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Exponential decay towards equilibrium and global classical solutions for nonlinear reaction-diffusion systems

We consider a system of reaction-diffusion equations describing the reversible reaction of two species $\mathcal{U}, \mathcal{V}$ forming a third species $\mathcal{W}$ and vice versa according to mass action law kinetics with arbitrary stochiometric coefficients (equal or larger than one). Firstly, we prove existence of global classical solutions via improved duality estimates under the assumption that one of the diffusion coefficients of $\mathcal{U}$ or $\mathcal{V}$ is sufficiently close to the diffusion coefficient of $\mathcal{W}$. Secondly, we derive an entropy entropy-dissipation estimate, that is a functional inequality, which applied to global solutions of these reaction-diffusion system proves exponential convergence to equilibrium with explicit rates and constants.