A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models
Year: 2020
Type: article
Abstract: In a bounded domain, we consider a thermoelastic plate with rotational forces. The rotational forces involve the spectral fractional Laplacian, with power parameter \begin{document}$ 0\le\theta\le 1 $\end{document} . The model includes both the Euler-Bernoulli ( \begin{document}$ \theta = 0 $\end{document} ) and Kirchhoff ( \begin{document}$ \theta = 1 $\end{document} ) models for thermoelastic plate as special cases. First, we show that the underlying semigroup is of Gevrey class \begin{document}$ \delta $\end{document} for every \begin{document}$ \delta>(2-\theta)/(2-4\theta) $\end{document} for both the clamped and hinged boundary conditions when the parameter \begin{document}$ \theta $\end{document} lies in the interval \begin{document}$ (0, 1/2) $\end{document} . Then, we show that the semigroup is exponentially stable for hinged boundary conditions, for all values of \begin{document}$ \theta $\end{document} in \begin{document}$ [0, 1] $\end{document} . Finally, we prove, by constructing a counterexample, that, under hinged boundary conditions, the semigroup is not analytic, for all \begin{document}$ \theta $\end{document} in the interval \begin{document}$ (0, 1] $\end{document} . 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.
DOI: 10.3934/dcds.2020152
Authors: Valentin Keyantuo, Louis Tebou, Mahamadi Warma
Institutions: Florida International University, George Mason University
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W3009213622