Regularity to Timoshenko's System with Thermoelasticity of Type III with Fractional Damping

• Published in: arXiv.org 2023-08
• Main Authors: Filomena Barbosa Rodrigues Mendes, Lesly Daiana Barbosa Sobrado, Fredy Maglorio Sobrado Suárez
• Format: Article
• Language: English
• Subjects: Conduction heating; Conductive heat transfer; Mathematical models; Regularity; Thermoelasticity; Timoshenko beams; Viscous damping
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2211.10816

The article, presents the study of the regularity of two thermoelastic beam systems defined by the Timoshenko beam model coupled with the heat conduction of Green-Naghdiy theory of type III, both mathematical models are differentiated by their coupling terms that arise as a consequence of the constitutive laws initially considered. The systems presented in this work have 3 fractional dampings: \(\mu_1(-\Delta)^\tau \phi_t\), \(\mu_2(-\Delta)^\sigma \psi_t\) and \(K(-\Delta)^\xi \theta_t\), where \(\phi,\psi\) and \(\theta\) are transverse displacement, rotation angle and empirical temperature of the bean respectively and the parameters \((\tau,\sigma,\xi)\in [0,1]^3\). It is noted that for values 0 and 1 of the parameter \(\tau\), the so-called frictional or viscous damping will be faced, respectively. The main contribution of this article is to show that the corresponding semigroup \(S_i(t)=e^{\mathcal{B}_it}\), with \(i=1,2\), is of Gevrey class \(s>\frac{r+1}{2r}\) for \(r=\min \{\tau,\sigma,\xi\}\) for all \((\tau,\sigma,\xi )\in R_{CG}:= (0, 1)^3\). It is also showed that \(S_1(t)=e^{\mathcal{B}_1t}\) is analytic in the region \(R_{A_1}:=\{(\tau,\sigma, \xi )\in [\frac{1}{2},1]^3\}\) and \(S_2(t)=e^{\mathcal{B}_2t}\) is analytic in the region \(R_{A_2}:=\{(\tau,\sigma, \xi )\in [\frac{1}{2},1]^3/ \tau=\xi\}\).