Fisher-KPP equation with advection on the half-line

We consider the Fisher–KPP equation with advection: ut=uxx−βux+f(u) on the half-line x∈(0,∞), with no-flux boundary condition ux−βu = 0 at x = 0. We study the influence of the advection coefficient −β on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when β∈(0,c0), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when β∈[c0,∞), the solution converges locally uniformly to 0, here c0 is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when β > 0, the asymptotic positions of the level sets on the right side of the solution are (β + c0)t + o(t), and (ii) when β≥c0, the asymptotic positions of the level sets on the left side are (β − c0)t + o(t). Copyright © 2015 John Wiley & Sons, Ltd.