Entire solutions of time periodic Fisher–KPP equation on the half line

Abstract

We study the entire solutions of the Fisher–KPP equation of the form ut=uxx+f(t,u) on [0,∞) with boundary condition u(t,0)=0, where f(t,u) is a Fisher–KPP type of nonlinearity, T-periodic in t. We obtain two types of entire solutions. (i) For any c≥2a¯ (with a¯≔1T∫0Tfu(s,0)ds), we obtain an entire solution Uc(t,x) connecting the periodic traveling wave solution ϕc(t,x+ct) at t=−∞ with the periodic solution V(t,x) at t=+∞; (ii) The second type of entire solution U(t,x) connecting the solution of the initial problems of ODE η′(t)=f(t,η) at t=−∞ and V(t,x) at t=+∞, with properties Uxx<0 and U(t,∞)=P(t), where P(t) is the periodic solution of η′(t)=f(t,η).