On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production

Abstract

<p style='text-indent:20px;'>This paper deals with a quasilinear chemotaxis system with nonlinear signal production</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(\phi(u)\nabla u)-\chi\nabla\cdot(\psi(u)\nabla v), &amp; (x, t)\in \Omega\times (0, \infty), \\ &amp; v_t = \Delta v-v+g(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} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \chi\in \mathbb{R} $\end{document}</tex-math></inline-formula>, the nonnegative nonlinearities <inline-formula><tex-math id="M3">\begin{document}$ \phi, \psi $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ g $\end{document}</tex-math></inline-formula> belong to <inline-formula><tex-math id="M5">\begin{document}$ C^{2}([0, \infty)) $\end{document}</tex-math></inline-formula> and satisfy <inline-formula><tex-math id="M6">\begin{document}$ \phi(u)\geq K_{0}(u+1)^{m}, \psi(u)\leq K_{1}u(u+1)^{\alpha-1} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ g(u)\leq K_{2}(u+1)^{\beta} $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M8">\begin{document}$ K_{0}, K_{1}, K_{2}, \beta&gt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \alpha, m\in\mathbb{R} $\end{document}</tex-math></inline-formula>. </p> <p style='text-indent:20px;'><inline-formula><tex-math id="M10">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> In the chemo-attractive setting, i.e. <inline-formula><tex-math id="M11">\begin{document}$ \chi&gt;0 $\end{document}</tex-math></inline-formula>, assume that <inline-formula><tex-math id="M12">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ \beta&gt;1 $\end{document}</tex-math></inline-formula>, it is shown that the solution of the above system is global and uniformly bounded provided that <inline-formula><tex-math id="M14">\begin{document}$ \alpha+\beta-m&lt;1+\dfrac{2}{n} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ m &gt;-\dfrac{2}{n} $\end{document}</tex-math></inline-formula>.</p> <p style='text-indent:20px;'><inline-formula><tex-math id="M16">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> In the chemo-repulsive setting, i.e. <inline-formula><tex-math id="M17">\begin{document}$ \chi&lt;0 $\end{document}</tex-math></inline-formula>, assume that <inline-formula><tex-math id="M18">\begin{document}$ n\geq3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ g'(u) \geq0 $\end{document}</tex-math></inline-formula>, it is proved that the solution of the above system is also global and uniformly bounded if <inline-formula><tex-math id="M20">\begin{document}$ \alpha-m+\dfrac{n-2}{n+2}\beta&lt;1 $\end{document}</tex-math></inline-formula>.</p>