Averaging principle and hyperbolic evolution equations

TL;DR

This paper develops an averaging principle for nonlinear evolution equations with almost periodic perturbations, extending classical results to hyperbolic equations and providing a framework for analyzing their long-term behavior.

Contribution

It generalizes Henry's averaging results to hyperbolic evolution equations, broadening the applicability of averaging methods in nonlinear dynamics.

Findings

Averaging principle established for hyperbolic evolution equations.

Main hypothesis verified for Kato's hyperbolic equations.

Extension of classical averaging results to a broader class of equations.

Abstract

An averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It generalizes classical Henry's results for perturbations of sectorial operators on fractional spaces. It is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by Kato.