Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production

The following degenerate chemotaxis system with flux limitation and nonlinear signal production \begin{document}$ \begin{equation*} \begin{cases} u_t = \nabla\cdot(\frac{u\nabla u}{\sqrt {u^{2}+|\nabla u|^{2}}})-\chi\nabla\cdot(\frac{u\nabla v}{\sqrt {1+|\nabla v|^{2}}}) \quad i'>is considered in balls \begin{document}$ B_R = B_R(0)\subset \mathbb{R}^n $\end{document} for \begin{document}$ n\geq 1 $\end{document} and \begin{document}$ R>0 $\end{document} with no-flux boundary conditions, where \begin{document}$ \chi>0, \kappa>0 $\end{document} . We obtained local existence of unique classical solution and extensibility criterion ruling out gradient blow-up, and moreover proved global existence and boundedness of solutions under some conditions for \begin{document}$ \chi, \kappa $\end{document} and \begin{document}$ \int_{B_R}u_{0} $\end{document} .