Long-time dynamics in plate models with strong nonlinear damping

TL;DR

This paper investigates the long-term behavior of nonlinear damped plate models, proving the existence of finite-dimensional attractors and exponential convergence to equilibria, with applications to various nonlinear wave equations.

Contribution

It establishes the existence of finite-dimensional and exponential attractors for nonlinear plate models with strong damping, using novel compactness and stability methods.

Findings

Existence of a compact finite-dimensional attractor.

Existence of a fractal exponential attractor.

Trajectories converge exponentially to equilibria under certain conditions.

Abstract

We study long-time dynamics of a class of abstract second order in time evolution equations in a Hilbert space with the damping term depending both on displacement and velocity. This damping represents the nonlinear strong dissipation phenomenon perturbed with relatively compact terms. Our main result states the 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 existence of a finite number of determining functionals. In the case when the set of equilibria is finite and hyperbolic we show that every trajectory is attracted by some equilibrium with exponential rate. Our arguments involve a recently developed method based on the "compensated" compactness and quasi-stability estimates. As an application we consider the nonlinear Kirchhoff, Karman and Berger plate models with different types of boundary conditions and strong damping terms. Our results can be also applied to the nonlinear wave equations.