Abstract and Applied Analysis

Volume 2014 (2014), Article ID 512520, 6 pages

http://dx.doi.org/10.1155/2014/512520

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

^{1}Department of Basic Courses, Beijing Union University, Beijing 100101, China^{2}Department of Mathematics and Physics, Beijing Institute of Petrochemical-Technology, Beijing 102617, China

Received 25 January 2014; Revised 12 March 2014; Accepted 14 April 2014; Published 30 April 2014

Academic Editor: Marco Squassina

Copyright © 2014 Qian Xu and Ye Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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*

*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*

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

*Acknowledgments*

*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).*

*References*

*References*

- N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,”
*Journal of Theoretical Biology*, vol. 79, no. 1, pp. 83–99, 1979. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. U. Kim, “Smooth solutions to a quasi-linear system of diffusion equations for a certain population model,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 8, no. 10, pp. 1121–1144, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - P. Deuring, “An initial-boundary-value problem for a certain density-dependent diffusion system,”
*Mathematische Zeitschrift*, vol. 194, no. 3, pp. 375–396, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with weak cross-diffusion,”
*Discrete and Continuous Dynamical Systems*, vol. 9, no. 5, pp. 1193–1200, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S.-A. Shim, “Uniform boundedness and convergence of solutions to cross-diffusion systems,”
*Journal of Differential Equations*, vol. 185, no. 1, pp. 281–305, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Lou, W.-M. Ni, and Y. Wu, “On the global existence of a cross-diffusion system,”
*Discrete and Continuous Dynamical Systems*, vol. 4, no. 2, pp. 193–203, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Yagi, “Global solution to some quasilinear parabolic system in population dynamics,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 21, no. 8, pp. 603–630, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Kuiper and L. Dung, “Global attractors for cross diffusion systems on domains of arbitrary dimension,”
*Rocky Mountain Journal of Mathematics*, vol. 37, no. 5, pp. 1645–1668, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. S. Choi, R. Lui, and Y. Yamada, “Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion,”
*Discrete and Continuous Dynamical Systems*, vol. 10, no. 3, pp. 719–730, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Le, L. V. Nguyen, and T. T. Nguyen, “Shigesada-Kawasaki-Teramoto model on higher dimensional domains,”
*Electronic Journal of Differential Equations*, vol. 2003, no. 72, pp. 1–12, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Le and T. T. Nguyen, “Global existence for a class of triangular parabolic systems on domains of arbitrary dimension,”
*Proceedings of the American Mathematical Society*, vol. 133, no. 7, pp. 1985–1992, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - P. V. Tuôc, “Global existence of solutions to shigesada-kawasaki-teramoto cross-diffusion systems on domains of arbitrary dimensions,”
*Proceedings of the American Mathematical Society*, vol. 135, no. 12, pp. 3933–3941, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - P. V. Tuoc, “On global existence of solutions to a cross-diffusion system,”
*Journal of Mathematical Analysis and Applications*, vol. 343, no. 2, pp. 826–834, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. K. Hale,
*Asymptotic Behavior of Dissipative Systems*, vol. 25 of*Mathematical Surveys and Monographs*, American Mathematical Society, Providence, RI, USA, 1988. View at MathSciNet

*
*