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

This paper deals with a quasilinear chemotaxis system with nonlinear signal production\begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} & u_t = \nabla\cdot(\phi(u)\nabla u)-\chi\nabla\cdot(\psi(u)\nabla v), & (x, t)\in \Omega\times (0, \infty), \\ & v_t = \Delta v-v+g(u), & (x, t)\in \Omega\times (0, \infty), \end{split} \right. \end{eqnarray*} $\end{document}under homogeneous Neumann boundary conditions in a smoothly bounded domain \begin{document}$ \Omega \subset \mathbb{R}^{n} $\end{document} , where \begin{document}$ \chi\in \mathbb{R} $\end{document} , the nonnegative nonlinearities \begin{document}$ \phi, \psi $\end{document} and \begin{document}$ g $\end{document} belong to \begin{document}$ C^{2}([0, \infty)) $\end{document} and satisfy \begin{document}$ \phi(u)\geq K_{0}(u+1)^{m}, \psi(u)\leq K_{1}u(u+1)^{\alpha-1} $\end{document} and \begin{document}$ g(u)\leq K_{2}(u+1)^{\beta} $\end{document} with some \begin{document}$ K_{0}, K_{1}, K_{2}, \beta>0 $\end{document} and \begin{document}$ \alpha, m\in\mathbb{R} $\end{document} .\begin{document}$ \bullet $\end{document} In the chemo-attractive setting, i.e. \begin{document}$ \chi>0 $\end{document} , assume that \begin{document}$ n\geq1 $\end{document} and \begin{document}$ \beta>1 $\end{document} , it is shown that the solution of the above system is global and uniformly bounded provided that \begin{document}$ \alpha+\beta-m<1+\dfrac{2}{n} $\end{document} and \begin{document}$ m >-\dfrac{2}{n} $\end{document} .\begin{document}$ \bullet $\end{document} In the chemo-repulsive setting, i.e. \begin{document}$ \chi<0 $\end{document} , assume that \begin{document}$ n\geq3 $\end{document} and \begin{document}$ g'(u) \geq0 $\end{document} , it is proved that the solution of the above system is also global and uniformly bounded if \begin{document}$ \alpha-m+\dfrac{n-2}{n+2}\beta<1 $\end{document} .