  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.
