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In this paper we study the existence of uniqueness global weak solutions for m × m reaction-diffusion systems for which two main properties hold: the positivity of the weak solutions and the total mass of the components are preserved with time. Moreover, we suppose that the non-linearities have critical growth with respect to the gradient. The technique we use here in order to prove global existence is in the same spirit of the method developed by Boccardo, Murat, and Puel for a single equation. Keywords: Semigroups, local weak solution, global weak solution, reaction- diffusion systems, invariant regions, matrice of diffusion.
Barrouk et al. (Fri,) studied this question.