We consider a free boundary problem: () with free boundary conditions and , where is a positive -periodic function, is a Fisher–KPP type of nonlinearity and -periodic in . 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 -periodic function which plays a key role in the dynamics. More precisely:
(i) When , we obtain a trichotomy result:
Spreading, i.e., and as , where is the periodic solution of the ODE .
Vanishing, i.e., and , where is some positive constant.
Transition, i.e., , and , where is a -periodic solution with compact support.
(ii) In the case , vanishing happens for any solution.
"Convergence of solutions for Fisher–KPP equation with a free boundary condition." Rocky Mountain J. Math. 50 (1) 55 - 68, Febuary 2020. https://doi.org/10.1216/rmj.2020.50.55