The correspondence between travelling-wave solutions of a nonlinear reaction-convection-diffusion equation and an integral equation
type: Item
instance of: scholarly article
title: The correspondence between travelling-wave solutions of a nonlinear reaction-convection-diffusion equation and an integral equation
zbMATH Open document ID: 0855.35061
author: Brian H. Gilding
published in: Differential and Integral Equations
publication date: 1996-08-22T00:00:00Z
review text: The author examines the nonlinear reaction-convection-diffusion equation \[ u_t= (a(u))_{xx}+ (b(u))_x+ c(u). \] Under mild continuity conditions on the coefficients \(a\), \(b\) and \(c\), it is shown that the existence of travelling wave solutions, of the form \(f(x- \lambda t)\) (for suitably restricted \(f\)), is equivalent to the existence of a solution \(\theta\) of the singular nonlinear integral equation \[ \theta(s)= \lambda s+ b(s)- \int^s_0 {c(r)\over \theta(r)} da(r). \] A number of consequences of this correspondence between the integral equation and travelling waves are discussed. In particular, an open question in the theory of finite speed of propagation for the reaction-convection-diffusion equation is resolved.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 896467
zbMATH Keywords: nonlinear reaction-convection-diffusion equation
zbMATH Keywords: travelling wave solutions
reviewed by: Wilson Lamb
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 896467