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