Revista Matemática Iberoamericana

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Volume 24, Issue 2, 2008, pp. 407–431
DOI: 10.4171/RMI/541

Published online: 2008-08-31

Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds

Laurent Desvillettes[1] and Klemens Fellner[2]

(1) CMLA-ENS, Cachan, France
(2) Universität Wien, Austria

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 $L^1$ to the steady state via a precise entropy-entropy dissipation estimate, 2) an explicit global $L^{\infty}$ bound via interpolation of a polynomially growing $H^1$ bound with the almost exponential $L^1$ convergence, and 3), finally, explicit exponential convergence to the steady state in all Sobolev norms.

Keywords: Reaction-diffusion, entropy method, exponential decay, slowly growing a-priori estimates

Desvillettes Laurent, Fellner Klemens: Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds. Rev. Mat. Iberoam. 24 (2008), 407-431. doi: 10.4171/RMI/541