Nonnegative weak solution to the degenerate viscous Cahn–Hilliard equation

The Cahn–Hilliard equation is a widely used model for describing phase separation processes in a binary mixture. In this paper, we investigate the viscous Cahn–Hilliard equation with a degenerate, phase-dependent mobility. We define the concept of a weak solution and establish the existence of such a solution by taking limits of solutions to the viscous Cahn–Hilliard equation with positive mobility. Additionally, assuming that the initial data is positive, we demonstrate that the weak solution remains nonnegative and is not identically zero. Finally, we prove that the weak solution satisfies an energy dissipation inequality.