Asymptotic behavior of solutions of Fisher–KPP equation with free boundary conditions

We study a free boundary problem for the Fisher–KPP equation modeling the spreading of a biological or chemical species. In this model, the free boundaries represent the spreading fronts of the species. We discuss the asymptotic behavior of bounded solutions and obtain a trichotomous result: spreading (the free boundaries amounts to the whole space and the solution converges to 1), transition (the free boundaries stay in a bounded interval and the solution converges to a stationary solution with positive compact support) and vanishing (the free boundaries converge to the same point and the solution tends to 0 within a finite time).