Special Session 13: Asymptotic Behavior of PDEs

The talk will report on recent progress in the study of the long-term behaviour of some composite systems of Partial Differential Equations (PDE) which arise in the modeling of fluid/structure interactions. Such PDE systems comprise a wave equation in a three dimensional bounded domain and an elastic (or thermoelastic) plate equation acting on a portion of its boundary. Previous work by the author (with Igor Chueshov and Irena Lasiecka) established, in particular, the existence of a global attractor of finite fractal dimension for a coupled wave/plate model with interior nonlinear dissipation and nonlinear perturbations acting on either component of the system. Significant recent advances in the study of wave equations made it possible to show that the aforementioned results can be actually improved, allowing a localized—rather than fully interior—wave damping. It is important to emphasize that the major difficulties come from the combination of nonlinear localized dissipation and (nonlinear) “critical” restoring forces. The main part of the talk will focus on results jointly obtained with Daniel Toundykov (University of Nebraska-Lincoln).