
GLOBAL WEAK SOLUTIONS FOR AN ATTRACTIONREPULSION CHEMOTAXIS SYSTEM WITH $p$LAPLACIAN DIFFUSION AND LOGISTIC SOURCE
Xiaoshan Wang, Zhongqian Wang, Zhe Jia
Acta mathematica scientia,Series B
2024, 44 (3):
909924.
DOI: 10.1007/s1047302403087
This paper is concerned with the following attractionrepulsion chemotaxis system with $p$Laplacian diffusion and logistic source: $$\left\{\begin{array}{ll}u_{t}=\nabla\cdot(\nabla u^{p2}\nabla u)\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u),\;\;\;x\in \Omega,\;t>0,\\v_{t}=\triangle v\beta v+\alpha u^{k_{1}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\0=\triangle w\delta w+\gamma u^{k_{2}},\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega,\;t>0,\\u(x,0)=u_{0}(x),\;\;v(x,0)=v_{0}(x),\;\;w(x,0)=w_{0}(x), \;\;\;\;\;\;\;\;\;\;\;\;\;x\in \Omega.\end{array}\right.$$ 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 $l>\max\{ 2k_{1}, \frac{2k_{1}n}{2+n}+\frac{1}{p1}\}$, then system possesses a global bounded weak solution and (ii) if $k_{2}>\max\{2k_{1}1, \frac{2k_{1}n}{2+n}+\frac{2p}{p1}\}$ with $l>2$, then system possesses a global bounded weak solution.
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