Construction and stability analysis of transition layer solutions in reaction-diffusion systems
type: Item
instance of: scholarly article
title: Construction and stability analysis of transition layer solutions in reaction-diffusion systems
zbMATH Open document ID: 0708.35006
author: Kunimochi Sakamoto
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1990-01-01T00:00:00Z
review text: The problem here considered is the construction and the stability analysis of transition layers associated to the system \[ u_ t=\epsilon^ 2u_{xx}+f(u,v),\quad 0<x<1,\quad t>0 \] \[ v_ t=\sigma^{-1}v_{xx}+g(u,v),\quad 0<x<1,\quad t>0 \] \[ u_ x=v_ x=0\quad at\quad x=0,1 \] (\(\epsilon\),\(\sigma\) being small positive parameters). Under some specific assumptions on the source terms f,g a careful analysis is performed about the existence and the behaviour of the equilibrium solutions as \(\epsilon\),\(\sigma\) approach zero, showing the occurrence of a transition boundary layer for u. In particular a result about the stability of such solutions is obtained by means of the spectral analysis of the operator which linearizes \(\epsilon^ 2u''+f(u,v)\), \(\sigma 'v''+g(u,v)\) with b.c. \(u'=v'=0\) at \(x=0,1\).
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35P99
zbMATH DE Number: 4163342
zbMATH Keywords: transition layers
zbMATH Keywords: equilibrium solutions
reviewed by: Antonio Fasano
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227692
OpenAlex ID: W1998106866
cites work: Stable transition layers in a semilinear boundary value problem
cites work: Q3948194
cites work: Large Time Behavior of Solutions of Systems of Nonlinear Reaction-Diffusion Equations
cites work: Transition layers in singular perturbation problems
cites work: Boundary and interior transition layer phenomena for pairs of second- order differential equations
cites work: Q3714441
cites work: Stability of Singularly Perturbed Solutions to Systems of Reaction-Diffusion Equations
cites work: Large diffusivity and asymptotic behavior in parabolic systems
cites work: Varying boundary conditions with large diffusivity
cites work: Existence and stability of transition layers
cites work: Multiple Solutions of Two-Point Boundary Value Problems of Neumann Type with a Small Parameter
cites work: Q4734203
cites work: Global Structure of Bifurcating Solutions of Some Reaction-Diffusion Systems
cites work: Q3784486
DOI: 10.2748/TMJ/1178227692
description: scientific article; zbMATH DE number 4163342