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.