| Abstract: |
| We consider a nonlinear diffusion equation $u_t = du_{xx}+f(u)$ in $[0, h(t)]$ with zero Dirichlet boundary condition at $x=0$ and a free boundary condition at $x=h(t)$. Here $f(u)$ belongs to a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ODE model. This problem models the invasion of a biological species and the free boundary $x=h(t)>0$ represents the spreading front of its habitant. Our main interest is to study large-time behaviors of solutions for the free boundary problem. We obtain a trichotomy result for vanishing and two types of spreading and sharp thresholds for the asymptotic behavior of the solutions by the phase plane analysis and the zero number arguments. |
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