Long-Time Behavior for a Class of Semi-linear Viscoelastic Kirchhoff Beams/Plates

Abstract

This is a complementation work of the paper referred in Jorge Silva, Munoz Rivera and Racke (Appl Math Optim 73:165–194, 2016) where the authors proposed a semi-linear viscoelastic Kirchhoff plate model. While in [28] it is presented a study on well-posedness and energy decay rates in a historyless memory context, here our main goal is to consider the problem in a past history framework and then analyze its long-time behavior through the corresponding autonomous dynamical system. More specifically, our results are concerned with the existence of finite dimensional attractors as well as their intrinsic properties from the dynamical systems viewpoint. In addition, we also present a physical justification of the model under consideration. Hence, our new achievements complement those established in [28] to the case of memory in a history space setting and extend the results in Jorge Silva and Ma (IMA J Appl Math 78:1130–1146, 2013, J Math Phys 54:021505, 2013) to the case of dissipation only given by the memory term.