Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

TL;DR

This paper investigates the stability and robustness of nonuniform hyperbolicity in infinite-dimensional differential equations, establishing conditions for persistence, uniqueness, and continuous dependence of hyperbolic structures under perturbations.

Contribution

It provides new results on the stability, uniqueness, and persistence of nonuniform hyperbolic structures in Banach space evolution processes, including nonlinear cases.

Findings

Nonuniform exponential dichotomy is stable under perturbations.

Conditions for uniqueness and continuous dependence of projections are established.

Persistence of nonuniform hyperbolic solutions in nonlinear systems is proven.

Abstract

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.