Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms
type: Item
instance of: scholarly article
title: Wave patterns, stability, and slow motions in inviscid and viscous hyperbolic equations with stiff reaction terms
zbMATH Open document ID: 1015.35065
DOI: 10.1016/S0022-0396(02)00057-8
author: Haitao Fan
author: Judith R. Miller
author: Shi Jin
published in: Journal of Differential Equations
publication date: 2003-04-28T00:00:00Z
review text: The qualitative behavior of solutions of a scalar conservation law with a stiff source term of the form \[ u_t + f(u)_x = \varepsilon A U_{xx} - \varepsilon^{-1}W'(u) \] is studied in the case of a convex flux \(f\) and a bistable source \(W'(u)\). The main results concern the inviscid case \(A=0\). Here it had been shown previously that in the zero reaction time limit \(\varepsilon\to 0\) the solution tends almost everywhere to a piecewise constant function \(u(x,t)\) where constant parts are separated by straight lines in the \((x,t)\)-plane. Now the authors study the asymptotic for \(\varepsilon>0\) small and show that shock layers move exponentially slowly while rarefaction layers are stationary. Another result shows that in the viscous case \(A\neq 0\) there exist traveling wave solutions correponding to either a single shock or a rarefaction layer and that these waves are asymptotically stable in \(W^{1,2}\).
reviewed by: Jörg Härterich
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 1901949
zbMATH Keywords: balance law
zbMATH Keywords: conservation law with stiff source
zbMATH Keywords: bistable source
zbMATH Keywords: traveling wave
zbMATH Keywords: exponentially slow motion
zbMATH Keywords: zero reaction time limit
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1901949