Regularity for the Timoshenko system with fractional damping

Abstract

We study, the Regularity of the Timoshenko system with two fractional dampings $(-\Delta {)}^{\tau }{u}_t$ and $(-\Delta {)}^{\sigma }{\psi }_t$; both of the parameters $(\tau ,\sigma )$ vary in the interval $[0,1]$. We note that ($\tau =0$ or $\sigma =0$) and ($\tau =1$ or $\sigma =1$) the dampings are called frictional and viscous, respectively. Our main contribution is to show that the corresponding semigroup $S(t)=e^{\mathcal{B}t}$, is analytic for $(\tau ,\sigma )\in R_A:=[1/2,1]\times [1/2,1]$ and determine the Gevrey's class $\nu >\frac{1}{\varphi }$, where $\varphi =\{\begin{array}{ccc}\frac{2\sigma }{\sigma +1}& \mathrm{for}& \sigma \le \tau ,\\ & & \\ \frac{2\tau }{\tau +1}& \mathrm{for}& \tau \le \sigma .\end{array}$ \quad and \quad $(\tau ,\sigma )\in R_{CG}:=(0,1{)}^2$.

17 pages. arXiv admin note: text overlap with arXiv:2211.10816