Long-time dynamics in plate models with strong nonlinear damping

Abstract

We study long-time dynamics of a class of abstract second order in time evolution equationsin a Hilbert space with the damping term depending both on displacementand velocity. This damping represents the nonlinear strong dissipationphenomenon perturbed with relatively compact terms.Our main result statesthe existence of a compact finite dimensional attractor.We study properties of this attractor.We also establish the existence of a fractal exponential attractor and give the conditions that guarantee the existenceof a finite number of determining functionals.In the case when the set of equilibria is finite and hyperbolicwe show that every trajectory is attracted by some equilibriumwith exponential rate.Our arguments involve a recentlydeveloped method based on the 'compensated' compactness andquasi-stability estimates. As an application we consider thenonlinear Kirchhoff, Karman and Berger plate models with differenttypes of boundary conditions and strong damping terms. Our resultscan be also applied to the nonlinear wave equations.