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

Abstract

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.