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