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

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 $(S(t){)}_{t\ge 0}$ is of Gevrey class $\delta >24$ for $t>0$, hence immediately differentiable. Moreover, we show that $(S(t){)}_{t\ge 0}$ is exponentially stable.