On an attraction-repulsion chemotaxis model involving logistic source

Abstract

This paper is concerned with the attraction-repulsion chemotaxis system involving logistic source: $u_{t}=\Delta u-\chi \nabla \cdot \left( u\nabla \upsilon \right) +\xi \nabla \cdot \left( u\nabla \omega \right) +f(u)$, $\rho \upsilon _{t}=\Delta \upsilon -\alpha_{1}\upsilon +\beta _{1}u$, $\rho \omega _{t}=\Delta \omega -\alpha_{2}\omega +\beta _{2}u$ under homogeneous Neumann boundary conditions with nonnegative initial data $(u_{0},\upsilon _{0},\omega _{0})\in $ $\left( W^{1,\infty }\left( \Omega \right) \right) ^{3}$, the parameters $\chi $, $\xi $, $\alpha _{1}$, $\alpha_{2}$, $\beta _{1}$, $\beta _{2}>0$, $\rho \geq 0$ subject to the non-flux boundary conditions in a bounded domain $\Omega \subset\mathbb{R}^{N}(N\geq 3)$ with smooth boundary and $f(u)\leq au-\mu u^{2}$ with $f(0)\geq 0$ and $a\geq 0$, $\mu >0$ for all $u>0$. Based on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a globally bounded classical solution provided that $\chi +\xi <\frac{\mu }{2}$ and there exists a constant $\beta _{\ast }>0$ is sufficiently small for all $\beta _{1}$, $\beta _{2}<\beta _{\ast }$.

Keywords

• attraction-repulsion
• chemotaxis
• logistic source
• global existence

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