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

Abstract

<p style='text-indent:20px;'>The following degenerate chemotaxis system with flux limitation and nonlinear signal production</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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 &in\quad B_{R}\times(0, +\infty), \\ 0 = \Delta v-\mu (t)+u^{\kappa}, \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}u^{\kappa}(\cdot, t) \quad &in\quad B_{R}\times(0, +\infty) \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>is considered in balls <inline-formula><tex-math id="M1">\begin{document}$ B_R = B_R(0)\subset \mathbb{R}^n $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ R>0 $\end{document}</tex-math></inline-formula> with no-flux boundary conditions, where <inline-formula><tex-math id="M4">\begin{document}$ \chi>0, \kappa>0 $\end{document}</tex-math></inline-formula>. We obtained local existence of unique classical solution and extensibility criterion ruling out gradient blow-up, and moreover proved global existence and boundedness of solutions under some conditions for <inline-formula><tex-math id="M5">\begin{document}$ \chi, \kappa $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \int_{B_R}u_{0} $\end{document}</tex-math></inline-formula>.</p>