An attraction–repulsion chemotaxis with logistic source involving the exponents depending on the spatial variables

We study the quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source involving the exponents depending on the spatial variables: \(u_{t}=\Delta u-\chi \nabla \cdot \left( u\left( u+1\right) ^{r-1}\nabla \upsilon \right) +\xi \nabla \cdot \left( u\left( u+1\right) ^{r-1}\nabla \omega \right) +au-bu^{m(x)}\) $u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }·\left(u,{\left(u,+,1\right)}^{r-1},\mathrm{\nabla },\upsilon \right)+\xi \mathrm{\nabla }·\left(u,{\left(u,+,1\right)}^{r-1},\mathrm{\nabla },\omega \right)+au-b{u}^{m(x)}$ , \( 0=\Delta \upsilon -\beta \upsilon +\alpha u\) $0=\mathrm{\Delta }\upsilon -\beta \upsilon +\alpha u$ , \(0=\Delta \omega -\delta \omega +\gamma u\) $0=\mathrm{\Delta }\omega -\delta \omega +\gamma u$ , where \(\alpha \) $\alpha$ , \(\beta \) $\beta$ , \(\delta \) $\delta$ , \(\gamma \) $\gamma$ , \(\chi \) $\chi$ , \(\xi \) $\xi$ , \(b>0\) $b>0$ , \(a\ge 0\) $a\ge 0$ , \(r\in \mathbb {R} \) $r\in \mathbb{R}$ and \(m:\Omega \rightarrow \left( 1,\infty \right) \) $m:\mathrm{\Omega }\to \left(1,,,\infty \right)$ is a measurable function, subject to the homogeneous Neumann boundary conditions in a bounded domain \( \mathbb {R} ^{N}\) ${\mathbb{R}}^N$ \(\left( N\ge 1\right) \) $\left(N,\ge ,1\right)$ with smooth boundary. We prove that this system possesses a unique global bounded classical solution, which is an extension of known results, if the repulsion cancels the attraction in the sense that (balance) \(\chi \alpha =\xi \gamma \) $\chi \alpha =\xi \gamma$ with \( ess\inf _{x\in \Omega }m\left( x\right) >\left\{ r+\frac{\left( N-2\right) _{+}}{N},1\right\} \) $ess{inf}_{x\in \mathrm{\Omega }}m\left(x\right)>\left\{r,+,\frac{{\left(N,-,2\right)}_{+}}{N},,,1\right\}$ , and if the attraction prevails over the repulsion in the sense that \(\chi \alpha >\xi \gamma \) $\chi \alpha >\xi \gamma$ with \(ess\inf _{x\in \Omega }m\left( x\right) >\max \left\{ r+1,1\right\} \) $ess{inf}_{x\in \mathrm{\Omega }}m\left(x\right)>max\left\{r,+,1,,,1\right\}$ , and if the repulsion prevails over the attraction in the sense that \(\chi \alpha <\xi \gamma \) $\chi \alpha <\xi \gamma$ .