In this paper we focus on this attraction-repulsion chemotaxis model with consumed signalsformulated in a bounded and smooth domain Ω of R n , with n ≥ 2, for some positive real numbers χ, ξ and with Tmax ∈ (0, ∞].Once equipped with appropriately smooth initial distributions u(x, 0) = u 0 (x) ≥ 0, v(x, 0) = v 0 (x) ≥ 0 and w(x, 0) = w 0 (x) ≥ 0, as well as Neumann boundary conditions, we establish sufficient assumptions on its data yielding global and bounded classical solutions; these are functions u, v and w, with zero normal derivative on ∂Ω × (0, Tmax), pointwisely satisfying the equations in problem (✸) with Tmax = ∞. This is proved: 1) for any such initial data, whenever χ and ξ belong to bounded and open intervals, depending respectively on v 0 L ∞ (Ω) and w 0 L ∞ (Ω) ; 2) for a wider (but also bounded) interval of ξ, provided a further largeness assumption on the minimum of w 0 in Ω is imposed; 3) for an interval of ξ arbitrarily extendable, whenever the mentioned largeness assumption on the minimum of w 0 in Ω is preserved, and as long as also w 0 L ∞ (Ω) is allowed to indefinitely increase. Lastly, we conclude the article discussing our results in the frame of others available in the literature; specifically, in the more restricted class of initial data u 0 , v 0 , w 0 such that χv 0 − ξw 0 ≥ 0, model (✸) may be also faced by considering a natural transformation and then adapting already known theorems. Again with regard to the boundeness of its solutions, the aforementioned transformation yields such a property for sharper conditions on χv 0 − ξw 0 L ∞ (Ω) than those directly obtained with our approach, when the initial data are taken in accordance to items 1) and 2). Conversely, for those data complying with restrictions in item 3), the situation changes and, making the most from the repulsive coefficient ξ, our strategy provides a milder condition.
