Asymptotic behavior of the Rao–Nakra sandwich beam model with Kelvin–Voigt damping

The Rao–Nakra sandwich beam is a coupled system consisting of two wave equations for the longitudinal displacements of the top and bottom layers and an Euler–Bernoulli beam equation for the transversal displacement. This paper concerns the system’s stability when the Kelvin–Voigt damping terms act on the first and third equations. Using the semigroup theory of linear operators, we prove the global well-posedness of the associated initial boundary value problem. And then, we prove the lack of exponential stability of the system. Because of this lack of exponential stability, we study the polynomial stability and prove that the system decays with rate t^1/4. We further prove that this decay rate is optimal.