A weak comparison principle for reaction-diffusion systems

TL;DR

This paper establishes a weak comparison principle for reaction-diffusion systems without requiring solution uniqueness, and applies it to ecological, chemical, and generalized models, providing new bounds and maximum principles.

Contribution

It introduces a novel weak comparison principle for reaction-diffusion systems lacking solution uniqueness and applies it to various complex models, including Lotka-Volterra and fractional-order systems.

Findings

Weak comparison principle proved for reaction-diffusion systems

Application to Lotka-Volterra, logistic, and fractional models

Establishment of a weak maximum principle and bounds in L-infinity

Abstract

In this paper we prove a weak comparison principle for a reaction-diffusion system without uniqueness of solutions. We apply the abstract results to the Lotka-Volterra system with diffusion, a generalized logistic equation and to a model of fractional-order chemical autocatalysis with decay. Morever, in the case of the Lotka-Volterra system a weak maximum principle is given, and a suitable estimate in the space of essentially bounded functions $L^{\infty}$ is proved for at least one solution of the problem.