Angular Region On The One-dimensional Two Phase Free Boundary Problem

In this paper we consider an one-dimensional solidification problem of a finite domain of a pure substance which is initially in liquid state. First, we prove the existence of the solution of parabolic equation in triangular domain ; Second, the existence of a local solution in global domain is established by the fixed point theorem; Third, by studying the property of the free boundary, we prove the free boundary dissipates in finite time; later, we obtain the and uniqueness.