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

Abstract

We consider a reaction–diffusion–advection equation of the form: $$u_t=u_{xx}-\beta (t)u_x+f(t,u)$$ for $$x\in (g(t),h(t))$$ , where $$\beta (t)$$ 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 $$\beta $$ 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 $$\beta $$ is small; a vanishing–transition–virtual spreading trichotomy result holds when $$\beta $$ is a medium-sized function; all solutions vanish when $$\beta $$ is large. Here the partition of $$\beta (t)$$ depends not only on the “size” $$\bar{\beta }:= \frac{1}{T}\int _0^T \beta (t) dt$$ of $$\beta (t)$$ but also on its “shape” $$\tilde{\beta }(t) := \beta (t) - \bar{\beta }$$ .