Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials
type: Item
instance of: scholarly article
title: Global existence of smooth solution to nonlinear thermoviscoelastic system with clamped boundary conditions in solid-like materials
zbMATH Open document ID: 0938.35003
author: Peicheng Zhu
author: Boling Guo
published in: Communications in Mathematical Physics
publication date: 2000-06-22T00:00:00Z
review text: The authors have proved the global existence and uniqueness of a smooth solution of a system of one-dimensional nonlinear thermoviscoelasticity which describes the thermomechanical process in a class of solid like material -- such as rubber etc. -- under the assumption, -- proposed by \textit{C. D. Dafermos} in 1982 [SIAM. J. Math. Anal. 13, 397-408 (1982; Zbl 0489.73124)] --, that both the ends of the one-dimensional solid are clamped, and there is no flux of heat through the boundary. A new approach is developed to obtain all important estimates for the \(L^\infty\)-norm of the strain function \(u(x,t)\). First uniform pointwise boundedness, from both below and above, of the unknown functions and their derivatives is proved. Finally the global existence and uniqueness of a smooth solution is proved by using the framework given in the paper of \textit{C. M. Dafermos} and \textit{L. Hsiao} [Nonlinear Anal., Theory Methods Appl. 6, 435-454 (1982; Zbl 0498.35015)]. Thus apriori bounds of strain, velocity and temperature in some Banach space is proved.
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35G30
Mathematics Subject Classification ID: 74F05
Mathematics Subject Classification ID: 74D05
zbMATH DE Number: 1334928
zbMATH Keywords: a priori bounds of strain
zbMATH Keywords: velocity and temperature
MaRDI profile type: Publication
full work available at URL: s002200050617
OpenAlex ID: W2059602530
DOI: 10.1007/S002200050617
description: scientific article; zbMATH DE number 1334928