文献类型：期刊文献

英文题名：THE EXISTENCE AND STABILITY OF NONTRIVIAL STEADY STATES FOR S-K-T COMPETITION MODEL WITH CROSS DIFFUSION

作者：Ni, Wei-Ming[1,2];Wu, Yaping[3,4];Xu, Qian[5]

第一作者：Ni, Wei-Ming

通讯作者：Ni, WM[1]

机构：[1]E China Normal Univ, Ctr PDE, Shanghai 200241, Peoples R China;[2]Univ Minnesota, Sch Math, Minneapolis, MN 55455 USA;[3]Capital Normal Univ, Coll Math Sci, Beijing 100048, Peoples R China;[4]Beijing Ctr Math & Informat Interdisciplinary Sci, Beijing 100048, Peoples R China;[5]Beijing Union Univ, Dept Basic Courses, Beijing 100101, Peoples R China

第一机构：E China Normal Univ, Ctr PDE, Shanghai 200241, Peoples R China

通讯机构：[1]corresponding author), E China Normal Univ, Ctr PDE, Shanghai 200241, Peoples R China.

年份：2014

卷号：34

期号：12

起止页码：5271-5298

外文期刊名：DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

收录：;Scopus(收录号：2-s2.0-84902668610);WOS:【SCI-EXPANDED(收录号:WOS:000338187000013)】；

基金：The authors would like to thank the referees for their many valuable comments and useful suggestions which helped improve the exposition of the current paper. The work is partially supported by NNSF of China (11071172, 11226178), Beijing Municipal Education Commission (KZ201310028030), Beijing NSF (1132003) and SRFDP (20101108110001); Wei-Ming Ni is supported by a special grant from East China Normal University and NSF Grant DMS-1210400; Qian Xu is also supported by Research Project of Beijing Union Univ. (ZK 201206).

语种：英文

外文关键词：Existence; stability; steady states; spectral analysis; cross diffusion; shadow system

摘要：This paper concerns with the existence and stability properties of non-constant positive steady states in one dimensional space for the following competition system with cross diffusion { ut = [(d(1) + rho(12v))(u)](xx) + u(a(1) - b(1)u - c(1)v), x is an element of (0, 1), t > 0, v(t) = d(2)v(xx) + v(a(2) - b(2)u - c(2)v), x E (0, 1), t > 0, (1) u(x) = v(x) = 0, x = 0,1,t > 0. First, by Lyapunov-Schmidt method, we obtain the existence and the detailed structure of a type of small nontrivial positive steady states to the shadow system of (1) as rho(12) -> infinity and when d(2) is near a(2)/pi(2), which also verifies some related existence results obtained earlier in [11] by a different method. Then, based on the detailed structure of the steady states, we further establish the stability of the small nontrivial positive steady states for the shadow system by spectral analysis. Finally, we prove the existence and stability of the corresponding nontrivial positive steady states for the original cross diffusion system (1) when rho(12) is large enough and d(2) is near a(2)/pi(2).