Existence and Uniqueness: Solutions of Thermoelastodynamics

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

$$ u(x,0)={u_0}(x),\,{u_t}(x,0)={u_1}(x)\, \quad \mathrm{ in}\ (0,\ell ) $$

(3)

$$ \theta (x,0)={\theta_0}(x)\, \quad \mathrm{...