Confluence of the nonlinear waves in the Stefan problem with   undercooling
  We assume that the Stefan problem with undercooling has a classical solution until the moment of contact of free boundaries and the free boundaries have continuous velocities until the moment of contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with undercooling, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution which is the solution of the heat equation. 
V.G.Danilov and V.Yu. Rudnev