Stabilization for Euler-Bernoulli beam equation with a local degenerated Kelvin-Voigt damping

TL;DR

This paper studies the stabilization of an Euler-Bernoulli beam with a localized Kelvin-Voigt damping that degenerates near a point, proving polynomial decay rates that depend on the degeneracy speed.

Contribution

It establishes polynomial stability for the beam with a degenerated damping coefficient, linking decay rates to the degeneracy parameter.

Findings

Semigroup is polynomially stable.

Decay rate depends on degeneracy speed α.

Damping effectiveness varies near the degeneracy point.

Abstract

We consider the Euler-Bernoulli beam equation with a local Kelvin-Voigt dissipation type in the interval $(-1,1)$. The coefficient damping is only effective in $(0,1)$ and is degenerating near the $0$ point with a speed at least equal to $x^{\alpha}$ where $\alpha\in(0,5)$. We prove that the semigroup corresponding to the system is polynomially stable and the decay rate depends on the degeneracy speed $\alpha$.