Stabilization of Coupled Wave Equations with Viscous Damping on Cylindrical and Non-regular Domains: Cases Without the Geometric Control Condition

Abstract

In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $$t^{-\frac{1}{2}}$$ if the two waves have the same speed of propagation, and with rate $$t^{-\frac{1}{3}}$$ if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order $$t^{-1}$$ .