Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics
type: Item
instance of: scholarly article
title: Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics
zbMATH Open document ID: 0792.35086
author: Yukio Kan-on
published in: Hiroshima Mathematical Journal
publication date: 1994-08-07T00:00:00Z
review text: The author is interested in the existence of spatially inhomogeneous equilibrium solutions of the following reaction diffusion system which is coming from a model equation in a problem of biology: \[ \begin{cases} u_ t = \varepsilon^ 2 u_{xx} + f(u,v) & \text{in \((0,\infty)\times I\)}\\ v_ t = D[(1 + \beta u)v]_{xx} + g(u,v) & \text{in \((0,\infty) \times I\)}\\ u_ x = [(1 + \beta u)v]_ x = 0 & \text{on \((0,\infty) \times \partial I\),}\end{cases}\tag{P} \] where \(f\) and \(g\) satisfy some conditions so that (P) is a competition system. The diffusion term is nonlinear and hence somewhat complicated. Such a thing is called the cross diffusion term which arises because of a certain biological effect. The main question is whether the spatially inhomogeneous equilibrium solution is stable or not. In the case \(\beta = 0\) (no cross diffusion), it is known that such a solution does not exist. This fact was proved by Kishimoto and Weinberger. The author finds a different situation from the case \(\beta = 0\), including the existence of spatially inhomogeneous stable solutions in the singularly perturbation situation, that is, \(\varepsilon\) is very small.
reviewed by: Shuichi Jimbo
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 558858
zbMATH Keywords: cross diffusion term
zbMATH Keywords: spatially inhomogeneous stable solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 558858