Global Dissipative Dynamics of the Extended Brusselator System

TL;DR

This paper proves the existence of a finite-dimensional global attractor for the extended Brusselator reaction-diffusion system, using novel estimation methods, with implications for biological and biochemical systems.

Contribution

It introduces a new grouping and re-scaling estimation method to establish the global attractor and its properties for a complex reaction-diffusion system.

Findings

Existence of a global attractor in $L^2$ space.

Finite Hausdorff and fractal dimensions of the attractor.

Regularity of the attractor in $L^ Infty$.

Abstract

The existence of a global attractor for the solution semiflow of the extended Brusselator system in the $L^2$ phase space is proved, which is a cubic-autocatalytic and partially reversible reaction-diffusion system with linear coupling between two compartments. The method of grouping and re-scaling estimation is developed to deal with the challenge in proving the absorbing property and the asymptotic compactness of this typical multi-component reaction-diffusion systems. It is also proved that the global attractor is an $(H, E)$ global attractor with the $L^\infty$ regularity and that the Hausdorff dimension and the fractal dimension of the global attractor are finite. The results and methodology can find many applications and further extensions in complex biological and biochemical dynamical systems.