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

Redouane Douaifia, Salem Abdelmalek, Mokhtar Kirane

TL;DR

This paper addresses global existence and uniform-in-time bounds for a three-component reaction-diffusion system with mass control and the linear intermediate weighted sum condition. It introduces an $L^p$-energy polynomial framework based on a homogeneous polynomial $\mathcal{H}_p$ to construct Lyapunov functionals that yield global classical solutions in arbitrary dimensions and under broad boundary conditions, even with arbitrary polynomial growth in the nonlinearities. Under slightly stronger assumptions and mixed boundary conditions, the framework provides uniform-in-time $L^\infty$ bounds, leveraging the regularizing effect of the parabolic operator. The results extend to general boundary conditions and to higher-order interactions, including three-species sub-skew-symmetric Lotka-Volterra models. This work broadens the class of RD systems for which global well-posedness and uniform bounds are known, with potential applications in chemical and biological modeling.

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.

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 addresses global existence and uniform-in-time bounds for a three-component reaction-diffusion system with mass control and the linear intermediate weighted sum condition. It introduces an -energy polynomial framework based on a homogeneous polynomial to construct Lyapunov functionals that yield global classical solutions in arbitrary dimensions and under broad boundary conditions, even with arbitrary polynomial growth in the nonlinearities. Under slightly stronger assumptions and mixed boundary conditions, the framework provides uniform-in-time bounds, leveraging the regularizing effect of the parabolic operator. The results extend to general boundary conditions and to higher-order interactions, including three-species sub-skew-symmetric Lotka-Volterra models. This work broadens the class of RD systems for which global well-posedness and uniform bounds are known, with potential applications in chemical and biological modeling.

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 -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.

Paper Structure

This paper contains 15 sections, 13 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. State of the art and motivation
  4. Main results and key ideas
  5. Structure of the paper
  6. Proof of main results
  7. Preliminaries
  8. Theorem \ref{['main_Theor']}: Global existence
  9. Theorem \ref{['main_Theor']}: Uniform-in-time bounds
  10. Proof of Theorem \ref{['main_Theor2']}
  11. Generalizations: Proof of Theorems \ref{['main_Theor3GBCs']} and \ref{['main_Theor4GBCs']}.
  12. Applications
  13. Example 1
  14. Example 2
  15. Example 3: HOIs three-species Lotka-Voltera model

Key Result

Theorem 1

Assume the conditions a1, a2, a3, a4, for some $\lambda _{1}\geq$$\theta ^{2(p-1)}$, $\lambda _{2}\geq$$\sigma ^{2(p-1)}$, with such that and $u_{0},v_{0},w_{0}\in L^{\infty }\left( \Omega ;\mathbb{R}_{+}\right)$. Then the system (main_system) possesses a unique nonnegative global classical solution. Moreover, if $K_i= 0$ for all $i=1,\dots,4$, then the solution is bounded uniformly in time, i.e

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Remark 1: Generalization
  • Theorem 3
  • Theorem 4
  • Definition 1: Classical solutions
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 16 more