New notion of nonuniform exponential dichotomy with applications to the theory of pullback and forward attractors

TL;DR

This paper introduces a new concept of nonuniform exponential dichotomy, explores its properties, and applies it to establish the existence of pullback and forward attractors in nonautonomous differential equations.

Contribution

It defines a novel form of nonuniform exponential dichotomy, analyzes its relation to existing notions, and demonstrates its robustness and applications to attractors in differential equations.

Findings

New nonuniform exponential dichotomy concept introduced

Established robustness under perturbations

Applied to existence of pullback and forward attractors

Abstract

In this work we study nonuniform exponential dichotomies and existence of pullback and forward attractors for evolution processes associated to nonautonomous differential equations. We define a new concept of nonuniform exponential dichotomy, for which we provide several examples, study the relation with the standard notion, and establish a robustness under perturbations. We provide a dynamical interpretation of admissibility pairs related with exponential dichotomies to obtain existence of pullback and forward attractors. We apply these abstract results for ordinary and parabolic differential equations.