Lyapunov-type characterisation of exponential dichotomies with applications to the heat and Klein-Gordon equations

TL;DR

This paper establishes a Lyapunov-type criterion for exponential dichotomies in linear dynamical systems within Banach spaces, with applications to heat and Klein-Gordon equations involving time-varying potentials.

Contribution

It introduces a new Lyapunov-based condition for exponential dichotomies applicable to non-invertible systems and demonstrates its utility in PDEs with moving potentials.

Findings

Established a sufficient Lyapunov-type condition for exponential dichotomies.

Applied the criterion to backward heat equations with time-varying potentials.

Extended the approach to Klein-Gordon equations with moving potentials.

Abstract

We give a sufficient condition for existence of an exponential dichotomy for a general linear dynamical system (not necessarily invertible) in a Banach space, in discrete or continuous time. We provide applications to the backward heat equation with a potential varying in time and to the heat equation with a finite number of slowly moving potentials. We also consider the Klein-Gordon equation with a finite number of potentials whose centres move at sub-light speed with small accelerations.