Convergence to a traveling wave for the Burgers-FKPP equation

We consider the long time behavior of solutions to the Burgers-FKPP equation $u_t +\beta u u_x = u_{xx} + u-u^2.$

The Burgers-FKPP equation solutions exhibit a phase transition phenomenon from being pulled to pushed as $\beta$ increases, and the analysis at the transition case $\beta=2$ is quite delicate. We show the convergence to a traveling wave for the whole spectrum of $\beta$. In particular, when $\beta\leq 2$, 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 $\beta=2$ from $\beta<2$, 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.