Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity
type: Item
instance of: scholarly article
title: Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity
zbMATH Open document ID: 0844.35021
author: Stuart S. Antman
author: Thomas I. Seidman
published in: Journal of Differential Equations
publication date: 1996-08-29T00:00:00Z
review text: The aim of this paper is to give sufficient conditions for a nonlinear stress function \(n= n(s, w_s, w_{st})\) guaranteeing global-in-time existence of solutions of initial-value problem for the quasilinear hyperbolic-parabolic equation \(\varrho w_{tt}= n(s, w_s, w_{st})_s+ f(s, t)\) describing longitudinal motions of a viscoelastic rod. Here \(w= w(s, t)\), \((s, t)\in (0, 1)\times \mathbb{R}^+\) denotes the position of a material point and therefore \(w_s(s, t)\) is the stretch at \((s, t)\). The function \(n= n(s, y, 0)\) is allowed to diverge to \(-\infty\) as \(y\to 0\). This causes the above equation to be singular. The proof of global existence of solutions is based on delicate estimates showing that total compression (i.e. the stretch \(w_s(s, t)\) vanishes) cannot occur in finite time.
reviewed by: Daniel Ševčovič
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 74Hxx
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 838713
zbMATH Keywords: global-in-time existence
zbMATH Keywords: longitudinal motions of a viscoelastic rod
zbMATH Keywords: total compression
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0005
OpenAlex ID: W2007000896
DOI: 10.1006/JDEQ.1996.0005
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description: scientific article; zbMATH DE number 838713