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

<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 &amp;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 &amp;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&gt;0 $\end{document}</tex-math></inline-formula> with no-flux boundary conditions, where <inline-formula><tex-math id="M4">\begin{document}$ \chi&gt;0, \kappa&gt;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>

Global boundedness and asymptotic behavior in a quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source

In this work, we consider the quasilinear attraction–repulsion chemotaxis model with nonlinear signal production and logistic-type source. We present the global existence of classical solutions under appropriate regularity assumptions on the initial data. In addition, the asymptotic behavior of the solutions is studied, and our results generalize and improve some well-known results in the literature.