Asymptotic behavior of KPP fronts in a periodic and time-varying medium

I'll describe the asymptotic behavior of solutions to a reaction-diffusion equation with KPP-type nonlinearity. One can interpret the solution to the PDE in terms of a branching Brownian motion. It is well-known that solutions to the Cauchy problem may behave asymptotically like a traveling wave. On the other hand, for certain initial data, M. Bramson proved that the solution to the Cauchy problem may lag behind the traveling wave by an amount that grows logarithmically in time. Using PDE arguments, we have extended this statement about the logarithmic delay to the case of periodic, spatially varying reaction rate. The PDE approach involves the study of the linearized equation with Dirichlet condition on a moving boundary. The method also gives refined asymptotics for the solution to the equation with time-varying coefficients. This is joint work with Francois Hamel, Jean-Michel Roquejoffre, and Lenya Ryzhik.