The existence, uniqueness, and instability of spherically symmetric solutions of a system of reaction-diffusion equations
type: Item
instance of: scholarly article
title: The existence, uniqueness, and instability of spherically symmetric solutions of a system of reaction-diffusion equations
zbMATH Open document ID: 0545.35049
DOI: 10.1016/0022-0396(84)90137-2
author: Gene A. Klaasen
author: W. C. Troy
published in: Journal of Differential Equations
publication date: 1984-01-01T00:00:00Z
review text: Existence, uniqueness and stability of (steady-state) radially symmetric solutions of the reaction-diffusion system \(\partial x/\partial t=\Delta x+F(x,y),\quad \partial y/\partial t=G(x,y)\) are studied under suitable assumptions on the nonlinear terms F and G. In particular, as is shown in Appendix A, they include the Fitzhugh-Nagumo system for nerve conduction and the Field-Noyes model for the Belousov-Zhabotinskij reaction. These results generalize, under slightly different hypotheses, previous work by C. Jones is the case of one single equation. Under the above mentioned assumptions, the authors prove that the problem (for space dimension \(n>1)\) has a unique radially symmetric solution which is linearly unstable. The proofs are quite complicated from the technical point of view and use nontrivial plane phase arguments, in particular ''backwards shooting'' and center manifolds.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 3867830
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: stability
zbMATH Keywords: steady-state
zbMATH Keywords: radially symmetric
zbMATH Keywords: reaction-diffusion
zbMATH Keywords: Fitzhugh-Nagumo system
zbMATH Keywords: nerve conduction
zbMATH Keywords: Field-Noyes model
zbMATH Keywords: Belousov-Zhabotinskij reaction
zbMATH Keywords: linearly unstable
zbMATH Keywords: backwards shooting
zbMATH Keywords: center manifolds
reviewed by: Jesús Hernández
MaRDI profile type: Publication
full work available at URL: 0022-0396(84)90137-2
OpenAlex ID: W1993462206
cites work: The Existence of Solitary Traveling Wave Solutions of a Model of the Belousov-Zhabotinskii Reaction
cites work: Single and Multiple Pulse Waves for the FitzHugh–Nagumo
cites work: Q5533735
description: scientific article; zbMATH DE number 3867830