Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem 'ut−dΔu=g(u)u=0andut=μ|∇xu|2u(0,x)=u0(x)forx∈Ω(t),t>0,forx∈ (t),t>0,forx∈Ω0', where Ω(t)⊂Rn ( n≧2 ) is bounded by the free boundary Γ(t) , with Ω(0)=Ω0 , μ and d are given positive constants. The initial function u 0 is positive in Ω0 and vanishes on ∂Ω0 . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary Γ(t) is smooth outside the closed convex hull of Ω0 , and as t→∞ , either Ω(t) expands to the entire Rn , or it stays bounded. Moreover, in the former case, Γ(t) converges to the unit sphere when normalized, and in the latter case, u→0 uniformly. When g(u)=au−bu2, we further prove that in the case Ω(t) expands to Rn , u→a/b as t→∞ , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists μ∗≧0 such that Ω(t) expands to Rn exactly when μ>μ∗.