  The qualitative behavior of a thermodynamically consistent two-phase Stefan
problem with surface tension and with or without kinetic undercooling is
studied. It is shown that these problems generate local semiflows in
well-defined state manifolds. If a solution does not exhibit singularities in a
sense made precise below, it is proved that it exists globally in time and its
orbit is relatively compact. In addition, stability and instability of
equilibria is studied. In particular, it is shown that multiple spheres of the
same radius are unstable, reminiscent of the onset of Ostwald ripening.
