Uniform in time <inline-formula><tex-math id="M1">$ L^\infty $</tex-math></inline-formula>-estimates for an attraction-repulsion chemotaxis model with double saturation

In this paper we focus on this attraction-repulsion chemotaxis model with consumed signals $\begin{equation}\label{problem_abstract}\tag{$\Diamond$}\begin{cases}u_t = \Delta u - \chi \nabla \cdot (u \nabla v)+\xi \nabla \cdot (u \nabla w) & \text{ in }~~ \Omega \times (0, T_{max}), \\v_t = \Delta v- uv & \text{ in }~~ \Omega \times (0, T_{max}), \\w_t = \Delta w- uw & \text{ in }~~ \Omega \times (0, T_{max}), \end{cases}\end{equation}$ formulated in a bounded and smooth domain $ \Omega $ of $ {\mathbb R}^n $, with $ n\geq 2 $, for some positive real numbers $ \chi, \xi $ and with $ {T_{max}}\in (0, \infty] $. Once equipped with appropriately smooth initial distributions $ u(x, 0) = u_0(x)\geq 0 $, $ v(x, 0) = v_0(x)\geq 0 $ and $ w(x, 0) = w_0(x)\geq 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 $ \partial \Omega\times (0, {T_{max}}) $, satisfying pointwise the equations in problem $\Diamond$ with $ {T_{max}} = \infty $. This is proved for any such initial data, whenever $ \chi $ and $ \xi $ belong to bounded and open intervals, depending respectively on $ \|v_0\|_{L^{\infty}(\Omega)} $ and $ \|w_0\|_{L^{\infty}(\Omega)} $. Finally, we illustrate some aspects of the dynamics present within the chemotaxis system by means of numerical simulations.