Close-to-equilibrium regularity for reaction-diffusion systems

The close-to-equilibrium regularity of solutions to a class of reaction-diffusion systems is investigated. The considered systems typically arise from chemical reaction networks and satisfy a complex balanced condition. Under some restrictions on spatial dimensions ($d\le 4$) and order of nonlinearities ($\mu =1+4\mathrm{/}d$), we show that if the initial data is close to a complex balanced equilibrium in $L^2$-norm, then classical solutions are shown global and converging exponentially to equilibrium in $L^{\mathrm{\infty }}$-norm. Possible extensions to higher dimensions and order of nonlinearities are also discussed. The results of this paper improve the recent work [M.J. C'aceres and J.A. Ca~nizo, Nonlinear Analysis: TMA 159 (2017): 62-84].