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Reaction–Diffusion Problems on Time-Periodic Domains

Jane Allwright

Journal of Dynamics and Differential Equations

Swansea University Author: Jane Allwright

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DOI (Published version): 10.1007/s10884-023-10308-9

Abstract

Reaction-diffusion equations are studied on bounded, time-periodic domains with zero Dirichlet boundary conditions. The long-time behaviour is shown to depend on the principal periodic eigenvalue of a transformed periodic-parabolic problem. We prove upper and lower bounds on this eigenvalue under a...

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Published in: Journal of Dynamics and Differential Equations
ISSN: 1040-7294 1572-9222
Published: Springer Science and Business Media LLC 2023
Online Access: Check full text

URI: https://cronfa.swan.ac.uk/Record/cronfa64113
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Abstract: Reaction-diffusion equations are studied on bounded, time-periodic domains with zero Dirichlet boundary conditions. The long-time behaviour is shown to depend on the principal periodic eigenvalue of a transformed periodic-parabolic problem. We prove upper and lower bounds on this eigenvalue under a range of different assumptions on the domain, and apply them to examples. The principal eigenvalue is considered as a function of the frequency, and results are given regarding its behaviour in the small and large frequency limits. A monotonicity property with respect to frequency is also proven. A reaction-diffusion problem with a class of monostable nonlinearity is then studied on a periodic domain, and we prove convergence to either zero or a unique positive periodic solution.
Keywords: Time-periodic domain · Principal periodic eigenvalue · Reaction–diffusion
College: Faculty of Science and Engineering
Funders: The author is grateful for an EPSRC-funded studentship: EPSRC DTP grant EP/R51312X/1, and research associate funding: EP/W522545/1.

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