The Gevrey class of the Euler-Bernoulli beam model with singularities

TL;DR

This paper analyzes a composite Euler-Bernoulli beam with singularities and Kelvin-Voigt damping, demonstrating that the associated semigroup is Gevrey class 4, leading to exponential stability and smoothing effects.

Contribution

It proves the semigroup is immediately differentiable and of Gevrey class 4 for a beam with singularities and Kelvin-Voigt damping, establishing stability and smoothing properties.

Findings

Semigroup is of Gevrey class 4

Model exhibits exponential stability

Initial data smoothing effect

Abstract

We study the Euler-Bernoulli beam model with singularities at the points $x=\xi_1$, $x=\xi_2$ and with localized viscoelastic dissipation of Kelvin-Voigt type. We assume that the beam is composed by two materials; one is an elastic material and the other one is a viscoelastic material of Kelvin-Voigt type. Our main result is that the corresponding semigroup is immediately differentiable and also of Gevrey class $4$. In particular, our result implies that the model is exponentially stable, has the linear stability property, and the smoothing effect property over the initial data.