A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models

Valentin Keyantuo; Louis Tebou; Mahamadi Warma

doi:10.3934/dcds.2020152

Article Contents

2020, Volume 40, Issue 5: 2875-2889. Doi: 10.3934/dcds.2020152

This issue Previous Article A functional CLT for nonconventional polynomial arrays Next Article Realization of big centralizers of minimal aperiodic actions on the Cantor set

A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models

1.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, Faculty of Natural Sciences, 17 University AVE. STE 1701 San Juan PR 00925-2537, USA
2.
Department of Mathematics and Statistics, Florida International University, Miami, FL 33199, USA
3.
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

^* Corresponding author: Louis Tebou
^* Corresponding author: Louis Tebou

Received: June 30, 2019

Revised: November 30, 2019

Published: March 2020

The work of V. Keyantuo and M. Warma is partially supported by the Air Force Office of Scientific Research under Award NO [FA9550-18-1-0242]

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Abstract

In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter $ 0\le\theta\le 1 $. The model includes both the Euler-Bernoulli ($ \theta = 0 $) and Kirchhoff ($ \theta = 1 $) models for thermoelastic plate as special cases. First, we show that the underlying semigroup is of Gevrey class $ \delta $ for every $ \delta>(2-\theta)/(2-4\theta) $ for both the clamped and hinged boundary conditions when the parameter $ \theta $ lies in the interval $ (0, 1/2) $. Then, we show that the semigroup is exponentially stable for hinged boundary conditions, for all values of $ \theta $ in $ [0, 1] $. Finally, we prove, by constructing a counterexample, that, under hinged boundary conditions, the semigroup is not analytic, for all $ \theta $ in the interval $ (0, 1] $. The main features of our Gevrey class proof are: the frequency domain method, appropriate decompositions of the components of the system and the use of Lions' interpolation inequalities.

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Mathematics Subject Classification: 35Q74, 35B65, 47D03, 74K20, 93D20.

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