Stability analysis of irreversible chemical reaction-diffusion systems with boundary equilibria

TL;DR

This paper analyzes the long-term behavior of irreversible reaction-diffusion systems, demonstrating conditions for convergence to equilibrium or instability of boundary equilibria, using entropy methods and spectral analysis.

Contribution

It provides a novel stability analysis framework for reaction-diffusion systems with boundary equilibria, combining entropy methods and spectral gap techniques.

Findings

Explicit convergence to equilibrium when no boundary equilibria are present

Boundary equilibria are shown to be unstable in Lyapunov sense

Positive equilibrium is nonlinearly stable due to spectral gap

Abstract

Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. Depending on initial masses, these networks possibly possess boundary equilibria, where some of the chemical concentrations are completely used up. In the absence of these equilibria, we show an explicit convergence to equilibrium by a modified entropy method, where it is shown that reactions in a measurable set with positive measure is sufficient to combine with diffusion and to drive the system towards equilibrium. When the boundary equilibria are present, we show that they are unstable (in Lyapunov sense) using some bootstrap instability technique from fluid mechanics, while the nonlinear stability of the positive equilibrium is proved by exploiting a spectral gap of the linearized operator and the uniform-in-time boundedness of solutions.