Remarks on spreading and vanishing for free boundary problems of some reaction-diffusion equations

We discuss a free boundary problem for a di¤usion equation in a onedimensional interval which models the spreading of invasive or new species. Moreover, the free boundary represents a spreading front of the species and its dynamical behavior is determined by a Stefan-like condition. This problem has been proposed by Du and Lin (2010) and, recently, Kaneko and Yamada have studied a free boundary problem for a general reaction-di¤usion equation under Dirichlet boundary conditions. The main purpose of this paper is to define ‘‘spreading’’ and ‘‘vanishing’’ of species for a free boundary problem with general nonlinearity and study the underlying principle to determine the spreading or vanishing behavior as time tends to infinity. It will be proved that vanishing occurs if and only if the free boundary stays in a bounded interval, and that, when vanishing occurs, the population decreases exponentially to zero in large time.