Well-posedness and polynomial decay for Rao–Nakra sandwich beam with memory and Kelvin–Voigt damping

Abstract

Abstract In this paper, we investigate the well-posedness and long-term behavior of a Rao–Nakra sandwich beam model with two different damping mechanisms. The top layer is subjected to memory-type damping, described by a relaxation kernel that satisfies standard positivity and decay assumptions, while one of the other two layers is subjected to Kelvin–Voigt-type internal damping. First, using semigroup theory, we prove the well-posedness of the associated initial-boundary value problem. Next, using a frequency domain approach and resolvent estimates, we prove that if Kelvin-Voigt dissipation were effective in the core and bottom layers, the system would be exponentially stable. Conversely, if only one of the Kelvin-Voigt dissipations is effective, exponential stability fails. In the latter case, we establish sharp polynomial decay rates for the associated semigroup. These results highlight the delicate interaction between memory-type damping and Kelvin–Voigt damping in the stabilization of Rao–Nakra sandwich beams.