This paper deals with the boundedness of solutions to the following quasilinear chemotaxis–haptotaxis model of parabolic–parabolic–ODE type:
{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−ur−1−w),x∈Ω,t>0,vt=Δv−v+uη,x∈Ω,t>0,wt=−vw,x∈Ω,t>0,\documentclass[12pt]{minimal}
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\begin{document}$$ \textstyle\begin{cases} u_{t}=\nabla \cdot (D(u)\nabla u)-\chi \nabla \cdot (u\nabla v)- \xi \nabla \cdot (u\nabla w)+\mu u(1-u^{r-1}-w),& x\in \varOmega , t>0, \\ v_{t}=\Delta v-v+u^{\eta },& x\in \varOmega , t>0, \\ w_{t}=-vw, &x\in \varOmega , t>0, \end{cases} $$\end{document} under zero-flux boundary conditions in a smooth bounded domain Ω⊂Rn(n≥2)\documentclass[12pt]{minimal}
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\begin{document}$\varOmega \subset \mathbb{R}^{n}(n\geq 2)$\end{document}, with parameters r≥2\documentclass[12pt]{minimal}
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\begin{document}$r\geq 2$\end{document}, η∈(0,1]\documentclass[12pt]{minimal}
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\begin{document}$\eta \in (0,1]$\end{document} and the parameters χ>0\documentclass[12pt]{minimal}
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\begin{document}$\chi >0$\end{document}, ξ>0\documentclass[12pt]{minimal}
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\begin{document}$\xi >0$\end{document}, μ>0\documentclass[12pt]{minimal}
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\begin{document}$\mu >0$\end{document}. The diffusivity D(u)\documentclass[12pt]{minimal}
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\begin{document}$D(u)$\end{document} is assumed to satisfy D(u)≥δu−α\documentclass[12pt]{minimal}
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\begin{document}$D(u)\geq \delta u^{-\alpha }$\end{document}, D(0)>0\documentclass[12pt]{minimal}
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\begin{document}$D(0)>0$\end{document} for all u>0\documentclass[12pt]{minimal}
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\begin{document}$u>0$\end{document} with some α∈R\documentclass[12pt]{minimal}
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\begin{document}$\alpha \in \mathbb{R}$\end{document} and δ>0\documentclass[12pt]{minimal}
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\begin{document}$\delta >0 $\end{document}. It is proved that if α<n+2−2nη2+n\documentclass[12pt]{minimal}
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\begin{document}$\alpha <\frac{n+2-2n\eta }{2+n}$\end{document}, then, for sufficiently smooth initial data (u0,v0,w0)\documentclass[12pt]{minimal}
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\begin{document}$(u_{0},v_{0},w_{0})$\end{document}, the corresponding initial-boundary problem possesses a unique global-in-time classical solution which is uniformly bounded.
