Finite-dimensionality of attractors for wave equations with degenerate nonlocal damping

TL;DR

This paper proves that the global attractors for certain wave equations with degenerate nonlocal damping have finite fractal dimension, despite challenges posed by degeneracy at specific points.

Contribution

It introduces a novel approach to analyze the attractor dimension near degenerate points, establishing finite dimensionality where traditional methods fail.

Findings

Finite fractal dimension of attractors established.

New method for analyzing degeneracy in wave equations.

Attractors remain finite-dimensional despite degeneracy.

Abstract

In this paper we study the fractal dimension of global attractors for a class of wave equations with (single-point) degenerate nonlocal damping. Both the equation and its linearization degenerate into linear wave equations at the degenerate point and the usual approaches to bound the dimension of the entirety of attractors do not work directly. Instead, we develop a new process concerning the dimension near the degenerate point individually and show the finite dimensionality of the attractor.