Abstract
AbstractReaction–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.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
Reference11 articles.
1. Allwright, J.: Exact solutions and critical behaviour for a linear growth-diffusion equation on a time-dependent domain. Proc. Edinb. Math. Soc. 65(1), 53–79 (2022)
2. Allwright, J.: Reaction–diffusion on a time-dependent interval: refining the notion of ‘critical length’. Commun. Contemp. Math. 25(09), 2250050 (2022)
3. Cantrell, R.S., Cosner, C.: Spatial Ecology via Reaction–Diffusion Equations. Wiley, Chichester (2003)
4. Castro, A., Lazer, A.C.: Results on periodic solutions of parabolic equations suggested by elliptic theory. Boll. Unione Mat. Ital., Series VI I–B(3), 1089–1104 (1982)
5. Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity (Pitman Research Notes in Mathematics). Longman Scientific and Technical, Harlow, Essex (1991)