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

Abstract: We study the Euler-Bernoulli beam model with singularities at the points $x=ξ_1$, $x=ξ_2$ 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.