Global solutions to the equation of viscoelasticity with fading memory
type: Item
instance of: scholarly article
title: Global solutions to the equation of viscoelasticity with fading memory
zbMATH Open document ID: 0818.35017
author: Shuichi Kawashima
published in: Journal of Differential Equations
publication date: 1993-05-16T00:00:00Z
review text: The author analyses the viscoelastic one-dimensional model \[ v_ t - u_ x = 0,\;u_ t - S_ x = 0,\;S = \Phi (v) + \int^ t_{-\infty} \dot a(t - \tau) \psi (v) (\tau)d \tau, \] where \(v\) and \(u\) are the deformation gradient and the velocity, respectively; \(S\) is the stress; \(\Phi (v)\) and \(\psi (v)\) are given smooth functions of \(v\) such that \(\Phi (0) = \psi (0) = 0\). When the kernel \(a(t)\) has integrable singularities of first order, then the problem has a unique global solution if the initial conditions are smooth and small. This solution is convergent to the equilibrium state, when the kernel is strongly positive definite. This result improves the results of \textit{W. J. Hrusa} and \textit{J. A. Nohel} [J. Differ. Equations 59, 388-412 (1985; Zbl 0564.35075)], who proved only the local existence of the solution.
Mathematics Subject Classification ID: 35F10
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 74Hxx
zbMATH DE Number: 166905
zbMATH Keywords: viscoelastic one-dimensional model
zbMATH Keywords: unique global solution
reviewed by: Ştefan Zanfir
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1017
OpenAlex ID: W4254403059
DOI: 10.1006/JDEQ.1993.1017
description: scientific article; zbMATH DE number 166905