# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021133
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## Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production

 1 College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China 2 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China 3 School of Sciences, Southwest Petroleum University, Chengdu 610500, China 4 College of Mathematics and Statistics, Yili Normal University, Yining 835000, China

* Corresponding author

Received  August 2020 Revised  June 2021 Early access August 2021

Fund Project: This work is supported in part by the NSFC under grants 11771062 and 11971082, the Fundamental Research Funds for the Central Universities under grants 2019CDJCYJ001, 2020CDJQY-Z001, Chongqing Key Laboratory of Analytic Mathematics and Applications, and Scientific Research Program of the Higher Education Institution of XinJiang under grant XJEDU2021Y043

The following degenerate chemotaxis system with flux limitation and nonlinear signal production
 $\begin{equation*} \begin{cases} u_t = \nabla\cdot(\frac{u\nabla u}{\sqrt {u^{2}+|\nabla u|^{2}}})-\chi\nabla\cdot(\frac{u\nabla v}{\sqrt {1+|\nabla v|^{2}}}) \quad &in\quad B_{R}\times(0, +\infty), \\ 0 = \Delta v-\mu (t)+u^{\kappa}, \quad \mu(t): = \frac{1}{|\Omega|}\int_{\Omega}u^{\kappa}(\cdot, t) \quad &in\quad B_{R}\times(0, +\infty) \end{cases} \end{equation*}$
is considered in balls
 $B_R = B_R(0)\subset \mathbb{R}^n$
for
 $n\geq 1$
and
 $R>0$
with no-flux boundary conditions, where
 $\chi>0, \kappa>0$
. 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
 $\chi, \kappa$
and
 $\int_{B_R}u_{0}$
.
Citation: Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021133
##### References:

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