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

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.