Existence of solution of a triangular degenerate reaction-diffusion system

TL;DR

This paper investigates the existence of global solutions for a degenerate triangular reaction-diffusion system with multiple species, demonstrating conditions under which weak and classical solutions exist across various dimensions.

Contribution

It establishes the existence of weak and classical global solutions for degenerate triangular reaction-diffusion systems in any dimension, including specific results for quadratic non-linearities and three-dimensional cases.

Findings

Weak global solutions exist for all degenerate cases in any dimension.

Classical global solutions exist in all but one degenerate case, up to dimension 2.

Analysis of quadratic non-linear rate functions and three-dimensional cases.

Abstract

In this article we study a chemical reaction-diffusion system with $m$ unknown concentration. The non-linearity in our study comes from a particular chemical reaction where one unit of a particular species generated from other $m-1$ species and disintegrates to generate all those $m-1$ species in the same manner, i.e., triangular in nature. Our objective is to find whether global in time solution exists for this system where one or more species stops diffusing. In particular weak global in time solution exists for all the degenerate cases in any dimension. We are further able to show classical global in time solution exists for all the degenerate cases in any dimension except one and this particular case too attain classical global in time solution up to dimension $2$. We also analyze global in time existence result for the case of quadratic non-linear rate functions and also analyze a three dimensional case.