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 Rᵈ (d\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 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 +.