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

Abstract

In this work, we use the theory of semigroups to study the well-posedness and the asymptotic behavior of the Timoshenko beam system with delayed viscoelastic damping acting only on the shear force. This system is different from all others related to time delay terms. We use the Lumer-Phillips Theorem to prove the well-posedness of the problem and then use the contrapositive of the Gearhart-Prüss-Huang Theorem to prove the lack of exponential decay independent of any relationship between the velocities. Finally, we use the Borichev-Tomilov Theorem for optimal rate polynomial decay.