Proof. In the process of the proof, we denote any positive constant by 𝑐 which may change from line to line and let 𝜏 be a small enough constant.
Step  1. Taking the inner product of the first equation of (1) with 𝑢 in 𝐿²(Ω), 12𝑑‖𝑢(𝑡)‖²_𝐿2+𝑑𝑡_Ω|∇𝑢|²=𝑑𝑥_Ω𝑢^𝛼∇𝑢⋅∇𝑔(𝑤)𝑑𝑥+_Ω𝑢𝑎𝑢−𝑏𝑢²≤1𝑑𝑥2_Ω|∇𝑢|²1𝑑𝑥+2_Ω𝑢^2𝛼|∇𝑤|²𝑎𝑑𝑥+²4𝑏_Ω𝑢𝑑𝑥. (32) By Hölder’s inequality, _Ω𝑢^2𝛼|∇𝑤|²𝑑𝑥≤_Ω𝑢^2𝑝𝛼𝑑𝑥^1/𝑝_Ω|∇𝑤|^2𝑝′𝑑𝑥^1/𝑝′=‖𝑢‖_𝐿^2𝛼_2𝑝𝛼‖∇𝑤‖²_𝐿′2𝑝1𝑝+1𝑝^′=1. (33) If 2𝑝𝛼(𝑛−2)≤2𝑛 and 2𝑝𝛼>1, then, from Lemma 3, there exists 𝑎=(𝑛−(𝑛/2𝑝𝛼))/(1+(𝑛/2))∈(0,1) such that ‖𝑢‖_𝐿^2𝛼_2𝑝𝛼≤‖𝑢‖_𝑊^2𝑎𝛼_1,2‖𝑢‖_𝐿^{2𝛼(1−𝑎)}₁. (34) Lemmas 9 and 11 imply that ‖∇𝑤(𝑡)‖_𝑞 is bounded for any 𝑞∈(1,𝑛/(𝑛−1)).
If 𝑞<𝑛/(𝑛−1)<2𝑝^′<2𝑛/(𝑛−2), then there exists 𝑏=((𝑛/𝑞)−(𝑛/2𝑝^′))/(1−(𝑛/2)+(𝑛/𝑞))∈(0,1) such that ‖∇𝑤‖²_𝐿′2𝑝≤𝑐‖∇𝑤‖_𝑊^2𝑏_1,2‖∇𝑤‖_𝐿^2(1−𝑏)_𝑞≤𝑐‖▵𝑤‖_𝐿^2𝑏₂. (35)
Let 𝑡=1/𝑝^′>(𝑛−2)/𝑛,𝑔(𝛼,𝑝,𝑞)=2𝑎𝛼+2𝑏 since 𝛼<1/𝑛, 𝑔𝑛𝛼,𝑝,=()−14𝑛𝛼−2𝑛1−𝑡+2𝑛+2𝑛()=()𝑛2𝑛−2−𝑛𝑡4𝛼+2²+4𝑛−8𝑛()+𝑛+2−4𝑡<()𝑛𝑛+24𝛼+2²𝑛(𝑛+2)<2. (36) Then there exist 𝑝, 𝑞 which satisfy the above condition such that 2𝑎𝛼+2𝑏<2. By Lemma 6, for any 𝜀>0, there is a constant 𝑐_𝜀 such that _Ω𝑢^2𝛼|∇𝑤|²≤‖(𝑡𝑑𝑥∇𝑢)‖_𝐿^2𝑎𝛼₂‖▵𝑤(𝑡)‖_𝐿^2𝑏₂≤𝜀‖∇𝑢(𝑡)‖²_𝐿2+‖▵𝑤(𝑡)‖²_𝐿2+𝑐_𝜀. (37) Taking the inner product of the second equation of (1) with ▵𝑤 in 𝐿²(Ω), for any 𝜀₁>0, 