Global Existence and Long time dynamics of a four compartment Brusselator Type system

TL;DR

This paper proves the global existence of solutions and the existence of a finite-dimensional attractor for a four-compartment Brusselator system with sign-changing nonlinearities, using Lyapunov functionals and numerical validation.

Contribution

It introduces a novel approach to establish global existence and attractor properties for a complex reaction-diffusion system with sign-changing nonlinearities.

Findings

Proved global existence of solutions using Lyapunov functional.

Established the existence of a finite-dimensional global attractor.

Numerically validated theoretical results with simulations and attractor reconstruction.

Abstract

In this work we consider a four compartment Brusselator system. The reaction terms of this system are of non constant sign, thus components of the solution are not bounded apriori, and functional means to derive apriori bounds will fail. We prove global existence of solutions, via construction of an appropriate lyapunov functional. Furthermore due to the sign changing nonlinearities, the asymptotic sign condition is also not satisfied, causing further difficulties in proving the existence of global attractor. These difficulties are circumvented via the use of the lyapunov functional constructed along with the use of the uniform gronwall lemma. We are able to prove the existence of an attractor for the system, improving previous results in the literature.The Hausdorff and fractal dimensions of the attractor are also shown to be finite.In particular we derive a lower bound on the Hausdorff dimension of the global attractor. We use numerical simulations,as well as numerical attractor reconstruction methods via nonlinear time series analysis, to validate our results.