Gevrey regularity for the Euler-Bernoulli beam equation with localized structural damping

TL;DR

This paper proves that a Euler-Bernoulli beam equation with localized damping generates a Gevrey class semigroup, which is immediately differentiable and exponentially stable, indicating strong regularity and stability properties.

Contribution

The paper establishes Gevrey regularity and exponential stability for the semigroup associated with a damped Euler-Bernoulli beam equation with localized damping.

Findings

The semigroup is of Gevrey class $ ext{delta}>24$ for $t>0$.

The semigroup is exponentially stable.

The results demonstrate immediate differentiability of the semigroup.

Abstract

We study a Euler-Bernoulli beam equation with localized discontinuous structural damping. As our main result, we prove that the associated $C_0$-semigroup $(S(t))_{t\geq0}$ is of Gevrey class $\delta>24$ for $t>0$, hence immediately differentiable. Moreover, we show that $(S(t))_{t\geq0}$ is exponentially stable.