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Asymptotic Behavior in Time

  • Reference work entry
Encyclopedia of Thermal Stresses

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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

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

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Authors and Affiliations

  1. Università degli Studi di Brescia, Via Valotti 9, Brescia, 25133, Italy

    Maria Grazia Naso

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  1. Maria Grazia Naso
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Correspondence to Maria Grazia Naso .

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Editors and Affiliations

  1. Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, NY, USA

    Richard B. Hetnarski

  2. Naples, FL, USA

    Richard B. Hetnarski

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Naso, M.G. (2014). Asymptotic Behavior in Time. In: Hetnarski, R.B. (eds) Encyclopedia of Thermal Stresses. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2739-7_531

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