Asymptotic behavior of solutions to the system of one-dimensional nonlinear thermoviscoelasticity
type: Item
instance of: scholarly article
title: Asymptotic behavior of solutions to the system of one-dimensional nonlinear thermoviscoelasticity
zbMATH Open document ID: 0922.35087
author: Ling Hsiao
author: Huaiyu Jian
published in: Chinese Annals of Mathematics. Series B
publication date: 1999-10-11T00:00:00Z
review text: The authors study the large time behavior of solutions to the system of equations modeling one-dimensional thermoviscoelasticity: \(u_t=v_x\), \(v_t=\sigma_x\), \((e+v^2/2)_t=(\sigma v)_x -q_x\). The system is supplemented with the constitutive relations \(e=c_0 \theta\), \(\sigma= - f(u)\theta +\mu(u)v_x\), \(q=-k(u,\theta)\theta_x\) and solutions are subject to initial and Dirichlet boundary conditions. The main result of the paper is to show that globally defined smooth solutions may possess phase transitions when the material function \(f\) is not monotone. In the case \(f\) is strictly decreasing it is furthermore shown that \(v(.,t)\to 0\), \(\theta(.,t)\to T_0\) and \(u(.,t)\to \widetilde z\) as \(t\to\infty\) in the space \(H^1(0,1)\), i.e. any solution tends to a stable state.
reviewed by: Daniel Ševčovič
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L50
zbMATH DE Number: 1159353
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: phase transitions
zbMATH Keywords: large time behavior
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1159353