12𝑑‖∇𝑤(𝑡)‖²_𝐿2+𝑑𝑡_Ω|▵𝑤|²𝑑𝑥+_Ω|∇𝑤|²𝑑𝑥=−_Ω≤𝜀𝑢▵𝑤𝑑𝑥₂₁2_Ω|∇𝑢|²1𝑑𝑥+2𝜀²_1Ω|∇𝑤|²𝑑𝑥. (38) From (32) and (38), there are 𝜀, 𝜀₁ such that 𝜐=min{1−𝜀−𝜀²₁,2−1/𝜀²₁}>0 and 𝑑‖∇𝑤(𝑡)‖²_𝐿2+‖𝑢(𝑡)‖²_𝐿2𝑑𝑡+𝜐_Ω|▵𝑤|²𝑑𝑥+_Ω|∇𝑤|²𝑑𝑥+_Ω|∇𝑢|²𝑑𝑥≤𝑐. (39) By the Poincaré inequality, _Ω|∇𝑢|²𝑑𝑥≥𝑐_Ω𝑢²𝑑𝑥,_Ω|▵𝑤|²𝑑𝑥≥𝑐_Ω|∇𝑤|²𝑑𝑥. (40) Gronwall's lemma implies that ‖𝑢(𝑡)‖²_𝐿2+‖𝑤(𝑡)‖²_𝑊1,2≤𝑐1+𝑒^−𝜐𝑡𝑢₀²_𝐿2+𝑤₀²_𝑊1,2,𝑡∈[0,𝑇]. (41)
Step  2. From the analysis in Step 1, ‖𝑢(𝑡)‖_𝐿² is bounded in [0,𝑇]. Furthermore, by using Lemma 9, for any 𝑝<2𝑛/(𝑛−2) and small enough constant 𝜏>0, there is a constant 𝑐 such that ‖∇𝑤(𝑡)‖_𝐿^𝑝≤𝑐 for all 𝑡∈[𝜏,𝑇] since 𝑤(𝑥,𝑡)∈𝐶^2,1(Ω;(𝜏,𝑇])∩𝐶((𝜏,𝑇];𝐶³(Ω)). Applying the operator ∇ on both sides of the second equation of (1), 𝜕∇𝑤()𝜕𝑡=∇▵𝑤−∇𝑤+∇𝑢. (42) Taking the inner product of (42) with −∇(▵𝑤), for any 𝜀₂>0, 12𝑑‖▵𝑤‖²_𝐿2+𝑑𝑡_Ω|∇(▵𝑤)|²𝑑𝑥+_Ω|▵𝑤|²𝑑𝑥=−_Ω()≤𝜀∇𝑢⋅∇▵𝑤𝑑𝑥₂₂2_Ω|∇(▵𝑤)|²1𝑑𝑥+2𝜀²_2Ω|∇𝑢|²𝑑𝑥. (43) From (32) and (43), there exist 𝜀=1/4, 𝜀₂=√2, and a constant 𝑐_𝜀,𝜀2 depending only on 𝑎,𝑏,|Ω| such that 12𝑑‖𝑢(𝑡)‖²_𝐿2+𝑑‖‖𝑑𝑡▵𝑤²_𝐿2+1𝑑𝑡8‖∇𝑢‖²_𝐿2+78‖▵𝑤‖²_𝐿2≤𝑐_𝜀,𝜀2. (44) By Poincaré inequality and Gronwall's lemma, then ‖𝑢‖²_𝐿2+‖▵𝑤‖²_𝐿2≤𝑐2𝑐_𝜀,𝜀2+𝑒^(−1/8)𝑡𝑢_𝜏²_𝐿2+▵𝑤_𝜏²_𝐿2,𝑡∈(𝜏,𝑇]. (45) Taking the inner product of the first equation of (1) with −▵𝑢 in 𝐿²(Ω), then 