Smooth attractors of finite dimension for von Karman evolutions with nonlinear frictional damping localized in a boundary layer

TL;DR

This paper proves the existence of smooth, finite-dimensional attractors for von Karman plate equations with localized nonlinear damping, demonstrating asymptotic stabilization of complex hyperbolic-like flows.

Contribution

It establishes the existence and smoothness of finite-dimensional global attractors for nonlinear von Karman plates with boundary-localized damping, advancing understanding of long-term dynamics.

Findings

Existence of a compact global attractor for the system

The attractor is shown to be smooth and finite dimensional

Hyperbolic-like flows stabilize asymptotically to the attractor

Abstract

In this paper dynamic von Karman equations with localized interior damping supported in a boundary collar are considered. Hadamard well-posedness for von Karman plates with various types of nonlinear damping are well-known, and the long-time behavior of nonlinear plates has been a topic of recent interest. Since the von Karman plate system is of "hyperbolic type" with critical nonlinearity (noncompact with respect to the phase space), this latter topic is particularly challenging in the case of geometrically constrained and nonlinear damping. In this paper we first show the existence of a compact global attractor for finite-energy solutions, and we then prove that the attractor is both smooth and finite dimensional. Thus, the hyperbolic-like flow is stabilized asymptotically to a smooth and finite dimensional set. Key terms: dynamical systems, long-time behavior, global attractors, nonlinear plates, nonlinear damping, localized damping