An asymptotic analysis for a generalized Cahn-Hilliard system with fractional operators

In the recent paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30 (2019), 437-478 -- see also arXiv:1804.11290), the same authors have studied viscous and nonviscous Cahn-Hilliard systems of two operator equations in which nonlinearities of double-well type, like regular or logarithmic potentials, as well as nonsmooth potentials with indicator functions, were admitted. The operators appearing in the system equations are fractional powers $A^{2r}$ and $B^{2\sigma}$ (in the spectral sense) of general linear operators $A$ and $B$, which are densely defined, unbounded, selfadjoint, and monotone in the Hilbert space $L^2(\Omega)$, for some bounded and smooth domain $\Omega\subset{\mathbb{R}}^3$, and have compact resolvents. Existence, uniqueness, and regularity results have been proved in the quoted paper. Here, in the case of the viscous system, we analyze the asymptotic behavior of the solution as the parameter $\sigma$ appearing in the operator $B^{2\sigma}$ decreasingly tends to zero. We prove convergence to a phase relaxation problem at the limit, and we also investigate this limiting problem, in which an additional term containing the projection of the phase variable on the kernel of $B$ appears.