Interface dynamics of the Fisher equation with degenerate diffusion
type: Item
instance of: scholarly article
title: Interface dynamics of the Fisher equation with degenerate diffusion
zbMATH Open document ID: 1173.35075
published in: Journal of Differential Equations
publication date: 2008-05-29T00:00:00Z
review text: The authors study the initial-boundary value problem \[ \begin{alignedat}{2} u_t&= \varepsilon\Delta u^m + \frac{1}{\varepsilon}u(1-u) &&\quad \text{in } \Omega\times [0,T],\\ \frac{\partial(u^m)}{\partial\nu}&= 0, &&\quad \text{on } \partial\Omega\times [0,T],\\ u(x,0)&= g(x), &&\quad x \in \Omega, \end{alignedat} \] where \(\Omega\subset\mathbb R^N\) \((N\geq 1)\) is a smooth bounded domain, \(\nu\) is the outward normal vector to \(\partial\Omega\), \(m \geq 2\) and \(\varepsilon\) is a small parameter. The initial data \(g\geq 0\) has a convex compact support in \(\Omega\). This problem arises in population dynamics where it is called population density-dependent dispersal of individuals. The results of the article are focused on the asymptotic behavior of the solution \(u^{\varepsilon}\) of the above-mentioned problem as \(\varepsilon \to 0\). The main result is that for \(m\geq 2\) the solution \(u^{\varepsilon}\) converges to \(0\) or \(1\) on both sides of an interface moving with constant normal velocity \(c_m\) for compactly supported initial data.
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35A07
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35A21
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 5280781
zbMATH Keywords: nonlinear diffusion
zbMATH Keywords: finite speed of propagation
zbMATH Keywords: sharp interface limit
author: Danielle Hilhorst
author: Robert Kersner
author: E. Logak
author: Masayasu Mimura
MaRDI profile type: Publication
full work available at URL: j.jde.2008.02.018
OpenAlex ID: W1984820376
Wikidata QID: Q125676124
cites work: Traveling Wave Solution for Some Nonlinear Diffusion Equations
cites work: Q4795648
cites work: Generation and propagation of interfaces for reaction-diffusion equations
cites work: Q3034136
cites work: Limit theorems for large deviations and reaction-diffusion equations
cites work: Travelling waves in nonlinear diffusion-convection reaction
cites work: Q4513902
cites work: Spatial distribution of dispersing animals
cites work: RANDOM DISPERSAL IN THEORETICAL POPULATIONS
DOI: 10.1016/J.JDE.2008.02.018
description: scientific article; zbMATH DE number 5280781