A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems
type: Item
instance of: scholarly article
title: A Lyapunov-Schmidt method for transition layers in reaction-diffusion systems
zbMATH Open document ID: 1093.35006
author: Jack K. Hale
author: Kunimochi Sakamoto
published in: Hiroshima Mathematical Journal
publication date: 2006-03-09T00:00:00Z
review text: This paper concerns the reaction-diffusion system \[ \left.\begin{aligned} u_t= \varepsilon^2u_{xx}&+ f(u,v)\\ v_t=Dv_{xx}&+ g(u,v)\end{aligned}\right\}\quad 0<x<1,\;t>0 \] with homogeneous Neumann boundary conditions and nonlinearities of propagator-controller type. The aim is existence and stability of a family (parametrized by small \(\varepsilon>0)\) of equilibria with one internal transition layer. The main techniques are: construction of approximate solutions with high degree of accuracy, estimation of the principal eigenvalue of the linearization around the approximate solution (SLEP method), Lyapunov-Schmidt reduction and two applications of the implicit function theorem.
reviewed by: Lutz Recke
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35C20
zbMATH DE Number: 5011027
zbMATH Keywords: SLEP method
zbMATH Keywords: one space dimension
zbMATH Keywords: homogeneous Neumann boundary conditions
zbMATH Keywords: propagator-controller type
zbMATH Keywords: existence
zbMATH Keywords: stability
zbMATH Keywords: implicit function theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1150998273
OpenAlex ID: W1915057131
Wikidata QID: Q128866241
DOI: 10.32917/HMJ/1150998273
description: scientific article; zbMATH DE number 5011027