Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (3): 909-924.doi: 10.1007/s10473-024-0308-7

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GLOBAL WEAK SOLUTIONS FOR AN ATTRACTION-REPULSION CHEMOTAXIS SYSTEM WITH $p$-LAPLACIAN DIFFUSION AND LOGISTIC SOURCE

Xiaoshan Wang1, Zhongqian Wang2, Zhe Jia3,*   

  1. 1. Department of Mathematics, Luoyang Normal University, Luoyang 471934, China;
    2. School of Mathematics Science, Jiangsu Second Normal University, Nanjing 210013, China;
    3. School of Mathematics and Statistics, Linyi University, Linyi 276005, China
  • Received:2022-10-10 Revised:2023-08-11 Online:2024-06-25 Published:2024-05-21
  • Contact: *Zhe Jia,E-mail:jiazhe@lyu.edu.cn
  • About author:Xiaoshan Wang,E-mail:xswang2017@126.com;Zhongqian Wang,E-mail:channing88@163.com
  • Supported by:
    National Natural Science Foundation of China(12301251,12271232), the Natural Science Foundation of Shandong Province, China (ZR2021QA038) and the Scientific Research Foundation of Linyi University, China (LYDX2020BS014).

Abstract: This paper is concerned with the following attraction-repulsion chemotaxis system with $p$-Laplacian diffusion and logistic source:
$$\left\{\begin{array}{ll}u_{t}=\nabla\cdot(|\nabla u|^{p-2}\nabla u)-\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u),\;\;\;x\in \Omega,\;t>0,\\v_{t}=\triangle v-\beta v+\alpha u^{k_{1}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\0=\triangle w-\delta w+\gamma u^{k_{2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\u(x,0)=u_{0}(x),\;\;v(x,0)=v_{0}(x),\;\;w(x,0)=w_{0}(x), \;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega.\end{array}\right.$$
The system here is under a homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^{n}(n\geq2) $, with $ \chi, \xi, \alpha,\beta,\gamma,\delta, k_{1}, k_{2} >0, p\geq 2$. In addition, the function $f$ is smooth and satisfies that $f(s)\leq\kappa-\mu s^{l}$ for all $s\geq0$, with $\kappa\in \mathbb{R}, \mu>0, l>1$. It is shown that (i) if $l>\max\{ 2k_{1}, \frac{2k_{1}n}{2+n}+\frac{1}{p-1}\}$, then system possesses a global bounded weak solution and (ii) if $k_{2}>\max\{2k_{1}-1, \frac{2k_{1}n}{2+n}+\frac{2-p}{p-1}\}$ with $l>2$, then system possesses a global bounded weak solution.

Key words: global weak solutions, attraction-repulsion, $p$-Laplacian, logistic source

CLC Number: 

  • 35Q92
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