The Existence of a Global Attractor for the S-K-T Competition Model with Self-Diffusion

and Applied Analysis 3 3. Proof of Theorem 1 Lemma 6. For any dimension n, any solution u of (4) has the following estimate:


Introduction and Statement of Main Result
Shigesada et al. [1] introduced the following competition model to describe the spatial segregation of two competing species under inter-and intraspecies population pressures: where Ω is a bounded smooth region in R  with  as its unit outward normal vector to the smooth boundary Ω. and V are the population densities of the two competing species.The constants   ,   ,   , and   ( = 1, 2) are all positive, and constants   (,  = 1, 2) are nonnegative. 1 and  2 are the random diffusion rates,  11 and  22 are the self-diffusion rates which represent intraspecific population pressures, and  12 and  21 are the so-called cross-diffusion rates which represent the interspecific population pressures.
If   = 0 (,  = 1, 2), system (1) is reduced to the classical Lotka-Volterra competition model with diffusion; it has been extensively studied in the past few decades.When initial value is nonnegative and bounded, it is easy to prove that (1) has a unique uniformly bounded global solution.
For  11 = 0, the global existence of solutions has been widely investigated by many authors.When  = 1,  1 =  2 ,  12 > 0,  21 > 0, and  11 =  22 = 0 hold, Kim [2] proved the global existence of classical solutions by energy method.For  ≥ 1,  11 =  22 = 0, Deuring [3] proved the global existence of solutions if  12 and  21 are small enough depending on the  2, norm of initial values  0 , V 0 .Choi et al. [4] improved Deuring's result and proved the global existence of solutions if the cross-diffusion coefficients are small depending only on the  ∞ norm of initial value V 0 .By applying more detailed interpolated estimates, especially Gagliardo-Nirenberg inequality, Shim [5] improved Kim and Deuring's results and established the uniform bounds of the global existence of solutions in time.For  = 2, Lou et al. [6] established the unique global existence of solutions for  21 = 0,  12 > 0,  11 = 0, and  22 ≥ 0.
For  11 > 0, (1) can be written as (2) Equation ( 2) has been investigated by many authors; we state the results as follows.
Le and his collaborators [10] have shown the existence of a global attractor for (2) in case  ≤ 5. Le and Nguyen [11] constructed a special test function to prove the global existence of solutions for any dimension  under some certain restrictions on coefficients.Tuôc [12] improved the results of Le and Nguyen by a nontrivial application of maximum principle.Recently, Tuoc [13] has established the  4 -estimate of ∇V; then by an iteration method, they show  ∈   for any  ≥ 1 and  < 10, which implies the global existence of solutions.
In this paper, we consider the uniform bounds of the global existence of solutions in time of system (2) for  21 = 0,  11 > 0, and  22 > 0. In Section 2, we show some preliminary knowledge used in this paper.In Section 3, we follow the arguments of Le et al. and improve their results.We will prove the uniform bounds of the global existence of solutions in time of system (2) for  < 8.
The main result in this paper is as follows.
Theorem 1. Assume  < 8 holds; for any  0 > , system (2) has a global attractor with finite Hausdorff dimension in the space X defined by

Preliminary Results
System (2) can be written in the divergence form as Definition 2 (see [10,Definition 2.1]).Assume that there exists a solution (, V) of system (4) defined on a subinterval  of R + .Let O be the set of function  on  such that there exists a positive constant  0 , which may generally depend on the parameters of the system and the  1, 0 norm of the initial value ( 0 , V 0 ), such that Furthermore, if  = (0, ∞), one says that  is in P if  ∈ O and there exists a positive constant  ∞ that depends only on the parameters of the system but does not depend on the initial value of ( 0 , V 0 ) such that lim If  ∈ P and  = (0, ∞), one says  is ultimately uniformly bounded.