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

• Published in: arXiv.org 2023-04
• Main Authors: Fredy Maglorio Sobrado Suárez, Lesly Daiana Barbosa Sobrado
• Format: Article
• Language: English
• Subjects: Arrays; Couplings; Linear operators; Mathematical analysis; Parameters; Plates; Regularity; Semigroups
• ISSN: 2331-8422

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 \(\omega\geq 0\).