Existence, Uniqueness, and Polynomial Decay Results for a Laminated Beam with a Logarithmic Damping

In this paper, we investigate a laminated beam model incorporating a nonlinear logarithmic damping mechanism. We first establish the global existence and uniqueness of weak solutions by means of the Faedo–Galerkin approximation method. The proof relies on uniform a priori estimates, compactness arguments, and standard tools from functional analysis, together with suitable properties of logarithmic functions. Furthermore, under the assumption of equal propagation speeds, we analyze the long-time behavior of solutions and prove a polynomial decay rate for the associated energy. The decay result is obtained by constructing an appropriate Lyapunov functional and combining it with energy methods and differential inequality techniques. Our results extend and improve several related works in the literature by addressing a laminated beam system with logarithmic damping, for which the analysis requires refined estimates due to the nonstandard nature of the dissipation.