Regularity for the Timoshenko system with fractional damping

TL;DR

This paper investigates the regularity and analyticity of the semigroup associated with the Timoshenko system with fractional damping, identifying parameter regions where the system exhibits specific regularity properties.

Contribution

It establishes the analyticity of the semigroup for fractional damping parameters in a specific region and determines the Gevrey class for the system.

Findings

Semigroup is analytic for ( au,\sigma) in [1/2,1] x [1/2,1]

Gevrey class determined based on damping parameters

Regularity results depend on fractional damping parameters

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 $\nu>\dfrac{1}{\phi}$, where $\phi=\left\{\begin{array}{ccc} \dfrac{2\sigma}{\sigma+1} &{\rm for} & \sigma\leq \tau,\\\\ \dfrac{2\tau}{\tau+1} &{\rm for} & \tau\leq \sigma. \end{array}\right.$ \quad and \quad $(\tau,\sigma)\in R_{CG}:= (0,1)^2$.