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}+ωAu_{tt}+A^2u-A^σθ=0,\\ θ_t+Aθ+A^σu_t=0. \end{array}\right. \end{equation} where $\omega \ge 0$.

arXiv comment30 pages. arXiv admin note: text overlap with arXiv:2208.01481