Regularity of the free boundary in a nonlocal one-dimensional parabolic   free boundary value problem
  We consider one-dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. Results on $C^m$-regularity of the free boundary are obtained. In particular, a necessary and sufficient condition for infinite differentiability of the free boundary is given. 
Rossitza Semerdjieva
math.AP