Simple proofs for the existence of smooth solutions to a reaction-diffusion system modeling reversible chemistry

We present in this work a very short proof for the existence, uniqueness and smoothness in dimensions $d\leq 3$ of the system of reaction diffusion $\partial\_t a\_i - d\_i \Delta a\_i = (-1)^i (a\_1 a\_3 - a\_2 a\_4)$, where $a\_i \geq 0$ model the concentrations of chemical species undergoing a chemical reaction and diffusing (each with its diffusion rate $d\_i > 0$) in a bounded container.