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

  • * Corresponding author: Louis Tebou

    * Corresponding author: Louis Tebou 

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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  • 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.

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


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