Abstract
We study a free boundary problem for the reaction-diffusion equation with nonlinear advection term u t = u xx − u u x + u p ( 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 σϕ , we have two sharp results. There is some σ ∗ > 0 , blow up happens when σ > σ ∗ , vanishing happens when σ < σ ∗ and the transition case happens when σ = σ ∗ . Additionally, we also have another trichotomy result: the solution is either blow up or converging to a small stationary state or converging to a big stationary state.
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