On the Global Existence of a Class of Strongly Coupled Parabolic Systems

TL;DR

This paper proves the existence of strong solutions for a broad class of coupled parabolic systems, generalizing the SKT model in population dynamics, under weaker conditions, and fully solves the planar case with cubic diffusions.

Contribution

It introduces the concept of strong-weak solutions and demonstrates their equivalence to strong solutions, extending existence results to more general cross diffusion systems.

Findings

Existence of strong solutions for a class of cross diffusion systems.

Introduction of strong-weak solutions concept.

Complete solution for the SKT model on planar domains with cubic diffusions.

Abstract

We establish the existence of strong solutions to a class of cross diffusion systems on $\RR^N$ consists of $m$ equations ($m,N\ge 2$). which generalizes the Shigesada-Kawasaki-Teramoto (SKT) model in population dynamics. We introduce the concept of a {\em strong-weak solution} of the systems and show that their existence can be established under weaker conditions. These {\em strong-weak solutions} coincide with strong solutions so that the existence of strong solutions is proved. The SKT model on planar domains ($N=2$) with cubic diffusions and advections is completely solved.