Explicit exponential convergence to equilibrium for nonlinear reaction-diffusion systems with detailed balance condition

TL;DR

This paper establishes explicit exponential convergence rates to equilibrium for certain reaction-diffusion systems satisfying detailed balance, using an entropy method that leverages conservation laws.

Contribution

It introduces a general entropy-based approach to quantify convergence rates in reaction-diffusion systems with detailed balance, including explicit decay estimates.

Findings

Proves exponential convergence to equilibrium with explicit rates.

Applies the method to single reversible reactions with multiple substances.

Analyzes enzyme reaction chains with explicit decay estimates.

Abstract

The convergence to equilibrium of mass action reaction-diffusion systems arising from networks of chemical reactions is studied. The considered reaction networks are assumed to satisfy the detailed balance condition and have no boundary equilibria. We propose a general approach based on the so-called entropy method, which is able to quantify with explicitly computable rates the decay of an entropy functional in terms of an entropy entropy-dissipation inequality based on the totality of the conservation laws of the system. As a consequence follows convergence to the unique detailed balance equilibrium with explicitly computable convergence rates. The general approach is further detailed for two important example systems: a single reversible reaction involving an arbitrary number of chemical substances and a chain of two reversible reactions arising from enzyme reactions.