Fast Reaction Limits and Convergence Rate for Nonlinear Bulk-Surface Reaction–Diffusion Systems Modelling Reversible Chemical Reactions

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 Ω , and the other chemical lies only on the boundary ∂ Ω 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 ( Ω ) × L p ( Γ ) ) for any 1 ≤ p < ∞ to the solution of a heat equation with a 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.