Analyticity and stability results for a plate‐membrane type transmission problem

Abstract In this paper, we consider a transmission problem for a system of a thermoelastic plate with (or without) a rotational inertia coupled with a membrane. On the plate a structural damping may or may not act, and on the membrane a damping of Kelvin–Voigt type may or may not be present; free boundary operators for the plate are considered in the coupling with the membrane. We prove well‐posedness of the problem and higher regularity of the solution. Depending on the damping and on the presence of the rotational term, we establish strong stability, exponential stability, lack of exponential stability, polynomial stability, and analyticity of the semigroup associated to the transmission problem.