The stability of travelling fronts for general scalar viscous balance law
type: Item
instance of: scholarly article
title: The stability of travelling fronts for general scalar viscous balance law
zbMATH Open document ID: 1073.35112
author: Yaping Wu
author: Xiuxia Xing
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-28T00:00:00Z
review text: The authors deal with the travelling front solutions of the viscous balance law \[ u_t+ (f(u))_x=\varepsilon u_{xx}+ g(u),\quad x\in\mathbb{R},\tag{1} \] where \(\varepsilon> 0\) is called the viscosity parameter. They are mainly interested in the existence and stability of different types of travelling waves for (1). Using shooting methods, the authors prove that there exist smooth monotone waves of (1) connecting adjacent zeroes of \(g\). Moreover, based on the existence results, by semigroup theory and detailed spectral analysis, the authors show that for every fixed \(\varepsilon> 0\) the travelling fronts for (1) are locally asymptotically stable in some suitably choosen weighted space, and especially for sufficiently small viscosity, one can choose the same weighted spaces in which the travelling fronts are uniformly exponentially stable.
reviewed by: Messoud Efendiev
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 2162032
zbMATH Keywords: Shooting methods
zbMATH Keywords: Smooth monotone waves
zbMATH Keywords: Weighted spaces
zbMATH Keywords: Existence
zbMATH Keywords: Phase plane method
zbMATH Keywords: Asymptotic stability
zbMATH Keywords: Spectral analysis
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.12.042
OpenAlex ID: W1999277523
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DOI: 10.1016/J.JMAA.2004.12.042
description: scientific article; zbMATH DE number 2162032