In the continuation of [Desvillettes, L., Fellner, K.: Exponential Decay toward Equilibrium via Entropy Methods for Reaction-Diffusion Equations. J. Math. Anal. Appl. 319 (2006), no. 1, 157-176], we study reversible reaction-diffusion equations via entropy methods (based on the free energy functional) for a 1D system of four species. We improve the existing theory by getting 1) almost exponential convergence in to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global bound via interpolation of a polynomially growing bound with the almost exponential convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.
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Laurent Desvillettes, Klemens Fellner, Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds. Rev. Mat. Iberoam. 24 (2008), no. 2, pp. 407–431DOI 10.4171/RMI/541