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

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

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