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

TL;DR

This paper investigates the regularity and analyticity of semigroups associated with fractional damping in Timoshenko thermoelastic systems of type III, revealing conditions for Gevrey class and analyticity based on damping parameters.

Contribution

It establishes the Gevrey class regularity and analyticity regions for semigroups of fractional damped Timoshenko systems with thermoelasticity of type III, extending understanding of damping effects.

Findings

Semigroup $S_i(t)$ is of Gevrey class $s>rac{r+1}{2r}$ for $r= ext{min}\{ au,\sigma,\xi ext{}$.

$S_1(t)$ is analytic in $[rac{1}{2},1]^3$ region.

$S_2(t)$ is analytic in $[rac{1}{2},1]^3$ with $ au=\xi$.

Abstract

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 $R_{A_2}:=\{(\tau,\sigma, \xi )\in [\frac{1}{2},1]^3/ \tau=\xi\}$.