Global Attractor for a Chemotaxis Model with Reaction Term

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 chemotaxismodel 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 concernedwith the following chemotaxis model:


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][2][3][4][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: where (, ) represents the density of a biological species, which could be a cell, a germ, or an insect, while (, ) denotes the concentration of a chemical substance on position  ∈ Ω and at time  ∈ [0, ∞). is a positive constant.
( + 1)  can be replaced by ( + )  , where  is an arbitrary positive constant.For convenience, we choose  = 1.(, ) describes the intrinsic rate of growth of cells population.For  = 1,  = 0, and () = , 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  ̸ = 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  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.
Proposition 1. Suppose that  is a metric space and () ≥0 is a semigroup of continuous operators in .If () ≥0 has a bounded absorbing set and is asymptotically compact, then () ≥0 possesses a global attractor which is a compact invariant set and attracts every bounded set in .

Local Existence and Uniqueness
The local existence of a solution to system (1) is discussed in this section.First, an estimate to  in (1) is shown.
Next, we prove that  is a contractive mapping from  into itself for  small enough and  sufficiently large.By Lemma 4, then where then Similarly, for  ≤ 3, there is +  1−(/4) .
(16) By Lemma 5, for any  ∈ [0, ), there is a  ∈ (1/2, 1) such that Equations ( 16) and ( 17) imply that  ⊂  for any fixed positive  large enough and  small enough.Now we show that  is a contractive operator from  to .For for all (, ), (, ) ∈ , Equation ( 18) implies that  is a contractive mapping if  is sufficiently small.By Banach's fixed point theorem, there exists a unique fixed point (, ) ∈  which is just a local solution to (1) in .

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.
Proof.In the process of the proof, we denote any positive constant by  which may change from line to line and let  be a small enough constant.
Using the same method as the above analysis, for any  1 satisfying  1 > /2, 2 where  depends only on Ω, which implies that  is a bounded absorbing set of the semigroup {()} ≥0 .Next, by the Sobolev embedding theorem, the asymptotical compactness of the semigroup {()} ≥0 is shown, and then the existence of a global attractor to system (1) is given.
Theorem 15.Assume that  < 1/.Then the problem (1) has a global attractor which is a compact invariant set and attracts every bounded set in .