We consider the long time behavior of solutions to the Burgers-FKPP equation
The Burgers-FKPP equation solutions exhibit a phase transition phenomenon from being pulled to pushed as increases, and the analysis at the transition case is quite delicate. We show the convergence to a traveling wave for the whole spectrum of . In particular, when , we introduce a weighted Hopf-Cole transform to construct upper and lower barriers in the self-similar variables for the linearized equation on the half line. This new transform differentiates the transition case from , as its boundary condition approaches a positive constant rather than zero. In that case, capturing the exact logarithmic delay in the reference frame is essential, and the problem boils down to providing a temporal decay estimate for a spatially inhomogeneous conservation law. I will describe how we get this temporal decay rate by combining a weighted dissipation inequality with a weighted Nash-type inequality, which probably is the most novel part of the work.
This is joint work with Chris Henderson and Lenya Ryzhik.