Instability of homogeneous periodic solutions of parabolic-delay equations
type: Item
instance of: scholarly article
title: Instability of homogeneous periodic solutions of parabolic-delay equations
zbMATH Open document ID: 0811.35156
published in: Journal of Differential Equations
publication date: 1995-03-14T00:00:00Z
review text: Parabolic systems of the form \[ {\partial u \over \partial t} = D \Delta u(x,t) + f \bigl( u_ t(x, \cdot) \bigr) \tag{1} \] are considered, where \(f : C([-r,0], \mathbb{R}^ N) \to \mathbb{R}^ N\) and \(D\) is an \(N \times N\) real matrix with spectrum in the right half plane. The first part is concerned mainly with fairly standard existence and uniqueness results for equation (1) using the abstract form \(\dot u = Au + f(u_ t)\) and the variation of constants formula. In the second part, the existence of travelling waves of (1) is proved under the assumption that the reaction equation \(\dot u(t) = f(u_ t)\) has a non-constant periodic solution. The destabilizing effect of the \(D\) matrix is shown by considering a variational form of (1) and the corresponding reaction equation. If a characteristic multiplier has absolute value greater than 1 then the travelling wave is unstable.
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 589981
zbMATH Keywords: existence and uniqueness results
zbMATH Keywords: existence of travelling waves
zbMATH Keywords: variational form
author: Luiz Augusto F. De Oliveira
reviewed by: S. P. Banks
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1044
OpenAlex ID: W2005132309
DOI: 10.1006/JDEQ.1994.1044
description: scientific article; zbMATH DE number 589981