Asymptotic analysis of a nonsimple thermoelastic rod

Moncef  Aouadi; Taoufik  Moulahi

doi:10.3934/dcdss.2016059

Article Contents

2016, Volume 9, Issue 5: 1475-1492. Doi: 10.3934/dcdss.2016059

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Asymptotic analysis of a nonsimple thermoelastic rod

Moncef Aouadi^1, and
Taoufik Moulahi^2,

1.
Ecole Nationale d'Ingénieurs de Bizerte, Université de Carthage, BP66, Campus Universitaire Menzel Abderrahman 7035
2.
Faculté des Sciences de Bizerte, 7021 Zarzouna, Université de Carthage

Received: September 30, 2014

Revised: June 30, 2015

Published: September 2016

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Abstract

The asymptotic analysis of a one-dimensional nonsimple thermoelastic problem is considered in this paper. By a detailed spectral analysis, the asymptotic expressions for eigenvalues and eigenfunctions of the considered system are developed. It is shown that the eigenfunctions form a Riesz basis on the Hilbert space and the eigenvalues asymptotically fall on two branches. One branch is along the negative horizontal axis in the complex plane and the other branch is asymptotic to a vertical line that is parallel to the imaginary axis. This gives the spectrum-determined growth condition for the $C_0-$semigroup associated to the system, and consequently, the asymptotic and the exponential stability of the solutions are deduced. The approach developed in this paper confirms the already-existing results; furthermore, it can be extended to a larger field of applications such as coupled system of rod or beam with diffusion equation. The method will be illustrated by an example of thermoelastic beam equations with Dirichlet boundary conditions.

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Mathematics Subject Classification: 65E99, 45M10.

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