Long-time convergence of solutions to a phase-field system

We prove that any global bounded solution of a phase field model tends to a single equilibrium state for large times though the set of equilibria may contain a nontrivial continuum of stationary states. The problem has a partial variational structure, specifically, only the elliptic part of the first equation represents an Euler-Lagrange equation while the second does not. This requires some modifications in comparison with standard methods used to attack this kind of problems.