# Close-to-equilibrium regularity for reaction-diffusion systems

## TL;DR

This paper proves that solutions to certain reaction-diffusion systems, which model chemical reactions, remain regular and exponentially converge to equilibrium when starting close to equilibrium in low dimensions, extending previous results.

## Contribution

It establishes global regularity and exponential convergence for reaction-diffusion systems near equilibrium under specific dimension and nonlinearity conditions, improving prior work.

## Key findings

- Solutions are globally regular near equilibrium in low dimensions.
- Solutions converge exponentially to equilibrium in the supremum norm.
- Results extend to certain higher dimensions and nonlinearities.

## Abstract

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 ($d\leq 4$) and order of nonlinearities ($\mu = 1 + 4/d$), 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 $L^{\infty}$-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].