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 $d\le 3$ of the system of reaction diffusion $\mathrm{\partial }\mathrm{\_}ta\mathrm{\_}i-d\mathrm{\_}i\mathrm{\Delta }a\mathrm{\_}i=(-1{)}^i(a\mathrm{\_}1a\mathrm{\_}3-a\mathrm{\_}2a\mathrm{\_}4)$, where $a\mathrm{\_}i\ge 0$ model the concentrations of chemical species undergoing a chemical reaction and diffusing (each with its diffusion rate $d\mathrm{\_}i>0$) in a bounded container.