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

Abstract Large time dynamics of reaction-diffusion systems modeling some irreversible reaction networks are investigated. These systems can have boundary equilibria, which are constant steady states at which some of the chemical concentrations are completely used up. In the absence of these equilibria, we show an explicit convergence to the positive equilibrium by a modified entropy method, where it is shown that reactions in a measurable set with positive measure are sufficient to combine with diffusion to drive the system towards equilibrium. When boundary equilibria are present, we show that they are nonlinearly unstable by using a bootstrap instability technique, 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.