Boundedness for reaction-diffusion systems with Lyapunov functions and intermediate sum conditions

TL;DR

This paper establishes conditions under which solutions to reaction-diffusion systems remain uniformly bounded over time, broadening previous results by allowing more general nonlinearities and including applications to chemical networks.

Contribution

It introduces an intermediate sum condition that generalizes mass dissipation, enabling boundedness results for systems with arbitrary polynomial growth nonlinearities.

Findings

Two-dimensional systems with quadratic sum conditions have globally bounded solutions.

Higher-dimensional systems are bounded if diffusion coefficients are quasi-uniform.

Applications include boundedness in chemical reaction networks with diffusion.

Abstract

We study the uniform boundedness of solutions to reaction-diffusion systems possessing a Lyapunov-like function and satisfying an {\it intermediate sum condition}. This significantly generalizes the mass dissipation condition in the literature and thus allows the nonlinearities to have arbitrary polynomial growth. We show that two dimensional reaction-diffusion systems, with quadratic intermediate sum conditions, have global solutions which are bounded uniformly in time. In higher dimension, bounded solutions are obtained under the condition that the diffusion coefficients are {\it quasi-uniform}, i.e. they are close to each other. Applications include boundedness of solutions to chemical reaction networks with diffusion.