Reaction-Diffusion Problems on Time-Periodic Domains
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.
Cite
@article{arxiv.2210.11516,
title = {Reaction-Diffusion Problems on Time-Periodic Domains},
author = {Jane Allwright},
journal= {arXiv preprint arXiv:2210.11516},
year = {2023}
}