Existence theorem for a partially parabolic cross-diffusion system

TL;DR

This paper proves an existence theorem for a partially parabolic cross-diffusion system in population dynamics, addressing the challenge posed by the changing sign of the coefficient matrix determinant.

Contribution

It introduces an approximation scheme that converges to a solution satisfying the system in the parabolic region, advancing understanding of such complex systems.

Findings

Approximate solutions converge to a limit satisfying the system in the parabolic region.

The system's partial parabolicity is rigorously addressed.

Open problem remains for solutions in both parabolic and anti-parabolic regions.

Abstract

We study an initial boundary value problem for a cross-diffusion system in population dynamics. The mathematical challenge is due to the fact that the determinant of the coefficient matrix of the system changes signs. As a result, the system is only partially parabolic. We design an approximation scheme. The sequence of approximate solutions generated by our scheme converges and its limit satisfies the original system in the parabolic region. It remains open if one can construct a vector-valued function that satisfies the system in both the parabolic region and the anti-parabolic one.