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.