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

We consider a reaction–diffusion–advection equation of the form: u t = u x x - β ( t ) u x + 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 ¯ : = 1 T ∫ 0 T β ( t ) d t of β ( t ) but also on its “shape” β ~ ( t ) : = β ( t ) - β ¯ .