Uniform stability of a laminated beam with structural damping and second sound

Well‐posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks

“…For stability of laminated Timoshenko beams with structural memory, one can refer to the previous studies. () For laminated Timoshenko beams with second sound, one can find a stability result in Apalara …”

Lack of Exponential Decay for a Laminated Beam with Structural Damping and Second Sound

“…In previous paper [1], Apalara considered the following laminated beam with structural damping and second sound: Here w(x, t) denotes the transverse displacement of the beam which departs from its equilibrium position, ψ(x, t) represents the rotation angle, s(x, t) is proportional to the amount of slip along the interface at time t and longitudinal spatial variable x, 3s − ψ denotes the effective rotation angle and the third equation of (1.1) describes the dynamics of the slip, θ(x, t) is the difference temperature, q(x, t) is the heat flux. The coefficients ρ, G, I ρ , D, γ, β > 0 denote the density of the beams, the shear stiffness, the mass moment of inertia, the flexural rigidity, the adhesive stiffness of the beams, and the adhesive damping parameter, respectively.…”

“…As for the previous results and developments of laminated beam, the authors have stated and summarized in great detail in [1], thus we just omit it here. The readers, for a better understanding of present work, are strongly recommended to see [1] and the references therein. For the lack of exponential decay, there are some results in recent time.…”

“…The aim of this paper is to give a positive answer to the open problem proposed by Apalara [1], that is we will prove the system is not exponentially stable unless stability number…”

On the Stability of a Laminated Beam With Structural Damping and Gurtin-Pipkin Thermal Law

“…The stabilization of system (1.4)-(1.5) has been analyzed in [1]. There, Apalara obtained the well-posedness and uniform stability results depending on the following stability number…”