On the global existence and uniform-in-time bounds for three-component reaction-diffusion systems with mass control and polynomial growth

TL;DR

This paper proves the global existence and uniform-in-time bounds for three-component reaction-diffusion systems with polynomial growth, mass control, and a new structural condition, applicable to complex biological models.

Contribution

It introduces a novel linear intermediate weighted sum condition and extends $L^p$-energy methods to establish results for systems with polynomial nonlinearities.

Findings

Global classical solutions exist in arbitrary dimensions.

Solutions can have uniform-in-time bounds under certain conditions.

Framework applies to complex Lotka-Volterra systems with higher-order interactions.

Abstract

We investigate a class of three-component reaction-diffusion systems subject to mass control and a newly introduced structural assumption, referred to as linear intermediate weighted sum condition. Under these hypotheses, we establish the global existence of classical solutions in arbitrary spatial dimensions and wide class of boundary conditions, even when the nonlinearities exhibit arbitrary polynomial growth. We establish also that, under slight-stronger assumptions and mixed boundary conditions, solutions admit uniform-in-time bounds. Our approach relies on the extension of $L^p$-energy polynomial functionals, together with the regularizing effect for parabolic equations. Furthermore, we demonstrate the applicability of our framework by analyzing three-species sub-skew-symmetric Lotka-Volterra systems with higher-order interactions.