Spreading Profile and Nonlinear Stefan Problems

Abstract

We report some recent progress on the study of the following nonlinear Stefan problem ut − ∆u = f(u) for x ∈ Ω(t), t > 0, u = 0 and ut = µ|∇xu| 2 for x ∈ Γ(t), t > 0, u(0, x) = u0(x) for x ∈ Ω0, where Ω(t) ⊂ R N (N ≥ 1) is bounded by the free boundary Γ(t), with Ω(0) = Ω0, µ is a given positive constant. The initial function u0 is positive in Ω0 and vanishes on ∂Ω0. The class of nonlinear functions f(u) includes the standard monostable, bistable and combustion type nonlinearities. When µ → ∞, it can be shown that this free boundary problem converges to the corresponding Cauchy problem ut − ∆u = f(u) for x ∈ R N , t > 0, u(0, x) = u0(x) for x ∈ R N . We will discuss the similarity and differences of the dynamical behavior of these two problems by closely examining their spreading profiles, which suggest that the Stefan condition is a stabilizing factor in the spreading process.