Spectral comparison of compound cocycles generated by delay equations in Hilbert spaces

TL;DR

This paper develops analytical methods to compare spectral properties of cocycles generated by delay equations in Hilbert spaces, with implications for stability analysis in nonlinear PDEs.

Contribution

It introduces new techniques for spectral comparison of compound cocycles and semigroups, enhancing stability analysis in delay differential equations.

Findings

Established criteria for absence of invariant contours on attractors

Provided robust conditions for global stability

Linked spectral properties to PDE regularity

Abstract

We study linear cocycles generated by nonautonomous delay equations in a suitable Hilbert space and their extensions, called compound cocycles, to exterior powers. Using a recent version of the frequency theorem, we develop analytical techniques for comparing spectral properties, such as uniform exponential dichotomies, between such cocycles and semigroups generated by stationary equations. These methods are based on properties related to regularity and structure in PDEs associated with delay equations. In particular, the developed machinery leads to effective robust criteria that guarantee the absence of closed invariant contours on global attractors arising in nonlinear problems and are expected to ensure global stability.