Fast reaction limits and convergence rate for nonlinear bulk-surface reaction-diffusion systems modeling reversible chemical reactions

TL;DR

This paper analyzes the limit behavior and convergence rate of solutions in nonlinear bulk-surface reaction-diffusion systems modeling reversible chemical reactions as the reaction rate becomes infinitely fast.

Contribution

It establishes the convergence of solutions to a heat equation with nonlinear boundary conditions and derives the convergence rate for equal stoichiometric coefficients.

Findings

Solutions converge in $L^p$ spaces to a heat equation with nonlinear boundary conditions.

Uniform a-priori estimates are obtained for solutions as reaction rate tends to infinity.

Convergence rate is explicitly derived for systems with equal stoichiometric coefficients.

Abstract

The fast reaction limit for a nonlinear bulk-surface reaction-diffusion system is investigated. This system describes a reversible reaction with arbitrary stoichiometric coefficients, where one chemical is present in a bounded vessel $\Omega$ and the other chemical lies only on the boundary $\partial\Omega$ where the reaction takes place. In the limit as the reaction rate constant tends to infinity, we prove that the solution converges in $L^p(0,T;L^p(\Omega))$ to the solution of a heat equation with nonlinear dynamical boundary condition. This is obtained by showing a-priori estimates of solutions which are uniform in the reaction rate constants. In order to overcome the difficulty caused by the bulk-surface coupling, we consider the limit in suitable product spaces where the Aubin-Lions lemma is applicable. Moreover, in the case of equal stoichiometric coefficients, we obtain the convergence rate of the fast reaction limit by exploiting suitable estimates of the limiting system.