We prove here well-posedness and convergence to equilibria for the solution trajectories associated to a model for solidification of a liquid content of a rigid container in a gravity field. We observe that the gravity effects, which can be neglected without considerable changes of the process on finite time intervals, have a substantial influence on the long time behavior of the evolution system. Without gravity, we find a temperature interval, in which all phase distributions with a prescribed total liquid contents are admissible equilibria, while, under the influence of gravity, the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.