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

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.