Optimal stability results for laminated beams with Kelvin-Voigt damping and delay

Asymptotic behavior of Timoshenko beams with delayed viscoelasticity acting on the shear force

“…Therefore the polynomial decay is obtained with rate t −1/2 . In [2,12], the authors obtained a similar result for the laminated beam system. Motivated by the works [2,7,8,12], we wonder what the asymptotic behavior of the Timoshenko system would be, considering a delayed viscoelastic damping acting only on the shear force, that is, we refer to the following dissipation mechanisms…”

“…On the other hand, models governed by partial differential equations that take into account some delay in time are generally more realistic from a physical point of view than those that do not consider the effect of delay. In the literature, we can find several results [2,7,10,11,12,15,16] on the well-posedness and the asymptotic behavior of solutions of systems that present time delay. In this context, we highlight the model of Timoshenko beams with delayed viscoelastic damping introduced by Makheloufi et al [7]…”

“…2 H . Consequently, for |λ| large enough we have1 λ 2 ∥U ∥ H ≤ C∥F ∥ H .From resolvent equation we obtain U = (iλI − A) −1 F , and therefore, 1λ 2 ∥(iλI − A) −1 ∥ L(H) ≤ C ⇒ ∥(iλI − A) −1 ∥ L(H) = O(|λ| 2), which implies that the condition of the Theorem (3.3) holds for α = 2.…”

Polynomial stability for Lord–Shulman porous elasticity with microtemperature and strong time delay

“…In Ref. [23], Cabanillas et al obtained an analogous result for the laminated beam system. Motivated by the works mentioned above, we remove the temperature from Equation (4) assuming 𝛽 0 = 𝛽 1 = 𝑘 1 = 0 we studied the following system with strong time delay in (𝑥, 𝑡) ∈ (0, 𝑙) × (0, ∞), 𝜌 0 𝑢 𝑡𝑡 − 𝜇𝑢 𝑥𝑥 − 𝑏𝜙 𝑥 = 0, 𝜌 0 𝜅𝜙 𝑡𝑡 − 𝛼𝜙 𝑥𝑥 + 𝑏𝑢 𝑥 + 𝜉𝜙 + 𝑑(𝜃 𝑥 + 𝜏 0 𝜃 𝑥𝑡 ) − 𝛾 1 𝜙 𝑥𝑥𝑡 − 𝛾 2 𝜙 𝑥𝑥𝑡 (𝑥, 𝑡 − 𝜏) = 0, 𝑏 0 (𝜃 𝑡 + 𝜏 0 𝜃 𝑡𝑡 ) − 𝜅 2 𝜃 𝑥𝑥 + 𝑑𝜙 𝑥𝑡 + 𝜅 3 𝜃 = 0, (6) where 𝜏 > 0 is the time delay and the above constitutive coefficients satisfy 𝜌 0 > 0, 𝜇 > 0, 𝑏 ≠ 0, 𝜅 > 0, 𝛼 > 0, 𝜉 > 0, 𝛾 1 > 0, 𝛾 2 ≠ 0, 𝑑 ≠ 0, 𝜏 0 > 0, 𝑏 0 > 0, 𝜅 2 > 0, 𝜅 3 > 0 and 𝑏 2 < 𝜇𝜉.…”