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

Gong Chen; Jacek Jendrej

doi:10.1090/tran/7923

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

Gong Chen, Jacek Jendrej

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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.

Original language	English
Pages (from-to)	7461-7496
Number of pages	36
Journal	Transactions of the American Mathematical Society
Volume	372
Issue number	10
DOIs	https://doi.org/10.1090/tran/7923
State	Published - Nov 15 2019

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2019 American Mathematical Society

Keywords

Exponential dichotomy
Lyapunov functional
Time-dependent potential

ASJC Scopus subject areas

Mathematics (all)
Applied Mathematics

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10.1090/tran/7923

Cite this

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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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N2 - 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.

AB - 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.

KW - Exponential dichotomy

KW - Lyapunov functional

KW - Time-dependent potential

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

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

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