We show that any global bounded solution of phase-field systems
converges to a single stationary state as time goes to infinity.
The main problem consists in the fact that the set of equilibria may be
a continuum, so that even if we prove convergence of trajectories to
this set, it is a nontrivial question whether any trajectory stabilizes to a
single point. To solve the problem, we will apply a generalized version
of the Lojasiewicz theorem where the idea of analyticity plays a key
role.