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Convergence to a traveling wave for the Burgers-FKPP equation

  • Jing An (MPI MiS, Leipzig)
E1 05 (Leibniz-Saal)

Abstract

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.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

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