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

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