期刊
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
卷 17, 期 8, 页码 2745-2769出版社
AMER INST MATHEMATICAL SCIENCES
DOI: 10.3934/dcdsb.2012.17.2745
关键词
cross-diffusion; population model; positive solution; limit system; a priori estimates; bifurcation
资金
- Japan Society for the Promotion of Science [B-21740129, C-21540229]
- Grants-in-Aid for Scientific Research [20340034, 24740101, 24540220] Funding Source: KAKEN
This paper deals with the following reaction-diffusion system (SP) {Delta[(1 + alpha nu)u] + u(a - u - cv) = 0 Delta[(1 + beta u)v] + v(b - du - v) = 0 in a bounded domain of R-N with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients alpha and beta. For this purpose, our strategy is to study limiting behavior of positive solutions when alpha or beta goes to infinity and derive the corresponding limit systems. We will obtain a priori estimates of u and v independently of beta (resp. alpha) with small alpha >= 0 (resp. beta >=> 0) in case 1 <= N <= 3 under Neumann boundary conditions, while we will obtain a priori estimates of u and v independently of alpha and beta in case 1 <= N <= 5 under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When alpha = 0 and beta -> infinity, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.
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