12𝑑‖∇𝑢‖²_𝐿2+𝑑𝑡_Ω|▵𝑢|²𝑑𝑥=_Ω∇⋅𝑢^𝛼∇𝑔(𝑤)▵𝑢𝑑𝑥+_Ω𝑓^′(𝑢)|∇𝑢|²≤𝜀𝑑𝑥₂₃2_Ω|▵𝑢|²1𝑑𝑥+2𝜀²_3Ω∇⋅𝑢^𝛼∇𝑔(𝑤)²𝑑𝑥+𝑎_Ω|∇𝑢|²𝑑𝑥. (46) Since 𝑔(𝑠) is a real smooth function and |𝑔^′(𝑠)|<𝜆 for all 𝑠∈[0,+∞), then there is a constant 𝑀>0, such that |𝑔^′′(𝑠)|≤𝑀 and ∇⋅(𝑢+1)^𝛼∇𝑔(𝑤)²=𝛼𝑔^′(𝑤)(𝑢+1)^𝛼−1∇𝑢⋅∇𝑤+𝑔^′′(𝑤)(𝑢+1)^𝛼|∇𝑤|²+𝑔^′(𝑤)(𝑢+1)^𝛼▵𝑤²≤𝑐|∇𝑢⋅∇𝑤|²+(𝑢+1)^2𝛼|▵𝑤|²+(𝑢+1)^2𝛼|∇𝑤|⁴. (47) By Hölder’s inequality, _Ω|∇𝑢⋅∇𝑤|²𝑑𝑥≤‖∇𝑢‖²_𝐿2𝑝‖∇𝑤‖²_𝐿2𝑞, (48) where 1/𝑝+1/𝑞=1. There is 𝑝∈(𝑛/2,𝑛/(𝑛−2)) for 𝑛=2,3 such that 2𝑞<2𝑛/(𝑛−2), 2𝑝(𝑛−2)<2𝑛. By using Lemma 9, there is a constant 𝑐 such that ‖∇𝑤‖_𝐿⁴≤𝑐. Then from Lemmas 3 and 7 and the Poincaré inequality, for any 𝜀₀>0, there exists a constant 𝑐_𝜀0 such that ‖∇𝑢‖²_𝐿2𝑝≤‖∇𝑢‖_𝑊^2𝑎_1,2‖∇𝑢‖_𝐿^2(1−𝑎)₂≤𝑐‖𝑢‖_𝐻^2𝑎₂‖𝑢‖_𝐻^2(1−𝑎)₁≤𝑐‖𝑢‖_𝐻^2𝑎₂‖𝑢‖_𝐻^1/2₂‖𝑢‖_𝐿^1/2₂^2(1−𝑎)≤𝜀₀‖𝑢‖²_𝐻2+𝑐_𝜀0‖𝑢‖²_𝐿2, (49) where 𝑎=𝑛/2−𝑛/2𝑝<1.
Using the same method as the above analysis, for any 𝑝₁ satisfying 𝑝₁>𝑛/2, 2𝑝₁𝛼≤2, and 𝑝^′₁<𝑛/(𝑛−2), there exist 𝑎=𝑛/2−𝑛/2𝑝₁∈(0,1) and arbitrary constant 𝛿>0 such that _Ω(𝑢+1)^2𝛼|▵𝑤|²𝑑𝑥≤‖𝑢+1‖_𝐿^2𝛼_1𝛼2𝑝‖▵𝑤‖²_{𝐿′12𝑝},‖▵𝑤‖²_{𝐿′12𝑝}≤‖▵𝑤‖_𝐻^2𝑎₁‖▵𝑤‖_𝐿^2(1−𝑎)₂≤𝛿‖∇▵𝑤‖²_𝐿2+𝑐_𝛿‖𝑤‖²_𝐻2. (50) Equation (45) implies that ‖▵𝑤‖_𝐿² and ‖𝑢‖_𝐿² are bounded, and then there exists a constant 𝑐 such that ‖𝑢+1‖_𝐿^2𝛼_2𝑝𝛼≤‖𝑢+1‖_𝐿^2𝛼₂≤𝑐 and ∫_Ω(𝑢+1)^2𝛼|▵𝑤|²𝑑𝑥≤𝑐𝛿‖∇▵𝑤‖²_𝐿2+𝑐𝑐_𝛿‖𝑤‖²_𝐻2.
