Separation property and asymptotic behavior for a transmission problem of the bulk-surface coupled Cahn–Hilliard system with singular potentials and its Robin approximation

We consider a class of bulk-surface coupled Cahn–Hilliard systems in a smooth, bounded domain $\mathrm{\Omega }\subset {ℝ}^d$$(d=2,3)$, where the bulk phase variable is connected to the surface phase variable via a Dirichlet boundary condition or its Robin approximation. For a general class of singular potentials (including the physically relevant logarithmic potential), we establish the regularity propagation of global weak solutions to the initial boundary value problem. In particular, when the spatial dimension is two, we prove the instantaneous strict separation property, which ensures that every global weak solution remains uniformly away from the pure states $\pm 1$ after any given positive time. In the three-dimensional case, we obtain the eventual strict separation property that holds for sufficiently large time. This strict separation property allows us to prove that every global weak solution converges to a single equilibrium as time goes to infinity using the Łojasiewicz–Simon approach. Finally, we study the double obstacle limit for the problem with logarithmic potentials in the bulk and on the boundary, showing that as the absolute temperature $\mathrm{\Theta }$ tends to zero, the corresponding weak solutions converge (for a suitable subsequence) to a weak solution of the problem with a double obstacle potential.