Exponential decay toward equilibrium via log convexity for a degenerate reaction-diffusion system

TL;DR

This paper demonstrates that exponential convergence to equilibrium in a reaction-diffusion system persists even when the reaction occurs only on a subset of the domain, using a novel log convexity approach.

Contribution

It extends known exponential convergence results to cases where the reaction is localized, employing an observation estimate derived from logarithmic convexity.

Findings

Exponential decay to equilibrium is achieved even with localized reactions.

Logarithmic convexity provides a key estimate for partial domain reactions.

The method applies to a class of degenerate reaction-diffusion systems.

Abstract

We consider a system of two reaction-diffusion equations coming out of reversible chemistry. When the reaction happens on the totality of the domain, it is known that exponential convergence to equilibrium holds. We show in this paper that this exponential convergence also holds when the reaction holds only on a given open set of a ball, thanks to an observation estimate deduced by logarithmic convexity.