Reference Work Entry

Encyclopedia of Thermal Stresses

pp 1471-1474

Existence and Uniqueness: Solutions of Thermoelastodynamics

  • Maria Grazia NasoAffiliated withUniversità degli Studi di Brescia Email author 

Synonyms

Well-posedness in thermoelasticity​

Overview

We are concerned with a linear one-dimensional thermoelastic system where the hyperbolic elastic system is joined with the parabolic heat equation. In order to study the well posedness of the associated initial boundary value problem, a basic procedure is analyzed by means of semigroup techniques. For a detailed study in more general cases, some references are given at the end of this section.

A Simple Model in Thermoelasticity

The One-Dimensional Linear Thermoelastic System

For \( T>0 \), we consider the following one-dimensional linear thermoelastic system
$$ {u_{tt }}-\alpha {\,}{u_{xx }}+\gamma {\,}{\theta_x}=0\, \quad\mathrm{ in}\ (0,\ell )\times (0,T) $$
(1)
$$ {\theta_t}-k{\,}{\theta_{xx }}+\gamma {\,}{u_{xt }}=0\,\quad \mathrm{ in}\ (0,\ell )\times (0,T) $$
(2)
supplemented with initial conditions ...
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