Fisher-KPP equation with free boundaries and time-periodic advections

Abstract

We consider a reaction-diffusion-advection equation of the form: ut = uxx - β(t)ux + f (t, u) for x ∈ (g(t), h(t)), where β(t) is a T-periodic function representing the intensity of the advection, f (t, u) is a Fisher-KPP type of nonlinearity, T periodic in t, g(t) and h(t) are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both β and f are independent of t) was recently studied by Gu et al. (J Funct Anal 269:1714-1768, 2015). In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing-spreading dichotomy result holds when β is small; a vanishing transition-virtual spreading trichotomy result holds when β is a medium-sized function; all solutions vanish when β is large. Here the partition of β(t) depends not only on the "size" β := 1T ∫ T0 β(t)dt of β(t) but also on its "shape" β(t):=β(t)-β.