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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Gong Chen and Jacek Jendrej
Journal: Trans. Amer. Math. Soc. 372 (2019), 7461-7496
MSC (2010): Primary 35B40; Secondary 37D99
Published electronically: August 28, 2019
MathSciNet review: 4024558
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Abstract: We give a sufficient condition for the 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 sublight speed with small accelerations.

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Additional Information

Gong Chen
Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada
MR Author ID: 1003678

Jacek Jendrej
Affiliation: CNRS and Université Paris 13, LAGA, UMR 7539, 99 avenue J.-B. Clément, 93430 Villetaneuse, France
MR Author ID: 1060238

Keywords: Exponential dichotomy, Lyapunov functional, time-dependent potential
Received by editor(s): January 2, 2019
Received by editor(s) in revised form: May 30, 2019
Published electronically: August 28, 2019
Additional Notes: Part of this work was completed when the second author was visiting the University of Chicago Mathematics Department and the University of Toronto Mathematics Department. He was also partially supported by the ANR-18-CE40-0028 project ESSED
The authors would like to thank the Beijing International Center for Mathematical Research, where this work was finished.
Article copyright: © Copyright 2019 American Mathematical Society