Some results on the stability and bifurcation of stationary solutions of delay-diffusion equations
type: Item
instance of: scholarly article
title: Some results on the stability and bifurcation of stationary solutions of delay-diffusion equations
zbMATH Open document ID: 0881.35120
author: Pedro Freitas
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-22T00:00:00Z
review text: Nonlinear delay-diffusion equations of the form \[ u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau)+ g(x,u(x,t)) \] are studied and stability and bifurcation results are obtained by studying the perturbation of the spectrum of the linearization \[ u_t= u_{xx}(x,t)+ a(x)u(x,t)+ bu(x,t-\tau) \] caused by \(g\). It is shown that a centre manifold of any dimension exists for some \(a\), \(b\), and \(\tau\). Moreover, it is shown that if \(\pm i\omega_0,\pm i\omega_1,\dots,\pm i\omega_{m-1}\) are the \(2m\) pure imaginary eigenvalues of the linearized equation, then the numbers \(\omega_0,\omega_1,\dots,\omega_{m-1}\) are rationally independent and as \(\tau\) increases through the critical value, the nonlinear equation undergoes a bifurcation from \(2\ell\) eigenvalues, where \(\ell\) is the sum of the multiplicities of the \(m\) eigenvalues.
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1005572
zbMATH Keywords: nonlinear delay equations
zbMATH Keywords: diffusion
zbMATH Keywords: bifurcation
reviewed by: S. P. Banks
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5196
OpenAlex ID: W2076055885
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DOI: 10.1006/JMAA.1997.5196
description: scientific article; zbMATH DE number 1005572