Global existence of solutions for an m-component cross-diffusion system with a 3-component case study

In this paper, we examine a general m-component reaction-diffusion matrix with a full diffusion matrix and polynomially growing reaction terms through its diagonalization. We establish the invariant regions of the system and derive the necessary conditions for the existence of solutions. The 3 × 3 case is taken as a case study, where we determine the exact conditions for the positivity of the eigenvalues, which is necessary for the diagonalization process. Numerical examples are used to illustrate and confirm the findings of this paper.