Strong Uniform Attractors for Non-Autonomous Dissipative PDEs with non translation-compact external forces

TL;DR

This paper studies strong uniform attractors for non-autonomous dissipative PDEs with external forces that are not translation-compact, introducing new classes of forces and a unified energy-based approach applicable to various physical equations.

Contribution

It introduces new classes of non-translation-compact external forces and develops a unified energy method to verify asymptotic compactness in such systems.

Findings

Established strong attraction in non-translation-compact cases

Developed a unified energy approach for asymptotic compactness

Applied methods to Navier-Stokes, wave, and reaction-diffusion equations

Abstract

We give a comprehensive study of strong uniform attractors of non-autonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces which are not translation compact, but nevertheless allow to verify the attraction in a strong topology of the phase space and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.