Travelling waves for reaction--diffusion equations with time depending nonlinearities
type: Item
instance of: scholarly article
title: Travelling waves for reaction--diffusion equations with time depending nonlinearities
zbMATH Open document ID: 1032.35089
DOI: 10.1016/S0022-247X(02)00632-7
author: Bogdan Przeradzki
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: This paper concerns the parabolic equation \[ v_t =\Delta v-f(v,\alpha \cdot x-ct),\tag{1} \] where \(v=(v^1,\dots ,v^k):\mathbb{R}^l\times \mathbb{R}\to \mathbb{R}^k\), \(x,\alpha\in \mathbb{R}^l\), \(|\alpha|=1\), \(t\in \mathbb{R}\), \(c>0\) and \(f:\mathbb{R}^k\times \mathbb{R}\to \mathbb{R}^k\) is a continuous function. Under appropriate assumptions the author proves that equation (1) admits a travelling wave solution with end points \(w_{\pm}\), i.e., a solution \(v\) of the form \(v(x,t)=w(\alpha\cdot x-ct)\) and such that \(\lim_{x\to\pm\infty}w(x)=w_{\pm}\in \mathbb{R}^k\). The proof is obtained by reduction of the equation (1) to an ordinary one and by using the Schauder fixed point theorem and a suitable diagonalization procedure.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K40
zbMATH DE Number: 1956852
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: parabolic system
zbMATH Keywords: travelling wave
zbMATH Keywords: Schauder fixed point theorem
zbMATH Keywords: diagonalization procedure
reviewed by: Panagiotis Ch. Tsamatos
MaRDI profile type: Publication
cites work: EXISTENCE OF MULTIDIMENSIONAL TRAVELLING WAVES AND SYSTEMS OF WAVES
cites work: Q4746129
cites work: Q5768532
cites work: Geometric theory of semilinear parabolic equations
cites work: A new continuation method for the study of nonlinear equations at resonance
description: scientific article; zbMATH DE number 1956852