A blow-up result of a free boundary problem with nonlinear advection term

We study a free boundary problem for the reaction-diffusion equation with nonlinear advection term ut = uxx - uux + up(p > 1) on [0, h(t)]. This model describes species that live in an uninhabitable area and tries to spread into a new area, the free boundary h(t) represents such a spreading front. By considering the initial data sf, we have two sharp results. There is some s * > 0, blow up happens whens > s *, vanishing happens when s < s * and the transition case happens when s = s *. Additionally, wealso have another trichotomy result: the solution is either blow up or converging to a small stationary state or converging to a big stationary state.