Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions

TL;DR

This paper establishes uniform-in-time bounds for classical solutions to reaction-diffusion systems with mass dissipation in higher dimensions, advancing understanding of global existence beyond low-dimensional cases.

Contribution

It extends uniform-in-time bounds to higher-dimensional reaction-diffusion systems with mass dissipation and quadratic growth nonlinearities, previously known only in low dimensions.

Findings

Uniform bounds for solutions in all space dimensions.

Application to skew-symmetric Lotka-Volterra systems.

Existence of unique, globally bounded solutions with compact trajectories.

Abstract

Uniform-in-time bounds of nonnegative classical solutions to reaction-diffusion systems in all space dimension are proved. The systems are assumed to dissipate the total mass and to have locally Lipschitz nonlinearities of at most (slightly super-) quadratic growth. This pushes forward the recent advances concerning global existence of reaction-diffusion systems dissipating mass in which a uniform-in-time bound has been known only in space dimension one or two. As an application, skew-symmetric Lotka-Volterra systems are shown to have unique classical solutions which are uniformly bounded in time in all dimensions with relatively compact trajectories in $C(\overline{\Omega})^m$.