Reaction-Diffusion Problems on Time-Periodic Domains

TL;DR

This paper investigates the long-term behavior of reaction-diffusion equations on time-periodic domains, focusing on the principal eigenvalue's role in determining solutions and their stability.

Contribution

It provides bounds and properties of the principal eigenvalue for reaction-diffusion problems on periodic domains, including its dependence on frequency and implications for solution stability.

Findings

Bounds on the principal eigenvalue under various domain assumptions

Monotonicity of the eigenvalue with respect to frequency

Convergence results for solutions with monostable nonlinearities

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.