Existence and Uniqueness of Weak Solutions to a Class of Degenerate Cross Diffusion Systems

TL;DR

This paper proves the existence and uniqueness of weak solutions for a class of degenerate cross diffusion systems inspired by biological models, overcoming challenges posed by the lack of maximum principles.

Contribution

It introduces a novel approach to establish weak solutions for degenerate systems without relying on maximum principles, extending the mathematical theory for such models.

Findings

Existence of weak solutions under mild conditions

Convergence of solutions from nondegenerate to degenerate systems

Uniqueness of weak solutions in the degenerate case

Abstract

We consider a class of cross diffusion systems with degenerate (or porous media type) diffusion which is inspired by models in mathematical biology/ecology with zero self diffusions. Known techniques for scalar equations are no longer available here as maximum/comparison principles are generally unavailable for systems. However, we will provide the existence of weak solutions to the degenerate systems under mild integrability conditions of strong solutions to nondegenerate systems and show that they converge to a weak solution of the degerate system. These conditions will be verified for the model introduced by Shigesada {\it et al.}. Uniqueness of limiting and unbounded weak solutions will also be proved.