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Comptes Rendus Mathématique
Volume 351, n° 13-14
pages 539-544 (juillet 2013)
Doi : 10.1016/j.crma.2013.07.014
Received : 5 July 2013 ;  accepted : 17 July 2013
Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions
Analyticité et décroissance exponentielle uniformes du semi-groupe lié a une équation de plaque thermo-élastique avec des conditions aux limites perturbées
 

Louis Tebou 1
 Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA 

Abstract

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lionsʼ interpolation inequalities.

The full text of this article is available in PDF format.
Résumé

Dans un domaine borné, on considère une équation de plaque thermo-elastique de type Euler–Bernoulli avec des conditions aux limites perturbées. Les conditions aux limites utilisées sont telles que lʼon retrouve une plaque simplement posée lorsque le paramètre de perturbation tend vers lʼinfini, alors que lʼon retrouve une plaque encastrée quand le paramètre de perturbation tend vers zéro. En nous appuyant sur des estimations de la résolvante, nous montrons que le semi-groupe associé est analytique et exponentiellement stable, uniformément par rapport au paramètre de perturbation. Les éléments principaux de notre démonstration sont des décompositions appropriées des composantes du système et lʼutilisation dʼinégalités dʼinterpolation de Lions.

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