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

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength β ∈ R. This equation exhibits a transition from pulled to pushed front behavior at βc = 2. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position mβ(t) and study the asymptotics of the front location mβ(t). When β < 2, it has the same form as for the standard Fisher-KPP equation established by Bramson [9, 10]: mβ(t) = 2t − (3/2) log(t) + x∞ + o(1) as t → +∞. This form is typical of pulled fronts. When β > 2, the front is located at the position mβ(t) = c∗(β)t + x∞ + o(1) with c∗(β) = β/2 + 2/β, which is the typical form of pushed fronts. However, at the critical value βc = 2, the expansion changes to mβ(t) = 2t − (1/2) log(t) + x∞ + o(1), reflecting the “pushmi-pullyu” nature of the front. The arguments for β < 2 rely on a new weighted Hopf-Cole transform that allows to control the advection term, when combined with additional steepness comparison arguments. The case β > 2 relies on standard pushed front techniques. The proof in the case β = βc 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 βc = 2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.