A transmission problem of a system of weakly coupled wave equations with Kelvin–Voigt dampings and non-smooth coefficient at the interface

Abstract

The purpose of this work is to investigate the stabilization of a system of weakly coupled wave equations with one or two locally internal Kelvin–Voigt damping and non-smooth coefficient at the interface. The main novelty in this paper is that the considered system is a coupled system and that the geometrical situations covered (see Remarks 5.6, 5.12) are richer than all previous results, even for simple wave equation with Kelvin–Voigt damping. Firstly, using a unique continuation result, we prove that the system is strongly stable. Secondly, we show that the system is not always exponentially stable, instead, we establish some polynomial energy decay estimates. Further, we prove that a polynomial energy decay rate of order $$t^{-1/2}$$ is optimal in some sense.