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 <= 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 <= 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. (C) 2017 Elsevier Ltd. All rights reserved.