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

Abstract

<p style='text-indent:20px;'>In this paper we focus on this attraction-repulsion chemotaxis model with consumed signals</p><p style='text-indent:20px;'><disp-formula><label/><tex-math id="FE1"> \begin{document}$\begin{equation}\label{problem_abstract}\tag{$\Diamond$}\begin{cases}u_t = \Delta u - \chi \nabla \cdot (u \nabla v)+\xi \nabla \cdot (u \nabla w) & \text{ in }~~ \Omega \times (0, T_{max}), \\v_t = \Delta v- uv & \text{ in }~~ \Omega \times (0, T_{max}), \\w_t = \Delta w- uw & \text{ in }~~ \Omega \times (0, T_{max}), \end{cases}\end{equation}$ \end{document}</tex-math></disp-formula></p><p style='text-indent:20px;'>formulated in a bounded and smooth domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^n $\end{document}</tex-math></inline-formula>, with <inline-formula><tex-math id="M4">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>, for some positive real numbers <inline-formula><tex-math id="M5">\begin{document}$ \chi, \xi $\end{document}</tex-math></inline-formula> and with <inline-formula><tex-math id="M6">\begin{document}$ {T_{max}}\in (0, \infty] $\end{document}</tex-math></inline-formula>. Once equipped with appropriately smooth initial distributions <inline-formula><tex-math id="M7">\begin{document}$ u(x, 0) = u_0(x)\geq 0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ v(x, 0) = v_0(x)\geq 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ w(x, 0) = w_0(x)\geq 0 $\end{document}</tex-math></inline-formula>, as well as Neumann boundary conditions, we establish sufficient assumptions on its data yielding global and bounded classical solutions; these are functions <inline-formula><tex-math id="M10">\begin{document}$ u, v $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ w $\end{document}</tex-math></inline-formula>, with zero normal derivative on <inline-formula><tex-math id="M12">\begin{document}$ \partial \Omega\times (0, {T_{max}}) $\end{document}</tex-math></inline-formula>, satisfying pointwise the equations in problem <inline-formula><tex-math id="M13">\begin{document}$\Diamond$\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ {T_{max}} = \infty $\end{document}</tex-math></inline-formula>. This is proved for any such initial data, whenever <inline-formula><tex-math id="M15">\begin{document}$ \chi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> belong to bounded and open intervals, depending respectively on <inline-formula><tex-math id="M17">\begin{document}$ \|v_0\|_{L^{\infty}(\Omega)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M18">\begin{document}$ \|w_0\|_{L^{\infty}(\Omega)} $\end{document}</tex-math></inline-formula>. Finally, we illustrate some aspects of the dynamics present within the chemotaxis system by means of numerical simulations.</p>