The Cahn--Hilliard equation is a common model to describe phase separation
processes of a mixture of two components. In this paper, we study the viscous
Cahn--Hilliard equation with degenerate phase-dependent mobility. We define a
notion of weak solutions and prove the existence of such weak solutions by
considering the limits of the viscous Cahn--Hilliard equation with positive
mobility. Also, we prove that such weak solutions satisfy an energy dissipation
inequality under some additional conditions