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

Abstract

This paper concerns the uniform bounds of the global existence of solutions in time for the S-K-T competition model with self-diffusion. We prove that the system has a global attractor for .

1. 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 is a bounded smooth region in with as its unit outward normal vector to the smooth boundary . and are the population densities of the two competing species. The constants , , , and are all positive, and constants are nonnegative. and are the random diffusion rates, and are the self-diffusion rates which represent intraspecific population pressures, and and are the so-called cross-diffusion rates which represent the interspecific population pressures.

If , 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 , the global existence of solutions has been widely investigated by many authors. When , , , , and hold, Kim [2] proved the global existence of classical solutions by energy method. For , , Deuring [3] proved the global existence of solutions if and are small enough depending on the norm of initial values . 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 . 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 , Lou et al. [6] established the unique global existence of solutions for , , , and .

For , (1) can be written as Equation (2) has been investigated by many authors; we state the results as follows.

For , either , or , ; Yagi [7] proved the global existence of solutions. For , , and , Kuiper and Dung [8] established the uniform bounds of global solutions for any when and are uniformly bounded. Choi et al. [9] applied more detailed interpolated estimates and energy methods to prove the global existence of solutions for , , and .

Le and his collaborators [10] have shown the existence of a global attractor for (2) in case . 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 -estimate of ; then by an iteration method, they show for any and , 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 , , and . 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 .

The main result in this paper is as follows.

Theorem 1. Assume holds; for any , system (2) has a global attractor with finite Hausdorff dimension in the space defined by

2. 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 of system (4) defined on a subinterval of . Let be the set of function on such that there exists a positive constant , which may generally depend on the parameters of the system and the norm of the initial value , such that Furthermore, if , one says that is in if and there exists a positive constant that depends only on the parameters of the system but does not depend on the initial value of such that If and , one says is ultimately uniformly bounded.

Lemma 3 (the uniform Gronwall inequality). Assume that , , and hold and that they are integrable in satisfying where , , and are positive constants. If , then one has

Lemma 4 (see [10, Lemmas 3.2-3.3]). For any dimension , one has the following estimates for the solutions of system (4):

Lemma 5 (see [10, Theorem 2.4]). For the system (4), if holds, with , satisfying where , then there exists such that

3. Proof of Theorem 1

Lemma 6. For any dimension , any solution of (4) has the following estimate:

Proof. Define then satisfies the following equation: Multiplying (19) by and integrating with respect to over , we have Integrating (20) over , we obtain In virtue of (9), there exist positive constants , , and such that Here (18) implies By (9)-(10) and (23), we have Hence (22) and Hölder’s inequality imply By (12) and (25), we get
Multiplying (19) by and integrating with respect to over , we have with .
By (27), we get Recall that (9) and (18) yield It follows from (28) and (29) that By Young’s inequality and (30) Since together with (31), we see from (31) that where is independent of .
Since together with (9) and (13), we have . This fact, together with (12) and (26), implies . Hence, in view of and (9), we get the desired result .

Lemma 7. For any dimension , any solution of (4) satisfies the following estimates:

Proof. Multiplying the first equation of (4) by and integrating, we get Young’s inequality and (36) imply Taking in (37), we have By the uniform Gronwall inequality, together with (12), (17), and (38), we obtain In virtue of (36), we have Integrating (40) over , we get By Young’s inequality, we have Taking in (41) and applying Hölder’s inequality, we see from (42) that By (12), (17), and (39), we get
Next we prove . Multiplying (4) by and integrating with respect to over , we get Apply the following inequalities: Use (46) with to get Choosing small positive numbers and in the above inequalities, we get By (17), (39), (44), (48), and uniform Gronwall’s inequality, we get the desired result

Proof of Theorem 1. It follows from (48) that In virtue of (17), (35), (44), and (50), we obtain For , , we see .
Let ; then Gagliardo-Nirenberg inequality gives which implies with .
For , in virtue of (53), we have Note thus , with , satisfying Let when holds; in order to satisfy (15) in Lemma 5, we need to check , , and . By (56), we have the following results: Since is compact in , by the attractor theory in [14], we complete the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his valuable comments and helpful suggestions which helped improve the exposition of the current paper. The authors are greatly indebted to Professor Yaping Wu for her encouragement, valuable suggestions, and helpful discussions. This work is partially supported by Beijing Natural Science Foundation (1132003, KZ201310028030, and KM201210017008).