Finite Fractal Dimension of uniform attractors for non-autonomous dynamical systems with infinite dimensional symbol space

TL;DR

This paper establishes an upper bound for the fractal dimension of uniform attractors in non-autonomous systems without requiring finite-dimensional symbol spaces, using semi-continuity conditions.

Contribution

It introduces a novel approach to bounding attractor dimensions without finite-dimensional symbol spaces, based on semi-continuity and exponential attractors.

Findings

Finite box-counting dimension of attractors despite infinite-dimensional symbol space

New semi-continuity conditions ensure attractor dimension bounds

Application to reaction-diffusion equations with specific forcing terms

Abstract

The aim of this paper is to find an upper bound for the box-counting dimension of uniform attractors for non-autonomous dynamical systems. Contrary to the results in literature, we do not ask the symbol space to have finite box-counting dimension. Instead, we ask a condition on the semi-continuity of pullback attractors of the system as time goes to infinity. This semi-continuity can be achieved if we suppose the existence of finite-dimensional exponential uniform attractors for the limit symbols. After showing these new results, we apply them to study the box-counting dimension of the uniform attractor for a reaction-diffusion equation, and we find a specific forcing term such that the symbol space has infinite box-counting dimension but the uniform attractor has finite box-counting dimension anyway.