Exponential stability for laminated beams with intermediate damping

In this manuscript, we consider a system composed of two identical Timoshenko beams joined by an adhesive layer of negligible thickness, producing an interfacial slip. We introduce a dissipative mechanism not previously considered given by α(- Δ) θut, 0 ≤ θ≤ 1 , α> 0 , which includes the frictional damping case (θ= 0) and the Kelvin–Voigt damping case (θ= 1). Using the semigroup theory of linear operators, we show the well-posedness of the system and demonstrate that its solutions decay exponentially.