Global smooth solutions for triangular reaction-cross diffusion systems

TL;DR

This paper proves the global existence of smooth solutions for a class of reaction-cross diffusion systems relevant in population dynamics, extending known models to include general power-law growth in reactions and diffusion.

Contribution

It establishes the global regularity of solutions for a broader class of triangular reaction-cross diffusion systems with power-law growth, building on Amann's local existence results.

Findings

Global smooth solutions are proven to exist for the specified class of systems.

The results extend the classical triangular SKT system to more general growth laws.

The systems are shown to be well-posed for all time under the given conditions.

Abstract

For a class of reaction cross-diffusion systems of two equations with a cross-diffusion term in the first equation and with self-diffusion terms, we prove that the unique local smooth solution given by Amann theorem is actually global. This class of systems arises in Population dynamics, and extends the triangular Shigesada-Kawasaki-Teramoto system when general power-laws growth are considered in the reaction and diffusion rates.