Summary: This article concerns with the solution to a heat equation with a free boundary in n-dimensional space. By applying the energy inequality to the solutions that depend not only on the initial value but also on the dimension of space, we derive the sufficient conditions under which solutions blow up at finite time. We then explore the long-time behavior of global solutions. Results show that the solution is global and fast when initial value is small, and the solution is global but slow for suitable initial value. Numerical simulations are also given to illustrate the effect of the initial value on the free boundary.