Spreading and Vanishing for Nonlinear Stefan Problems in High Space Dimensions

Abstract

We classify the long-time behavior of solutions to nonlinear diffusive equations of the form ut − Δu = f(u) for t > 0 and x over a variable domain Ω(t) ⊂ ℝN, with a Stefan condition for u over the free boundary Γ(t) = ∂Ω(t), and u(0,x) > 0 in Ω(0) = Ω0. For monostable type of f and bistable type of f, we obtain a rather complete classification of the long-time dynamical behavior of the solution to this nonlinear Stefan problem, and examine how the behavior changes when u(0, x) takes initial functions of the form σφ(x) and σ > 0 is varied.