A nonlocal phase-field system with inertial term

We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter is a nonlocal and nonlinear second-order ODE. We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of omega-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the omega-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.