The Cahn-Hilliard-Oono equation with singular potential

We consider the so-called Cahn-Hilliard-Oono equation with singular (e.g. logarithmic) potential in a bounded domain of Rd, d <= 3. The equation is subject to an initial condition and Neumann homogeneous boundary conditions for the order parameter as well as for the chemical potential. However, contrary to the Cahn-Hilliard equation, the total mass might not be conserved. The existence of a global finite energy solution to such a problem was proven by Miranville and Temam. We first establish some regularization properties in finite time of the (unique) solution. Then, in dimension two, we prove the so-called strict separation property, namely, we show that any finite energy solution stays away from pure phases, uniformly with respect to the initial energy and the total mass. Taking advantage of these results, we study the long-time behavior of solutions. More precisely, we establish the existence of the global attractor in both two and three dimensions. Due to the strict separation property in dimension two, we also prove the existence of exponential attractors and we show that a finite energy solution always converges to a single equilibrium even though the mass is not conserved.