Asymptotic behavior of solutions of free boundary problem with logistic reaction term

We study a free boundary problem for a reaction diffusion equation modeling the spreading of a biological or chemical species. in this model,the free boundary represents the spreading front of the species. we discuss the asymptotic behavior of bounded solutions and obtain a trichotomy result: spreading (the free boundary tends to + ∞ and the solution converges to a stationary solution defined on [ 0 + ∞)),transition (the free boundary stays in a bounded interval and the solution converges to a stationary solution with positive compact support),and vanishing (the free boundary converges to 0 and the solution tends to 0 within a finite time). © 2014 jingjing cai.