A time-periodic reaction–diffusion–advection equation with a free boundary and sign-changing coefficients

Abstract

We consider a reaction–diffusion–advection equation of the form: ut=uxx−β(t)ux+f(t,u) for x∈[0,h(t)), where β(t) is a T-periodic function, f(t,u) is a T-periodic Fisher–KPP type of nonlinearity with a(t)≔fu(t,0) changing sign, h(t) is a free boundary satisfying the Stefan condition. We study the long time behavior of solutions and find that there are two critical numbers c̄ and B(β̃) with B(β̃)>c̄>0, β̄≔1T∫0Tβ(t)dt and β̃(t)≔β(t)−β̄, such that a vanishing–spreading dichotomy result holds when |β̄|<c̄; a vanishing–transition–virtual spreading trichotomy result holds when β̄∈[c̄,B(β̃)); all solutions vanish when β̄⩾B(β̃) or β̄⩽−c̄.