Passage to the limit over small parameters in the viscous Cahn-Hilliard equation

We study singular passage to the limit over different small parameters for the viscous Cahn-Hilliard equation under weak growth assumptions on the nonlinearity φ. A rigorous proof of convergence to solutions of either the Cahn-Hilliard equation, or of the Allen-Cahn equation, or of the Sobolev equation, depending on the choice of the parameter, is provided. We also study the singular limit of the Cahn-Hilliard equation as the parameter in the fourth-order term goes to zero. In particular, we show that a Radon measure-valued solution of the limiting ill-posed problem can arise, depending on the behavior of the nonlinearity φ at infinity.