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We consider Cahn-Hilliard equations with external forcing terms. Energydecreasing and mass conservation might not hold. We show that level surfaces ofthe solutions of such generalized Cahn-Hilliard equations tend to the solutionsof a moving boundary problem under the assumption that classical solutions ofthe latter exist. Our strategy is to construct approximate solutions of thegeneralized Cahn-Hilliard equation by the Hilbert expansion method used inkinetic theory and proposed for the standard Cahn-Hilliard equation, by Carlen,Carvalho and Orlandi, \cite {CCO}. The constructed approximate solutions allowto derive rigorously the sharp interface limit of the generalized Cahn-Hilliardequations. We then estimate the difference between the true solutions and theapproximate solutions by spectral analysis, as in \cite {A-B-C}

A Hilbert expansions method for the rigorous sharp interface limit of the generalized Cahn-Hilliard Equation