Hyperbolic relaxation of a sixth-order Cahn-Hilliard equation

This work explores the solvability of a sixth-order Cahn--Hilliard equation with an inertial term, which serves as a relaxation of a higher-order variant of the classical Cahn--Hilliard equation. The equation includes a source term that disrupts the conservation of the mean value of the order parameter. The incorporation of additional spatial derivatives allows the model to account for curvature effects, leading to a more precise representation of isothermal phase separation dynamics. We establish the existence of a weak solution for the associated initial and boundary value problem under the assumption that the double-well-type nonlinearity is globally defined. Additionally, we derive uniform stability estimates, which enable us to demonstrate that any family of solutions satisfying these estimates converges in a suitable topology to the unique solution of the limiting problem as the relaxation parameter approaches zero. Furthermore, we provide an error estimate for specific norms of the difference between solutions in terms of the relaxation parameter.