Spreading and vanishing in a free boundary problem for nonlinear diffusion equations with a given forced moving boundary

We will study a free boundary problem of the nonlinear diffusion equations of the form u_t=u_xx+f(u),t>0,ct<x<h(t), where f is C¹ function satisfying f(0)=0, c>0 is a given constant and h(t) is a free boundary which is determined by a Stefan-like condition. This model may be used to describe the spreading of a new or invasive species with population density u(t,x) over a one dimensional habitat. The free boundary x=h(t) represents the spreading front. In this model, we impose zero Dirichlet boundary condition at left moving boundary x=ct. This means that the left boundary of the habitat is a very hostile environment for the species and that the habitat is eroded away by the left moving boundary at constant speed c. In this paper we will extend the results of a trichotomy result obtained in [23] to general monostable, bistable and combustion types of nonlinearities. We show that the long-time dynamical behavior of solutions can be expressed by unified fashion, that is, for any initial data, the unique solution exhibits exactly one of the behaviors, spreading, vanishing and transition. We also give the asymptotic profile of the solution over the whole domain when spreading happens. The approach here is quite different from that used in [23].