Stability of a String with Local Kelvin--Voigt Damping and Nonsmooth Coefficient at Interface

Abstract

In this paper, we study the stability of an elastic string system with local Kelvin--Voigt damping. Few results for this system are known in the literature. Under the assumption that the damping coefficient has a singularity at the interface of the damped and undamped regions and behaves like $x^\alpha$ near the interface, we prove that the semigroup corresponding to the system is polynomially or exponentially stable and the decay rate depends on the parameter $\alpha\in(0,1]$.