Propagation in a Fisher-KPP equation with non-local advection

We investigate the influence of a general non-local advection term of the form K⁎u to propagation in the one-dimensional Fisher-KPP equation. This model is a generalization of the Keller-Segel-Fisher system. When K∈L¹(R), we obtain explicit upper and lower bounds on the propagation speed which are asymptotically sharp and more precise than previous works. When K∈L^p(R) with p>1 and is non-increasing in (−∞,0) and in (0,+∞), we show that the position of the “front” is of order O(t^p) if p<∞ and O(e^λt) for some λ>0 if p=∞ and K(+∞)>0. We use a wide range of techniques in our proofs.