  We consider a class of six-order Cahn-Hilliard equations with logarithmic
type potential. This system is closely connected with some important
phase-field models relevant in different applications, for instance, the
functionalized Cahn-Hilliard equation that describes phase separation in
mixtures of amphiphilic molecules in solvent, and the Willmore regularization
of Cahn-Hilliard equation for anisotropic crystal and epitaxial growth. The
singularity of the configuration potential guarantees that the solution always
stays in the physical relevant domain [-1,1]. Meanwhile, the resulting system
is characterized by some highly singular diffusion terms that make the
mathematical analysis more involved. We prove existence and uniqueness of
global weak solutions and show their parabolic regularization property for any
positive time. Besides, we investigate long-time behavior of the system,
proving existence of the global attractor for the associated dynamical process
in a suitable complete metric space.
