Zusammenfassung für den Vortrag am 28.07.2021 (11:00 Uhr)

\(u_t +\beta u u_x = u_{xx} + u-u^2.\)

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.