Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems
type: Item
instance of: scholarly article
title: Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems
zbMATH Open document ID: 1204.35033
author: Jing Wang
author: Feng Xie
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-08-26T00:00:00Z
review text: There is considered the singularly perturbed problem with unknown vector \(u^\varepsilon(t,x) \in \mathbb R^N\), \(\partial_tu^\varepsilon+A(t,x,u^\varepsilon)\partial_xu^\varepsilon =\varepsilon B(t,x,u^\varepsilon)\partial^2_x u^\varepsilon\), \(x>0\), \(0<t<T\), \(u^\varepsilon|_{t=0}=u^\varepsilon_0(x)\), \(u^\varepsilon|_{x=0}=0\), where \(B\) is a symmetric, positive definite matrix, \(A(t,x,u^\varepsilon)|_{x=0}=0\) for all \(t\), \(u^\varepsilon\). Using asymptotic expansion of the initial function \(u^\varepsilon_0(x)\) with respect to \(\varepsilon\), the authors approximate the solution \(u^\varepsilon\) by the solutions of suitable problems. With the help of weighted estimates with powers of \(x\) weights, the convergence of the solution \(u^\varepsilon\) (as \(\varepsilon\to 0\)) to the solution \(u\) of the unperturbed problem is proved, i.e., of the degenerate hyperbolic problem \(\partial_tu+A(t,x,u)\partial_xu =0\), \(x>0\), \(0<t<T\), \(u|_{t=0}=u^0_0(x)\), where \(u^0_0(x)=u^\varepsilon_0(x)|_{\varepsilon=0}\).
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35L80
zbMATH DE Number: 5774964
zbMATH Keywords: boundary value problem
zbMATH Keywords: quasilinear parabolic equation
zbMATH Keywords: singular perturbation
zbMATH Keywords: degenerate hyperbolic equation
zbMATH Keywords: asymptotic expansion
reviewed by: Galina Bizhanova
MaRDI profile type: Publication
full work available at URL: j.na.2010.06.022
OpenAlex ID: W2040927758
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DOI: 10.1016/J.NA.2010.06.022
description: scientific article; zbMATH DE number 5774964