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

TL;DR

This paper extends Fife-McLeod's theorem to degenerate diffusion equations in periodic environments, demonstrating exponential approximation of spreading solutions by periodic traveling waves through novel construction and analysis.

Contribution

It introduces a new periodic traveling sharp wave construction and extends the Fife-McLeod theorem to degenerate, periodic diffusion equations with free boundaries.

Findings

Existence of a periodic traveling sharp wave with a free boundary.

Exponential decay of spreading solutions to a periodic steady state.

Extension of Fife-McLeod's theorem to degenerate periodic environments.

Abstract

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.