Fife-McLeod's Theorem for Spatially Periodic Degenerate Diffusion Equations

For one dimensional homogeneous bistable diffusion equations, Fife-McLeod ([Arch. Ration. Mech. Anal., 65 (1977), 335-361]) gave a well-known theorem which says that spreading solutions starting from compactly supported initial data can be exponentially approximated by traveling wave solutions. We will extend this theorem to {\it degenerate diffusion equations in periodic environments}. First, we construct a {\it periodic traveling sharp wave} to the equation, which has a positive profile on the left half-line and a right free boundary governed by the Darcy's law. To achieve this we use a renormalization approach in which crucial uniform gradient estimates near the free boundary are derived via delicate asymptotic analysis. Next we show that the central part of any spreading solution decays exponentially to a periodic steady state. Based on these results, we can construct super- and sub-solutions to prove the Fife-McLeod's theorem for our equation: any spreading solution with compactly supported initial data can be exponentially approximated by the periodic traveling sharp wave.