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A thermodynamically consistent two-phase Stefan problem withtemperature-dependent surface tension and with or without kinetic undercoolingis studied. It is shown that these problems generate local semiflows inwell-defined state manifolds. If a solution does not exhibit singularities, itis proved that it exists globally in time and converges towards an equilibriumof the problem. In addition, stability and instability of equilibria isstudied. In particular, it is shown that multiple spheres of the same radiusare unstable if surface heat capacity is small; however, if kineticundercooling is absent, they are stable if surface heat capacity issufficiently large.

On thermodynamically consistent Stefan problems with variable surface energy