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 β_c=2. We prove convergence of the solutions to a traveling wave in a reference frame centered at a position m_β(t) and study the asymptotics of the front location m_β(t). When β < 2, it has the same form as for the standard Fisher-KPP equation established by Bramson : m_β(t) = 2t - (3/2)log(t) + x_∞ + o(1)

more »

... as t→+∞. This form is typical of pulled fronts. When β > 2, the front is located at the position m_β(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 β_c = 2, the expansion changes to m_β(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=β_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 β_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.