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

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