Traveling waves for the Keller-Segel system with Fisher birth terms
type: Item
instance of: scholarly article
title: Traveling waves for the Keller-Segel system with Fisher birth terms
zbMATH Open document ID: 1154.35459
DOI: 10.4171/IFB/200
published in: Interfaces and Free Boundaries
publication date: 2009-01-30T00:00:00Z
full work available at URL: http://www.ems-ph.org/journals/show_pdf.php
review text: Summary: We consider the traveling wave problem for the one dimensional Keller-Segel system with a birth term of either a Fisher/KPP type or with a truncation for small population densities. We prove that there exists a solution under some stability conditions on the coefficients which enforce an upper bound on the solution and \(\dot H^1(\mathbb R)\) estimates. Solutions in the KPP case are built as a limit of traveling waves for the truncated birth rates (similar to ignition temperature in combustion theory). We also discuss some general bounds and long time convergence for the solution of the Cauchy problem and in particular linear and nonlinear stability of the non-zero steady state.
Mathematics Subject Classification ID: 35Q80
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 92C17
zbMATH DE Number: 5500718
zbMATH Keywords: chemotaxis
zbMATH Keywords: traveling waves
zbMATH Keywords: Keller-Segel system
zbMATH Keywords: reaction-diffusion systems
zbMATH Keywords: nonlinear stability
author: Grégoire Nadin
author: Ben{ôı}t Perthame
author: Lenya Ryzhik
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5500718