The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation

Abstract. In this paper we consider the transmission problem of a material composed of three components; one of them is a Kelvin–Voigt viscoelastic material, the second is an elastic material (no dissipation), and the third is an elastic material inserted with a frictional damping mechanism. The main result of this paper is that the rate of decay will depend on the position of each component. When the viscoelastic component is not in the middle of the material, then there exists exponential stability of the solution. Instead, when the viscoelastic part is in the middle of the material, there is not exponential stability. In this case we show that the corresponding solution decays polynomially as 1/t2. Moreover we show that the rate of decay is optimal over the domain of the infinitesimal generator. Finally, using a second order scheme that ensures the decay of energy (Newmark-β method), we give some numerical examples which demonstrate this asymptotic behavior.