We propose an abstract variational inequality formulation of the Cahn-Hilliard equation with a time-dependent constraint. We introduce notions of strong and weak solutions, and prove that a strong solution, if it exists, is a weak solution, and that the existence of a unique weak solution holds under an appropriate time-dependence condition on the constraint. We also show that the weak solution is a strong solution under appropriate assumptions on the data. Our abstract results can be applied to various concrete problems. © 2012 Elsevier Ltd. All rights reserved.
Kubo, M. (2012). The Cahn-Hilliard equation with time-dependent constraint. Nonlinear Analysis, Theory, Methods and Applications, 75(14), 5672–5685. https://doi.org/10.1016/j.na.2012.05.015