Asymptotic Behavior of Solutions of Free Boundary Problem with Logistic Reaction Term

Abstract

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 $+\infty$ and the solution converges to a stationary solution defined on $[0+\infty )$), 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).