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We are concerned with the study of the well-posedness of a nonlineardiffusion equation with a monotonically increasing multivalued time-dependentnonlinearity derived from a convex continuous potential having a superlineargrowth to infinity. The results in this paper state that the solution of thenonlinear equation can be retrieved as the null minimizer of an appropriateminimization problem for a convex functional involving the potential and itsconjugate. This approach, inspired by the Brezis-Ekeland variational principle,provides new existence results under minimal growth and coercivity conditions.

Existence for weakly coercive nonlinear diffusion equations via a variational principle