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

We consider a reaction-diffusion-advection equation of the form: u(t) = u(xx) - beta(t)u(x) + f(t, u) for x is an element of (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" (beta) over bar := 1/T integral(T)(0) beta(t) dt of beta(t) but also on its "shape" (beta) over tilde (t) := beta(t) - (beta) over bar.