Evans function and blow-up methods in critical eigenvalue problems
type: Item
instance of: scholarly article
title: Evans function and blow-up methods in critical eigenvalue problems
zbMATH Open document ID: 1068.35080
author: B. Sandstede
author: Arnd Scheel
published in: Discrete and Continuous Dynamical Systems
publication date: 2004-11-01T00:00:00Z
review text: The goal of this paper is to investigate the stability properties of certain nonlinear waves that arise in dissipative, pattern-forming partial differential equations. A reaction-diffusion system \[ U_t = DU_{xx}+F(U), x\in{\mathbb R}\tag{1} \] are considered. Here \(U\in{\mathbb R}^N\) and \(D\) is a diagonal positive diffusion matrix. It is assumed that (1) supports a family of wave-train solutions \(U(x,t) = U_{wt}(kx-\omega t;k)\) that are \(2\pi\)-periodic in \(\phi = kx-\omega t\). The authors show that the Evans function can be extended across this branch point and discuss the smoothness properties of the extension. The construction utilizes blowup techniques and is quite general in nature. They also comment on known relations between roots of the Evans function and the temporal asymptotics of Green's functions, and discuss applications to algebraically decaying solitons.
reviewed by: Andrey Ivanovic Sedov
Mathematics Subject Classification ID: 35P05
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 37L15
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 2113143
zbMATH Keywords: algebraic decay
zbMATH Keywords: radial Laplacian
zbMATH Keywords: stability of nonlinear waves
MaRDI profile type: Publication
full work available at URL: dcds.2004.10.941
OpenAlex ID: W2157278438
DOI: 10.3934/DCDS.2004.10.941
description: scientific article; zbMATH DE number 2113143