Nonlinear stability of travelling wave solutions for viscoelastic materials with fading memory
type: Item
instance of: scholarly article
title: Nonlinear stability of travelling wave solutions for viscoelastic materials with fading memory
zbMATH Open document ID: 0857.35083
author: Harumi Hattori
author: Shuichi Kawashima
published in: Journal of Differential Equations
publication date: 1996-07-09T00:00:00Z
review text: The authors give a proof of existence and stability of shock-wave solutions to the system of equations describing wave propagation in a viscoelastic medium with memory, \[ v_t=u_x, \qquad u_t=[\sigma(v)]_x+ \int^t_{-\infty} a'(t-\tau)[\eta(v)]_x d\tau. \] They consider a class of smooth initial conditions satisfying the Rankine-Hugoniot boundary conditions at infinity. Using a priori estimates for the solutions and the continuation principle, they prove that any initial condition from this class gives rise to a unique smooth global solution and, provided that the initial profile is taken sufficiently close to a traveling wave \(u=u(x-ct)\), \(v=v(x-ct)\), the solution asymptotically coincides with the traveling (shock) wave at \(t\to \infty\) in the sense of the maximum norm.
reviewed by: B. A. Malomed
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 74Hxx
zbMATH DE Number: 881707
zbMATH Keywords: Rankine-Hugoniot boundary conditions
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0067
OpenAlex ID: W2023781581
DOI: 10.1006/JDEQ.1996.0067
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description: scientific article; zbMATH DE number 881707