Continuity and averaging for parabolic evolution systems

TL;DR

This paper extends averaging principles to non-autonomous parabolic evolution equations, including nonlinear cases, broadening the applicability of Henry's averaging technique to time-dependent linear operators and dissipative equations.

Contribution

It introduces a generalized averaging method for non-autonomous parabolic systems, covering both linear and nonlinear cases, and extends previous results to more complex time-dependent operators.

Findings

Established averaging principle for non-autonomous linear parabolic equations.

Extended averaging results to nonlinear perturbations.

Broadened the class of equations where averaging applies.

Abstract

Averaging principle for abstract non-autonomous parabolic evolution equations governed by time-dependent family of positive sectorial operators is proved. Apart from linear case also a nonlinear version for continuous perturbations is provided. These results extend Henry's averaging technique to the case where linear part of the equation depends on time and results due to Ilyin that were for dissipative equations.47D06, 34G05, 34G20