ON A CLASS OF SIXTH-ORDER CAHN-HILLIARD-TYPE EQUATIONS WITH LOGARITHMIC POTENTIAL

We consider a class of sixth-order Cahn-Hilliard-type equations with logarithmic potential. This system is closely connected to 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 the Cahn-Hilliard equation for anisotropic crystal and epitaxial growth. The singularity of the configuration potential guarantees that the solution always stays in the physically 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. In addition, 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.