Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

Abstract

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength $\beta \in \mathbb{R}$. This equation exhibits a transition from pulled to pushed front behavior at ${\beta }_c=2$. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position $m_{\beta }(t)$ and study the asymptotics of the front location $m_{\beta }(t)$. When $\beta <2$, it has the same form as for the standard Fisher-KPP equation established by Bramson: $m_{\beta }(t)=2t-(3/2)\log t+x_{\infty }+o(1)$ as $t\to \infty$. This form is typical of pulled fronts. When $\beta >2$, the front is located at the position $m_{\beta }(t)=c_{\ast }(\beta )t+x_{\infty }+o(1)$ with $c_{\ast }(\beta )=\beta /2+2/\beta$, which is the typical form of pushed fronts. However, at the critical value ${\beta }_c=2$, the expansion changes to $m_{\beta }(t)=2t-(1/2)\log t+x_{\infty }+o(1)$, reflecting the “pushmi-pullyu” nature of the front. The arguments for $\beta <2$ rely on a new weighted Hopf–Cole transform that allows one to control the advection term, when combined with additional steepness comparison arguments. The case $\beta >2$ relies on standard pushed front techniques. The proof in the case $\beta ={\beta }_c$ is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at ${\beta }_c=2$ and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.