数学
有界函数
Neumann边界条件
领域(数学分析)
正多边形
边界(拓扑)
数学分析
反应扩散系统
人口
纯数学
扩散
边值问题
应用数学
指数稳定性
非线性系统
几何学
热力学
物理
量子力学
社会学
人口学
作者
Youshan Tao,Michael Winkler
摘要
Abstract This work studies the two‐species Shigesada–Kawasaki–Teramoto model with cross‐diffusion for one species, as given by with positive parameters and , and nonnegative constants and . Beyond some statements on global existence, the literature apparently provides only few results on qualitative behavior of solutions; in particular, questions related to boundedness as well as to large time asymptotics in seem unsolved so far. In the present paper it is inter alia shown that if and is a bounded convex domain with smooth boundary, then whenever and are nonnegative, the associated Neumann initial‐boundary value problem for possesses a global classical solution which in fact is bounded in the sense that Moreover, the asymptotic behavior of arbitrary nonnegative solutions enjoying the boundedness property is studied in the general situation when is arbitrary and no longer necessarily convex. If , then in both cases and , an explicit smallness condition on is identified as sufficient for stabilization of any nontrivial solutions toward a corresponding unique nontrivial spatially homogeneous steady state. If and , then without any further assumption all nonzero solutions are seen to approach the equilibrium (0,1). As a by‐product, this particularly improves previous knowledge on nonexistence of nonconstant equilibria of .
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