WIAS Preprint No. 2326, (2016)

Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics



Authors

  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888

2010 Mathematics Subject Classification

  • 35K57 35B40 35Q79 92E20

Keywords

  • Reaction-diffusion systems, mass-action law, log-Sobolev inequality, exponential decay of relative entropy, energy-dissipation estimate, complex balance condition, detailed balance condition, convexity method

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 author's 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.

Appeared in

  • Patterns of Dynamics, P. Gurevich, J. Hell, B. Sandstede, A. Scheel, eds., Proceedings in Mathematics & Statistics, Springer, 2018, pp. 149--171, DOI 10.1007/978-3-319-64173-7_10 .

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