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

Source 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_{*}(\beta)t+x_{\infty}+o(1) with c_{*}(\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.