By Lemma 9 and 𝛼<1/𝑛, for any 𝑝₂ satisfying 𝑝₂>𝑛/(4−𝑛), 2𝑝₂𝛼≤2, and 4𝑝^′₂<2𝑛/(𝑛−2), there exists a constant 𝑐 such that _Ω(𝑢+1)^2𝛼|∇𝑤|⁴𝑑𝑥≤‖𝑢+1‖_𝐿^2𝛼_2𝛼2𝑝‖∇𝑤‖⁴_{𝐿′24𝑝}≤‖𝑢+1‖²_𝐿2‖∇𝑤‖⁴_{𝐿′24𝑝}≤𝑐, (51) where 1/𝑝₁+1/𝑝^′₁=1.
From the above analysis, if 𝜀₂=1, 𝜀₃=1 in (43), (46) and 𝜀₀, 𝛿 are small enough, then there exists positive constant 𝜐>0 such that 12𝑑‖▵𝑤‖²_𝐿2+‖∇𝑢‖²_𝐿2+𝑑𝑡_Ω|∇(▵𝑤)|²𝑑𝑥+𝜐_Ω|▵𝑤|²+|∇𝑢|²𝑑𝑥≤𝑐‖𝑢‖_𝐿²,‖𝑤‖_𝐻¹. (52) By Gronwall's lemma, ‖𝑤‖_𝐻²+‖𝑢‖_𝐻¹≤𝑐𝑐𝜀,𝜀₁+𝑒^−𝜐𝑡𝑤_𝜏𝐻2+𝑢_𝜏𝐻1,𝑡∈(𝜏,𝑇]. (53)
Equations (45), (53), and the choice of 𝑇 (in Theorem 10) depending on ‖𝑢₀‖_𝐿2(Ω)+‖𝑤₀‖_𝑊1,2(Ω) imply that (𝑢(𝑥,𝑇),𝑤(𝑥,𝑇))∈𝑆. It is clear by a standard argument that the solution (𝑢,𝑤) to (1) can be extended up to some 𝑇_max≤∞. With the same method as in the proof of Lemma 12, for any finite 𝑇_max, 𝑢𝑇_max²_𝐿2+𝑤𝑇_max²_𝑊1,2≤𝑐1+𝑒^{−𝜐𝑇max}𝑢₀²_𝐿2+𝑤₀²_𝑊1,2𝑢𝑇_max𝐻1+𝑤𝑇_max𝐻2≤𝐶 (54) which implies that 𝑇_max=+∞. The global existence of the solution to (1) is obtained as the following theorem.

Theorem 13. Suppose that nonnegative functions 𝑢₀∈𝐿²(Ω), 𝑤₀∈𝑊^1,2(Ω), and 𝛼<1/𝑛, and then there is a unique nonnegative global solution (𝑢,𝑤) to (1) satisfying 𝑢∈𝐶[0,∞);𝐿²(Ω)𝐶¹(0,∞);𝐿²(Ω)𝐶(0,∞);𝐻¹(Ω),𝑤∈𝐶[0,∞);𝑊^1,2(Ω)𝐶¹(0,∞);𝑊^1,2(Ω)𝐶(0,∞);𝐻²(Ω),‖𝑢(𝑡)‖²_𝐿2+‖𝑤(𝑡)‖²_𝑊1,2≤𝑐1+𝑒^−𝜐𝑡𝑢₀²_𝐿2+𝑤₀²_𝑊1,2,‖𝑢(𝑡𝑡≥0,)‖_𝐻¹+‖𝑤(𝑡)‖_𝐻²≤𝐶,𝑡>𝑇₁+1, (55) where 𝐶 depends only on Ω,𝑢_𝑇1,𝑤_𝑇1, and 𝑇₁=(1/𝜐)ln(‖𝑢₀‖²_𝐿2+‖𝑤₀‖²_𝑊1,2).

