| Summary: | Abstract In this paper, we deal with the following quasilinear attraction–repulsion model: {ut=∇⋅(D(u)∇u)−∇⋅(S(u)χ(v)∇v)+∇⋅(F(u)ξ(w)∇w)+f(u),x∈Ω,t>0,vt=Δv+βu−αv,x∈Ω,t>0,0=Δw+γu−δw,x∈Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω $$ \textstyle\begin{cases} u_{t}=\nabla \cdot (D(u)\nabla u)-\nabla \cdot ( S(u)\chi (v)\nabla v)+ \nabla \cdot ( F(u)\xi (w)\nabla w)+f(u), &x\in \varOmega , t>0, \\ v_{t}=\Delta v+\beta u-\alpha v, &x\in \varOmega ,t>0, \\ 0=\Delta w+\gamma u-\delta w, &x\in \varOmega , t>0, \\ u(x,0)=u_{0}(x), \quad\quad v(x,0)= v_{0}(x), &x\in \varOmega \end{cases} $$ with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn $\varOmega \subset R^{n}$ ( n≥2 $n\geq 2$). Let the chemotactic sensitivity χ(v) $\chi (v)$ be a positive constant, and let the chemotactic sensitivity ξ(w) $\xi (w)$ be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.
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