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

Hong Yi; Chunlai Mu; Shuyan Qiu; Lu Xu

doi:10.3934/cpaa.2021133

Article Contents

2021, Volume 20, Issue 11: 3825-3849. Doi: 10.3934/cpaa.2021133 open access

This issue Previous Article Continuous solution for a non-linear eikonal system Next Article Liouville-type theorem for higher-order Hardy-Hénon system

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
^* Corresponding author

Received: August 2020

Revised: June 2021

Early access: August 2021

Published: November 2021

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

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Abstract

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} $.

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Mathematics Subject Classification: Primary: 35K65, 35B51; Secondary: 39A22.

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