Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface

TL;DR

This paper investigates the stability of a star-shaped elastic string network with local Kelvin-Voigt damping, showing polynomial decay rates influenced by damping coefficient singularities near the interface.

Contribution

It establishes polynomial stability for the system with singular damping coefficients, improving previous decay rate results for wave equations with similar damping.

Findings

Semigroup is polynomially stable.

Decay rates depend on degeneracy speed.

Improves previous decay rate estimates.

Abstract

In this paper, we study the stability problem of a star-shaped network of elastic strings with a local Kelvin-Voigt damping. Under the assumption that the damping coefficients have some singularities near the transmission point, we prove that the semigroup corresponding to the system is polynomially stable and the decay rates depends on the speed of the degeneracy. This result improves the decay rate of the semigroup associated to the system on an earlier result of Z.~Liu and Q.~Zhang in \cite{LZ} involving the wave equation with local Kelvin-Voigt damping and non-smooth coefficient at interface.