  We consider one-dimensional reaction-diffusion equations for a large class of
spatially periodic nonlinearities (including multistable ones) and study the
asymptotic behavior of solutions with Heaviside type initial data. Our analysis
reveals some new dynamics where the profile of the propagation is not
characterized by a single front, but by a layer of several fronts which we call
a terrace. Existence and convergence to such a terrace is proven by using an
intersection number argument, without much relying on standard linear analysis.
Hence, on top of the peculiar phenomenon of propagation that our work
highlights, several corollaries will follow on the existence and convergence to
pulsating traveling fronts even for highly degenerate nonlinearities that have
not been treated before.
