Uniform exponential decay for reaction-diffusion systems with
complex-balanced mass-action kinetics : dedicated to Bernold Fiedler on the occasion of his sixtieth birthday
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
Doi
Abstract
We consider reaction-diffusion systems on a bounded domain with no-flux
boundary conditions. All reactions are given by the mass-action law and are
assumed to satisfy the complex-balance condition. In the case of a diagonal
diffusion matrix, the relative entropy is a Liapunov functional. We give an
elementary proof for the Liapunov property as well a few explicit examples
for the condition of complex or detailed balancing. We discuss three methods
to obtain energy-dissipation estimates, which guarantee exponential decay of
the relative entropy, all of which rely on the log-Sobolev estimate and
suitable handling of the reaction terms as well as the mass-conservation
relations. The three methods are (i) a convexification argument based on the
authors joint work with Haskovec and Markowich, (ii) a series of analytical
estimates derived by Desvillettes, Fellner, and Tang, and (iii) a compactness
argument of developed by Glitzky and Hünlich