Variational Solutions to Nonlinear Diffusion Equations with Singular Diffusivity

We provide existence results for nonlinear diffusion equations with multivalued time-dependent nonlinearities related to convex continuous not coercive potentials. The results in this paper, following a variational principle, state that a generalized solution of the nonlinear equation can be retrieved as a solution of an appropriate minimization problem for a convex functional involving the potential and its conjugate. In the not coercive case, this assertion is conditioned by the validity of a relation between the solution and the nonlinearity. A sufficient condition, under which this relation is true, is given. At the end, we present a discussion on the solution existence for a particular equation describing a self-organized criticality model.