Singular limit of a degenerate chemotaxis-Fisher equation
type: Item
instance of: scholarly article
title: Singular limit of a degenerate chemotaxis-Fisher equation
zbMATH Open document ID: 1063.35094
author: Fathi Dkhil
published in: Hiroshima Mathematical Journal
publication date: 2004-10-28T00:00:00Z
review text: The paper deals with a degenerate parabolic equation that arises in chemotaxis-growth models \[ \frac{\partial u}{\partial t} = \varepsilon \Delta (u^m)- \nabla . (u \nabla v) + \frac{1}{\varepsilon} u(1-u^p) \] in \(\Omega \times (0,T]\) with zero Neumann boundary conditions, initial condition \(u(x,0)=g^\varepsilon (x)\) in \(\Omega \subset \mathbb R^N\), \(m>1\), and a given smooth function \(v(x,t)\). The author studies the behaviour of the (weak) solution to this problem as \(\varepsilon \to 0\). Using the method of sub- and supersolutions, he proves that for suitable initial data \(g^\varepsilon\) with a compact support in \(\Omega\), \(u^\varepsilon \to u^0\) on \([0,T]\) for some \(T>0\). The limiting function \(u^0\) is the characteristic function of a smooth moving domain \(\Omega_t \subset\subset \Omega\), whose motion law is related to \(v\) through the critical velocity of a certain degenerate travelling Fisher wave problem. An example of radially symmetric case with non-trivial solutions is discussed.
reviewed by: David S. Boukal
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 35K45
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 2111509
zbMATH Keywords: zero Neumann boundary conditions
zbMATH Keywords: subsolutions
zbMATH Keywords: supersolutions
zbMATH Keywords: Keller-Segel model
zbMATH Keywords: travelling wave
zbMATH Keywords: interface motion
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1150998073
OpenAlex ID: W1559257309
Wikidata QID: Q128865673
DOI: 10.32917/HMJ/1150998073
description: scientific article; zbMATH DE number 2111509