Remark 14. From the estimates in Lemma 12, there exists fixed constant 𝑀>0 and 𝑇₁=(1/𝜐)ln(‖𝑢₀‖²_𝐿2+‖𝑤₀‖²_𝑊1,2) such that ‖𝑢(𝑡)‖_𝐿²+‖𝑤(𝑡)‖_𝑊^1,2≤𝑀,𝑡>𝑇₁. (56) Denote the set 𝐵={(𝑢,𝑤)∈𝐿²(Ω)×𝐻¹(Ω)∶‖𝑢,𝑤‖_𝐿²_×𝐻¹≤𝑀;𝑢≥0,𝑤≥0}, where 𝑀 is the constant in (56). The results of Theorem 13 imply that the existence of a dynamical system {𝑆(𝑡)}_𝑡≥0 which maps 𝑉={(𝑢,𝑤)∶(𝑢,𝑤)∈𝐿²×𝐻¹;𝑢≥0,𝑤≥0} into itself and satisfying (𝑢(𝑡),𝑤(𝑡))=𝑆(𝑡)(𝑢₀,𝑤₀). Since 𝐵 is bounded, by Lemma 12, there exists 𝑇(𝐵) depending only on 𝐵 and |Ω| such that 𝑆(𝑡)𝐵⊂𝐵,𝑡≥𝑇(𝐵) (57) which implies that 𝐵 is a bounded absorbing set of the semigroup {𝑆(𝑡)}_𝑡≥0.
Next, by the Sobolev embedding theorem, the asymptotical compactness of the semigroup {𝑆(𝑡)}_𝑡≥0 is shown, and then the existence of a global attractor to system (1) is given.

Theorem 15. Assume that 𝛼<1/𝑛. Then the problem (1) has a global attractor which is a compact invariant set and attracts every bounded set in 𝑉.

Proof. If (𝑢_𝑛,𝑤_𝑛) is bounded in 𝑉, assume that there exists 𝑅 such that ‖(𝑢_𝑛,𝑤_𝑛)‖_𝑉≤𝑅. Then by Lemma 12, there is a constant 𝑇₁(𝑅) (depending on 𝑅) such that 𝑢^𝑛(𝑡),𝑤^𝑛(𝑡)=𝑆(𝑡)𝑢_𝑛,𝑤_𝑛⊂𝐵,𝑡≥𝑇₁(𝑅), (58) where 𝐵 is the absorbing set given in (57) and 𝑛=1,2,…. For any sequence 𝑡_𝑛 (𝑡_𝑛→+∞ as 𝑛→∞), there exists 𝑁 such that, for any 𝑛>𝑁(𝑅), 𝑡_𝑛≥𝑇₁(𝑅)+1, 𝑢^𝑛𝑡_𝑛,𝑤^𝑛𝑡_𝑛=𝑆𝑡_𝑛−𝑇₁(𝑅)−1𝑆𝑇₁(𝑅)+1𝑢_𝑛,𝑤_𝑛. (59)
Since the two embedding 𝐻¹(Ω)↪𝐿²(Ω),𝐻²(Ω)↪𝐻¹(Ω) are compact, from the estimates in Theorem 13, it is clear that {𝑢^𝑛(𝑡_𝑛)} lies in a compact set in 𝐿²(Ω) and {𝑤^𝑛(𝑡_𝑛)} lies in a compact set in 𝐻¹(Ω). Hence, {𝑆(𝑡_𝑛)(𝑢_𝑛,𝑤_𝑛)|_Ω} is precompact in 𝐿²(Ω)×𝐻¹(Ω), which implies that {𝑆(𝑡)}_𝑡≥0 is asymptotically compact. In Remark 14, we have obtained the bounded absorbing set of {𝑆(𝑡)}_𝑡≥0. Then by Proposition 1, we obtain the existence of the global attractor to (1).

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 11272277), Program for New Century Excellent Talents in University (NCET-10-0238), the Key Project of Chinese Ministry of Education (no. 211105) and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (134100510013), Innovative Research Team in University of Henan Province (13IRTSTHN019), Foundation of Henan Educational Committee (no. 13A110737), and Hall of Henan Province Science and Technology Project (no. 12A110020).