# Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients

## TL;DR

This paper investigates nonlinear parabolic equations with discontinuous diffusion coefficients using variational and optimal transport techniques, establishing existence, uniqueness, and detailed solution characterizations.

## Contribution

It introduces a novel analytical framework for such equations, combining gradient flows and Sobolev space solutions, and links to free boundary problems.

## Key findings

- Existence and uniqueness of solutions in Sobolev spaces.
- Characterization of critical regions in solutions.
- Connection to three-phase free boundary problems.

## Abstract

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.