  This paper is devoted to the study of some nonlinear parabolic equations with
discontinuous diffusion intensities. Such problems appear naturally in physical
and biological models. Our analysis is based on variational techniques and in
particular on gradient flows in the space of probability measures equipped with
the distance arising in the Monge-Kantorovich optimal transport problem. The
associated internal energy functionals in general fail to be differentiable,
therefore classical results do not apply directly in our setting. We study the
combination of both linear and porous medium type diffusions and we show the
existence and uniqueness of the solutions in the sense of distributions in
suitable Sobolev spaces. Our notion of solution allows us to give a fine
characterization of the emerging critical regions, observed previously in
numerical experiments. A link to a three phase free boundary problem is also
pointed out.
