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

  • * Corresponding author

    * Corresponding author 
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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  • 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} $.

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


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