Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping
zbMATH Open document ID: 0881.35076
author: Kenji Nishihara
published in: Journal of Differential Equations
publication date: 1998-02-22T00:00:00Z
review text: The Cauchy problem for \(V_{tt}- a(V_x)_x+\alpha V_t= 0\), \((x,t)\in\mathbb{R}\times (0,\infty)\), \(\alpha>0\), \(a'(0)>0\), is considered. It is shown that the asymptotic profile is given by the solution \(\varphi\) to \((*)\) \(\alpha\varphi_t- a'(0)\varphi_{xx}= 0\), \(\varphi(x,0)= V(x,0)+ 1/\alpha V_t(x,0)\), and that \[ |(V-\varphi), (V- \varphi)_x, (V- \varphi)_t(t))|_{L^\infty}={\mathcal O}(t^{-1}, t^{-3/2}, t^{-2}). \] The proof uses the energy method and the fundamental solution for \((*)\).
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 1038940
zbMATH Keywords: asymptotic profile
zbMATH Keywords: energy method
zbMATH Keywords: fundamental solution
reviewed by: Reinhard Racke
MaRDI profile type: Publication
OpenAlex ID: W1985006985
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DOI: 10.1006/JDEQ.1997.3268
description: scientific article; zbMATH DE number 1038940