Uniform energy estimates for a semilinear evolution equation of the Mindlin-Timoshenko beam with memory

In this paper, we investigate a mathematical model for viscoelastic beams, based on the Mindlin-Timoshenko assumptions. The resulting constitutive equations are derived in the framework of well-established theory of linear viscoelasticity, and according to the approximation procedure due to Lagnese for the Kirchhoff viscoelastic beams and plates. Assuming a nonlinear body force acting on the beam, we show that this model generates a strongly continuous semigroup which acts on the appropriate phase space. Uniform energy estimates are then given. The existence of an absorbing set for the solution of the problem is also studied. Furthermore, we remark the necessary conditions to extend this approach to the study of the longtime behavior of the Mindlin-Timoshenko plates.