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Jing An (MPI MIS, Leipzig): Convergence to a traveling wave for the Burgers-FKPP equation

Ort: MPI für Mathematik in den Naturwissenschaften Leipzig, Inselstr. 22, E1 05 (Leibniz-Saal)

Vortrag in der Reihe: Arbeitsgemeinschaft ANGEWANDTE ANALYSIS 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.

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Beginn: July 28, 2021, 11 a.m.

Ende: July 28, 2021, 12:30 p.m.