Existence of periodic travelling wave solutions of reaction-diffusion systems with fast-slow dynamics
type: Item
instance of: scholarly article
title: Existence of periodic travelling wave solutions of reaction-diffusion systems with fast-slow dynamics
zbMATH Open document ID: 0704.35076
DOI: 10.1016/0022-0396(88)90066-6
author: Eiji Yanagida
published in: Journal of Differential Equations
publication date: 1988-01-01T00:00:00Z
review text: The paper is concerned with the problem to prove (or disprove) the existence of periodic travelling wave solutions of reaction-diffusion systems \[ (1)\quad u_ t=Su_{xx}+f(u,v),\quad v_ t=Tv_{xx}+\mu g(u,v), \] \(x\in {\mathbb{R}}\), \(u(x,t)\in {\mathbb{R}}^ m\), \(v(x,t)\in {\mathbb{R}}^ n\); S,T are diffusion matrices; \(\mu\) is a small parameter; f,g are smooth functions (representing chemical kinetics). In order to overcome the difficulty of this problem a perturbation problem around \(\mu =0\) is considered, so that a spatially homogeneous solution \(v(x,t)=\alpha\) is obtained, where \(\alpha \in {\mathbb{R}}^ n\) is an arbitrary constant vector. Then the paper assumes that an L-periodic travelling solution or stationary wave solution is obtained for the equation \(u_ t=Su_{xx}+f(u,\alpha)\) and four theorems on the existence (or non existence) of L-periodic travelling solutions of the system (1) are proved by a perturbation method for sufficiently small \(\mu\).
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35B20
zbMATH DE Number: 4155023
zbMATH Keywords: existence of periodic travelling wave solutions
zbMATH Keywords: reaction-diffusion systems
zbMATH Keywords: chemical kinetics
zbMATH Keywords: perturbation problem
zbMATH Keywords: perturbation method
reviewed by: Rudolf Heersink
MaRDI profile type: Publication
full work available at URL: 0022-0396(88)90066-6
OpenAlex ID: W2021537706
cites work: Q4146134
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cites work: Instability of certain traveling wave solutions to the Fitz Hugh-Nagumo nerve axon equations
cites work: Q3853967
description: scientific article; zbMATH DE number 4155023