Convergence of solutions for Fisher–KPP equation with a free boundary condition

Abstract

We consider a free boundary problem: ut=uxx+f(t,u) (g(t)<x<h(t)) with free boundary conditions h′(t)=−ux(t,h(t))−α(t) and g′(t)=−ux(t,g(t))+α(t), where α(t) is a positive T-periodic function, f(t,u) is a Fisher–KPP type of nonlinearity and T-periodic in t. Such a problem can model the spreading of a biological or chemical species in time-periodic environment, where free boundaries mimic the spreading fronts of the species. We mainly study the convergence of bounded solutions. There is a T-periodic function α0(t) which plays a key role in the dynamics. More precisely: (i) When 0<α<α0, we obtain a trichotomy result: Spreading, i.e., h(t),−g(t)→+∞ and u(t,⋅)→P(t) as t→∞, where P(t) is the periodic solution of the ODE ut=f(t,u). Vanishing, i.e., limt→𝒯h(t)= limt→𝒯g(t) and limt→𝒯maxg(t)≤x≤h(t)u(t,x)=0, where 𝒯 is some positive constant. Transition, i.e., 0< limt→∞h(t),limt→∞g(t)<+∞, 0< limt→∞[h(t)−g(t)]<+∞ and u(t,⋅)→V(t,⋅), where V is a T-periodic solution with compact support. (ii) In the case α≥α0, vanishing happens for any solution.