Global existence and blow-up of positive solutions of a parabolic problem with free boundaries

The purpose of this paper is to investigate a parabolic problem with double free boundaries. By the contraction mapping theorem, we establish the local existence and uniqueness of positive solutions. We present some sufficient conditions with respect to blow-up in finite time of the solution, and the existence of the fast solution and the slow solution, respectively. Finally, we obtain a trichotomy conclusion by considering the size of parameter σ.