Global attraction of the positive equilibrium in an irreversible reaction--diffusion system

An irreversible reaction--diffusion system can have both a positive equilibrium and a stationary state at which the catalyst is absent. For the three-species system studied by Nguyen and Tang, we prove that every strictly positive compatible initial triple converges uniformly to the positive equilibrium whenever the conserved masses satisfy \(M_2\le M_1<2M_2\). The result holds on bounded connected smooth domains in every dimension and for arbitrary positive diffusion coefficients. Quadratic dissipation first removes the spatial oscillations of two species. The total catalyst mass then satisfies a logistic equation with a relative error tending to zero, which identifies the positive limit and closes the strictly positive form of the global-attraction question.