Weak asymptotic solution of the phase field system in the case of   confluence of free boundaries 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 finite velocities until the 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 conduction equation. 
V.G.Danilov