Regularity and long-time behavior of global weak solutions to a coupled Cahn-Hilliard system: the off-critical case

We consider a diffuse interface model that describes the macro- and micro-phase separation processes of a polymer mixture. The resulting system consists of a Cahn-Hilliard equation and a Cahn-Hilliard-Oono type equation endowed with the singular Flory-Huggins potential. For the initial boundary value problem in a bounded smooth domain of $\mathbb{R}^d$ ($d\in\{2,3\}$) with homogeneous Neumann boundary conditions for the phase functions as well as chemical potentials, we study the regularity and long-time behavior of global weak solutions in the off-critical case, i.e., the mass is not conserved during the micro-phase separation of diblock copolymers. By investigating an auxiliary system with viscous regularizations, we show that every global weak solution regularizes instantaneously for $t>0$. In two dimensions, we obtain the instantaneous strict separation property under a mild growth condition on the first derivative of potential functions near pure phases $\pm 1$, while in three dimensions, we establish the eventual strict separation property for sufficiently large time. Finally, we prove that every global weak solution converges to a single equilibrium as $t\to +\infty$.