Reference Work Entry

Encyclopedia of Thermal Stresses

pp 251-257

Asymptotic Behavior in Time

  • Maria Grazia NasoAffiliated withUniversità degli Studi di Brescia Email author 

Synonyms

Decay rate; Longtime behavior in thermoelasticity​

Overview

We are interested to study the longtime behavior for a linear one-dimensional thermoelastic system where the hyperbolic elastic system is joined with the parabolic heat equation. By some results in semigroup theory, we prove the exponential decay of the solutions related to the associated initial boundary value problem. 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:
$$ {u_{tt }}-\alpha {\,}{u_{xx }}+\gamma {\,}{\theta_x}=0\quad \quad \mathrm{ in}\,\,(0,\ell )\times (0,T) $$
(1)
$$ {\theta_t}-k{\,}{\theta_{xx }}+\gamma {\,}{u_{xt }}=0\quad \quad \mathrm{ in}\,(0,\ell )\times (0,T) $$
(2)
supplemented with initial conditions ...
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