Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials
type: Item
instance of: scholarly article
title: Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials
zbMATH Open document ID: 0723.45017
DOI: 10.1007/BF00380814
published in: Archive for Rational Mechanics and Analysis
publication date: 1991-01-01T00:00:00Z
review text: Some equations arising in nonlinear viscoelasticity are studied. These equations can be transformed in the form \(u_{tt}(x,t)=div_ x\sigma (x,t)+f(x,t)\text{ on } \Omega \times (0,T],\sigma (x,t)=g(\nabla u(x,t))+\int^{t}_{-\infty}a(t-s)u(\nabla u(x,t),\nabla u(x,s))ds,\) with homogeneous boundary conditions and suitable initial conditions. The following types of results are proved: (i) there is global existence of solutions satisfying an entropy inequality for arbitrary data; (ii) local perturbations from the rest state propagate with finite speed; (iii) the solutions gain regularity if a is sufficiently singular at zero; (iv) solutions converge to a rest state as time goes to infinity in some cases. No uniqueness result is included. Space does not permit to go into details, but the paper is well written and interesting to read.
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 74Hxx
zbMATH DE Number: 4192387
zbMATH Keywords: weak solutions
zbMATH Keywords: quasilinear hyperbolic integro-differential equations
zbMATH Keywords: nonlinear viscoelasticity
zbMATH Keywords: global existence
zbMATH Keywords: entropy inequality
zbMATH Keywords: local perturbations
author: Hans Engler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00380814
OpenAlex ID: W2005647031
description: scientific article; zbMATH DE number 4192387