On an exponentially decaying diffusive chemotaxis system with indirect signals
Authors: Zheng Pan , Xing Jie 
Published: March 2022

<p style='text-indent:20px;'>This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &amp;u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &amp;(x,t)\in \Omega\times (0,\infty), \\ &amp;v_t = \Delta v+h(v,w), &amp;(x,t)\in \Omega\times (0,\infty), \\ &amp;w_t = \Delta w- w+u, &amp;(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, where the nonlinear diffusivity <inline-formula><tex-math id="M3">\begin{document}$ D $\end{document}</tex-math></inline-formula> and chemosensitivity <inline-formula><tex-math id="M4">\begin{document}$ S $\end{document}</tex-math></inline-formula> are supposed to satisfy</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ K_{1}e^{-\beta^{-}s}\leq D(s) \leq K_{2}e^{-\beta^{+}s} \;\;\;{\rm{and}}\;\;\;\frac{D(s)}{S(s)}\geq K_{3}s^{-\alpha}+\gamma, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with the constants <inline-formula><tex-math id="M5">\begin{document}$ \beta^{-}\geq \beta^{+}&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K_{1},K_{2},K_{3}&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \alpha,\gamma\geq0 $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M8">\begin{document}$ h(v,w) = -v+w $\end{document}</tex-math></inline-formula>, we study the global existence and boundedness of solutions for the above system provided that <inline-formula><tex-math id="M9">\begin{document}$ \alpha\in[0,\frac{2}{n}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \beta^{-}\geq \beta^{+}&gt;\frac{n}{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ \gamma&gt;1 $\end{document}</tex-math></inline-formula> and the initial mass of <inline-formula><tex-math id="M12">\begin{document}$ u_{0} $\end{document}</tex-math></inline-formula> is small enough. Moreover, it is proved that the global bounded solution <inline-formula><tex-math id="M13">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> converges to <inline-formula><tex-math id="M14">\begin{document}$ (\overline{u_{0}},\overline{u_{0}},\overline{u_{0}}) $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M15">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>-norm as <inline-formula><tex-math id="M16">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M17">\begin{document}$ \overline{u_{0}} = \frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)dx $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M18">\begin{document}$ h(v,w) = -vw $\end{document}</tex-math></inline-formula>, it is shown that this system possesses a unique uniformly bounded classical solution if <inline-formula><tex-math id="M19">\begin{document}$ \alpha\geq0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M20">\begin{document}$ \gamma&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M21">\begin{document}$ \beta^{-}\geq \beta^{+}&gt;\frac{n}{2} $\end{document}</tex-math></inline-formula>. Furthermore, if <inline-formula><tex-math id="M22">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M23">\begin{document}$ \alpha\geq0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M24">\begin{document}$ \gamma\geq0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M25">\begin{document}$ \beta^{-}\geq \beta^{+}&gt;\varepsilon $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M26">\begin{document}$ \varepsilon&gt;0 $\end{document}</tex-math></inline-formula>, we only obtain the global existence of solutions for the above system.</p>