Dynamical behavior of solutions of a free boundary problem

Abstract

This paper is concerned with the spreading properties for a reaction-diffusion equation with free boundary condition. We obtained a complete description of the long-time dynamical behavior of this problem. By introducing a parameter $ \sigma $ in the initial data, we revealed a threshold value $ \sigma^* $ such that spreading happens when $ \sigma > \sigma^* $ and vanishing happens when $ \sigma\leq \sigma^* $. There exists a unique $ L^* > 0 $ independent of the initial data such that $ \sigma^* = 0 $ if and only if the length of initial occupying interval is no smaller than $ 2L^* $. These theoretical results may have important implications for prediction and prevention of biological invasions.

Keywords

• reaction-diffusion equation
• free boundary problem
• long time behavior
• spreading phenomena