Global existence of solutions for an $m$-component reaction--diffusion system with a tridiagonal 2-Toeplitz diffusion matrix and polynomially growing reaction terms

TL;DR

This paper proves the global existence of solutions for a complex reaction-diffusion system with a special diffusion matrix, using eigenanalysis, invariant regions, and Lyapunov functionals, supported by numerical validation.

Contribution

It introduces a novel approach to analyze an $m$-component reaction-diffusion system with a tridiagonal 2-Toeplitz diffusion matrix, establishing conditions for global solutions.

Findings

Eigenvalues and eigenvectors of the diffusion matrix are derived.

Parabolicity conditions are established for system diagonalization.

Global existence of solutions is confirmed through Lyapunov functionals.

Abstract

This paper is concerned with the local and global existence of solutions for a generalized $m$-component reaction--diffusion system with a tridiagonal $2$--Toeplitz diffusion matrix and polynomial growth. We derive the eigenvalues and eigenvectors and determine the parabolicity conditions in order to diagonalize the proposed system. We, then,determine the invariant regions and utilize a Lyapunov functional to establish the global existence of solutions for the proposed system. A numerical example is used to illustrate and confirm the findings of the study.