Close-to-equilibrium behaviour of quadratic reaction-diffusion systems with detailed balance

We study general quadratic reaction-diffusion systems with detailed balance, in space dimension $d\le 4$. We show that close-to-equilibrium solutions (in an $L^2$ sense) are regular for all times, and that they relax to equilibrium exponentially in a strong sense. That is: all detailed balance equilibria are exponentially asymptotically stable in all $L^p$ norms, at least in dimension $d\le 4$. The results are given in detail for the four-species reaction-diffusion system, where the involved constants can be estimated explicitly. The main novelty is the regularity result and exponential relaxation in $L^p$ norms for $p>1$, which up to our knowledge is new in dimensions 3 and 4.