  We consider a modification of the so-called phase-field crystal (PFC)
equation introduced by K.R. Elder et al. This variant has recently been
proposed by P. Stefanovic et al. to distinguish between elastic relaxation and
diffusion time scales. It consists of adding an inertial term (i.e. a
second-order time derivative) into the PFC equation. The mathematical analysis
of the resulting equation is more challenging with respect to the PFC equation,
even at the well-posedness level. Moreover, its solutions do not regularize in
finite time as in the case of PFC equation. Here we analyze the modified PFC
(MPFC) equation endowed with periodic boundary conditions. We first prove the
existence and uniqueness of a solution with initial data in a bounded energy
space. This solution satisfies some uniform dissipative estimates which allow
us to study the global longtime behavior of the corresponding dynamical system.
In particular, we establish the existence of an exponential attractor. Then we
demonstrate that any trajectory originating from the bounded energy phase space
does converge to a unique equilibrium. This is done by means of a suitable
version of the {\L}ojasiewicz-Simon inequality. A convergence rate estimate is
also given.
