Attractors for Second-Order Evolution Equations with a Nonlinear Damping

Abstract

We study long-time dynamics of abstract nonlinear second-order evolution equations with a nonlinear damping. Under suitable hypotheses we prove existence of a compact global attractor and finiteness of its fractal dimension. We also show that any solution is stabilized to an equilibrium and estimate the rate of the convergence which, in turn, depends on the behaviour at the origin of the function describing the dissipation. If the damping is bounded below by a linear function, this rate is exponential. Our approach is based on far reaching generalizations of the Ceron–Lopes theorem on asymptotic compactness and Ladyzhenskaya’s theorem on the dimension of invariant sets. An application of our results to nonlinear damped wave and plate equations allow us to obtain new results pertaining to structure and properties of global attractors for nonlinear waves and plates.