Uniform in time L∞ estimates for an attraction-repulsion chemotaxis model with double saturation

In this paper we focus on this attraction-repulsion chemotaxis model with consumed signals {ut = ?u - chi & nabla; middot (u & nabla;v) + xi & nabla; middot (u & nabla;w) in ? x (0, T-max),vt = ?v - uv in ? x (0, T-max),wt = ?w - uw in ? x (0, T-max), (?) formulated in a bounded and smooth domain ? of R-n, with n >= 2, for some positive real numbers chi, xi and with T-max is an element of (0, infinity]. 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 & part;? x (0, T-max), satisfying pointwise the equations in problem (? ) with T-max = infinity. This is proved for any such initial data, whenever chi and xi belong to bounded and open intervals, depending respectively on Ilv(0)Il(L infinity()?) and Ilw0Il(L infinity)(?). Finally, we illustrate some aspects of the dynamics present within the chemotaxis system by means of numerical simulations.