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

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