  We consider a phase-field model where the internal energy depends on the
order parameter in a nonlocal way. Therefore, the resulting system consists of
the energy balance equation coupled with a nonlinear and nonlocal ODE for the
order parameter. Such system has been analyzed by several authors, in
particular when the configuration potential is a smooth double-well function.
More recently, in the case of a potential defined on (-1,1) and singular at the
endpoints, the existence of a finite-dimensional global attractor has been
proven. Here we examine both the case of smooth potentials as well as the case
of physically realistic (e.g., logarithmic) singular potentials. We prove
well-posedness results and the eventual global boundedness of solutions
uniformly with respect to the initial data. In addition, we show that the
separation property holds in the case of singular potentials. Thanks to these
results, we are able to demonstrate the existence of a finite-dimensional
attractors in the present cases as well.
