Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

We study the following nonlinear Stefan problem <Equation ID="Equa"> <MediaObject> </MediaObject> </Equation>where () is bounded by the free boundary , with , mu and d are given positive constants. The initial function u (0) is positive in and vanishes on . The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary is smooth outside the closed convex hull of , and as , either expands to the entire , or it stays bounded. Moreover, in the former case, converges to the unit sphere when normalized, and in the latter case, uniformly. When , we further prove that in the case expands to , as , and the spreading speed of the free boundary converges to a positive constant; moreover, there exists such that expands to exactly when .