Global strong solutions for the triangular Shigesada-Kawasaki-Teramoto cross-diffusion system in three dimensions and parabolic regularisation for increasing functions

TL;DR

This paper establishes the existence of global strong solutions for a complex three-dimensional cross-diffusion system, employing novel parabolic regularisation techniques for increasing functions, with implications for reaction-diffusion models.

Contribution

It introduces a new approach to prove global solutions for the SKT cross-diffusion system in three dimensions, focusing on parabolic regularisation for increasing functions.

Findings

Proved existence of global strong solutions in 3D for the SKT system.

Developed estimates for reaction-diffusion systems modeling reversible chemistry.

Analyzed parabolic equations with rough coefficients and monotonicity conditions.

Abstract

We prove the existence of global strong solutions to the triangular Shigesada-Kawasaki-Teramoto (SKT) cross-diffusion system with Lokta-Volterra reaction terms in three dimensions. A key part is the independent careful study of the parabolic equation $a\partial_t w - \Delta w = f$ with a rough coefficient $a$, homogeneous Neumann boundary conditions, and the special assumption $\partial_t w \ge 0$. By the same method, we obtain estimates for solutions to reaction-diffusion systems modelling reversible chemistry.