A free boundary problem for a reaction-diffusion system with nonlinear memory

We consider a integro-partial differential equation with a free boundary which appears in the theory of the nuclear dynamics. First, local existence and uniqueness are obtained by using the contraction mapping theorem. Then, the behavior of the free boundary and the blow-up criteria are obtained. Finally, we examine the long-time behavior of the global solution. We show that the solution is global and fast if the initial data are small.