  We study general quadratic reaction-diffusion systems with detailed balance,
in space dimension $d \leq 4$. 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
$d \leq 4$. 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.
