### abstract

- Both one-dimensional two-phase Stefan problems with the thermodynamic equilibrium condition θ(R(t),t)=0, and with the kinetic rule θ(R(t),t)=-ε̇R(t) at the moving boundary x=R(t) are considered. We study the properties of the regular solutions of the problem with equilibrium condition. They are obtained as a limit of solutions of the problem with the kinetic law as ε→0. The peculiarity of our problem is the partial supercooling of the liquid phase (θ<0) at the initial state. First, we show that the simply connected supercooled liquid phase disappears in a finite time, and after this the solution becomes the classical one. Second, under appropriate structural assumptions on the initial data, we prove the smoothness of the free boundary x=R(t) everywhere except at a point t=̄t. At this point the function R may have a jump R(̄t+0)-R(̄t-0)>0 exactly equal to the interval in which θ(x,̄t-0)≦-L. Here L is …