Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production

This paper deals with a quasilinear parabolic-elliptic chemo-repulsion 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(u(u+1)^{\alpha-1}\nabla v)+f(u), & (x,t)\in \Omega\times (0,\infty), \\ & 0 = \Delta v-v+u^{\beta}, & (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}(n\geq1), $\end{document} where \begin{document}$ \chi,\beta>0,\alpha\in\mathbb{R}, $\end{document} the nonlinear diffusion \begin{document}$ \phi\in C^{2}([0,\infty)) $\end{document} satisfies \begin{document}$ \phi(u)\geq(u+1)^{m} $\end{document} with \begin{document}$ m\in\mathbb{R}, $\end{document} and the function \begin{document}$ f\in C^{1}([0,\infty)) $\end{document} is a generalized growth term. \begin{document}$ \bullet $\end{document} When \begin{document}$ f\equiv0, $\end{document} it is shown that the solution of the above system is global and uniformly bounded for all \begin{document}$ \chi,\beta>0 $\end{document} and \begin{document}$ m,\alpha\in\mathbb{R} $\end{document} . \begin{document}$ \bullet $\end{document} When \begin{document}$ f\not\equiv0 $\end{document} and assume that \begin{document}$ f(u)\leq ku-bu^{\gamma+1} $\end{document} with \begin{document}$ k,b,\gamma>0, $\end{document} it is proved that the solution of the above system is also global and uniformly bounded for all \begin{document}$ \chi,\beta>0 $\end{document} and \begin{document}$ m,\alpha\in\mathbb{R}. $\end{document}