Regularity of Euler-Bernoulli and Kirchhoff-Love Thermoelastic Plates with Fractional Coupling

Abstract

I In this work, we present the study of the regularity of the solutions of the abstract system\eqref{Eq1.10} that includes the Euler-Bernoulli($\omega=0$) and Kirchoff-Love($\omega>0$) thermoelastic plates, we consider for both fractional couplings given by $A^\sigma\theta$ and $A^\sigma u_t$, where $A$ is a strictly positive and self-adjoint linear operator and the parameter $\sigma\in[0,\frac{3}{2}]$. Our research stems from the work of \cite{MSJR}, \cite{OroJRPata2013}, and \cite{KLiuH2021}. Our contribution was to directly determine the Gevrey sharp classes: for $\omega=0$, $s_{01}>\frac{1}{2\sigma-1}$ and $s_{02}> \sigma$ when $\sigma\in (\frac{1}{2},1)$ and $\sigma\in (1,\frac{3}{2})$ respectively. And $s_\omega>\frac{1}{4(\sigma-1)}$ for case $\omega>0$ when $\sigma\in (1,\frac{5}{4})$. This work also contains direct proofs of the analyticity of the corresponding semigroups $e^{t\mathbb{A}_\omega}$: In the case $\omega=0$ the analyticity of the semigroup $e^{t\mathbb{A}_0}$ occurs when $\sigma=1$ and for the case $\omega>0$ the semigroup $e^{t\mathbb{A}_\omega}$ is analytic for the parameter $\sigma\in[5/4, 3/2]$. The abstract system is given by: \begin{equation}\label{Eq1.10} \left\{\begin{array}{c} u_{tt}+\omega Au_{tt}+A^2u-A^\sigma\theta=0,\\ \theta_t+A\theta+A^\sigma u_t=0. \end{array}\right. \end{equation} where $\omega\geq 0$.