Global fast and slow solutions of a localized problem with free boundary

In this paper, we consider a localized problem with free boundary for the heat equation in higher space dimensions and heterogeneous environment. For simplicity, we assume that the environment and solution are radially symmetric. First, by using the contraction mapping theorem, we prove that the local solution exists and is unique. Then, some sufficient conditions are given under which the solution will blow up in finite time. Our results indicate that the blowup occurs if the initial data are sufficiently large. Finally, the long time behavior of the global solution is discussed. It is shown that the global fast solution does exist if the initial data are sufficiently small, while the global slow solution is possible if the initial data are suitably large.