This paper is devoted to the following chemotaxis system
ut=∇·(D(u)∇u)-∇·(S(u)∇v),x∈Ω,t>0,vt=Δv-uv,x∈Ω,t>0,\documentclass[12pt]{minimal}
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\begin{document}$$\left\{
\begin{array}{llll}u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v),\quad &x\in \Omega,\quad t>0,\\
v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0,\end{array}
\right.$$\end{document}under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$${\Omega\subset \mathbb{R}^n}$$\end{document} (n≥2\documentclass[12pt]{minimal}
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\begin{document}$${n\geq2}$$\end{document}), not necessarily being convex. There are some constants cD>0\documentclass[12pt]{minimal}
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\begin{document}$${c_D > 0}$$\end{document}, cS>0\documentclass[12pt]{minimal}
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\begin{document}$${c_S > 0}$$\end{document}, m∈R\documentclass[12pt]{minimal}
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\begin{document}$${m\in\mathbb{R}}$$\end{document} and q∈R\documentclass[12pt]{minimal}
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\begin{document}$${q\in\mathbb{R}}$$\end{document} such that
D(u)≥cD(u+1)m-1andS(u)≤cS(u+1)q-1forallu≥0.\documentclass[12pt]{minimal}
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\begin{document}$$D(u) \geq c_D(u+1)^{m-1}
\quad\text{and}
\quad S(u)\leq c_S(u+1)^{q-1}\quad for all
\,\,\,u\geq0.$$\end{document}If q<m+n+22n\documentclass[12pt]{minimal}
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\begin{document}$${q < m+\frac{n+2}{2n}}$$\end{document}, it is shown that the model possesses a unique global classical solution which is uniformly bounded; if q<m2+n+22n\documentclass[12pt]{minimal}
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\begin{document}$${q < \frac{m}{2}+\frac{n+2}{2n}}$$\end{document}, the global existence of solution is established.
