A singular Cahn--Hilliard--Oono phase-field system with hereditary memory

We consider a phase-field system modeling phase transition phenomena, where the Cahn--Hilliard--Oono equation for the order parameter is coupled with the Coleman--Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.