Boundedness in a quasilinear attraction–repulsion chemotaxis system with variable logistic source

This paper deals with a quasilinear attraction–repulsion chemotaxis system with a source term of variable logistic type $u_t=\mathrm{\nabla }\cdot (\varphi (u)\mathrm{\nabla }u)-\mathrm{\nabla }\cdot (\psi (u)\mathrm{\nabla }v)+\mathrm{\nabla }\cdot (\phi (u)\mathrm{\nabla }w)+g(u)$, ${\tau }_1{v}_t=\mathrm{\Delta }v-v+u$, $-\mathrm{\Delta }w=-w+u$ in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( $n\ge 1$), and endowed with nonnegative initial data and homogeneous Neumann boundary conditions. Moreover, the logistic source verifies $g(x,s)\le \eta s^{k(x)}-\mu s^{m(x)}$, $s>0$ with $g(x,0)\ge 0$, $x\in \mathrm{\Omega }$, where $\eta \ge 0$, $\mu >0$ are constants, $k,m$ are measurable functions fulfilling $0\le k^{-}:=\underset{x\in \mathrm{\Omega }}{essinf}k(x)\le k(x)\le k^{+}:=\underset{x\in \mathrm{\Omega }}{esssup}k(x)<+\mathrm{\infty }$ and $1<m^{-}:=\underset{x\in \mathrm{\Omega }}{essinf}m(x)\le m(x)\le m^{+}:=\underset{x\in \mathrm{\Omega }}{esssup}m(x)<+\mathrm{\infty }$, as well as $\varphi ,\psi$, and $\phi$ are regular functions satisfying $c_1{s}^p\le \varphi (s)$, $\psi (s)\le c_2{s}^q$, and ${\underset{\_}{c}}_3{s}^l\le \phi (s)\le c_3{s}^l$ with $p,q,l\in \mathbb{R}$, $c_1,c_2,{\underset{\_}{c}}_3,c_3>0$ and $s\ge s_0>1$. We show that when $q=m^{-}-1$ and $l\le m^{-}-1$, there exists ${\mu }^{\ast }>0$ such that if $\mu >{\mu }^{\ast }$, then the corresponding initial-boundary value problem possesses a unique globally bounded classical solution. Moreover, the same conclusion holds true provided that $q<m^{-}-1$ and $l\le m^{-}-1$ for any $\mu >0$.