This paper deals with the Neumann boundary value problem for the system
ut=∇·D(u)∇u-∇·S(u)∇v+f(u),x∈Ω,t>0,vt=Δv-v+u,x∈Ω,t>0\documentclass[12pt]{minimal}
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\begin{document}$$\left\{\begin{array}{lll}u_t = \nabla \cdot \left(D(u) \nabla u\right) - \nabla \cdot \left(S(u) \nabla v\right) + f(u), &\quad x \in \Omega, \, t > 0,\\ v_t = \Delta v - v + u, &\quad x \in \Omega, \, t > 0\end{array}\right.$$\end{document}in a smooth bounded domain Ω⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$${\Omega\subset{\mathbb{R}}^n}$$\end{document}(n≥1)\documentclass[12pt]{minimal}
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\begin{document}$${(n\geq1)}$$\end{document}, where the functions D(u) and S(u) are supposed to be smooth satisfying D(u)≥Mu-α\documentclass[12pt]{minimal}
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\begin{document}$${D(u)\geq Mu^{-\alpha}}$$\end{document} and S(u)≤Muβ\documentclass[12pt]{minimal}
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\begin{document}$${S(u)\leq Mu^{\beta}}$$\end{document} with M > 0, α∈R\documentclass[12pt]{minimal}
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\begin{document}$${\alpha\in{\mathbb{R}}}$$\end{document} and β∈R\documentclass[12pt]{minimal}
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\begin{document}$${\beta\in{\mathbb{R}}}$$\end{document} for all u≥1\documentclass[12pt]{minimal}
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\begin{document}$${u\geq1}$$\end{document}, and the logistic source f(u) is smooth fulfilling f(0)≥0\documentclass[12pt]{minimal}
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\begin{document}$${f(0)\geq0}$$\end{document} as well as f(u)≤a-μuγ\documentclass[12pt]{minimal}
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\begin{document}$${f(u)\leq a-\mu u^{\gamma}}$$\end{document} with a≥0\documentclass[12pt]{minimal}
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\begin{document}$${a\geq0}$$\end{document}, μ>0\documentclass[12pt]{minimal}
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\begin{document}$${\mu > 0}$$\end{document} and γ≥1\documentclass[12pt]{minimal}
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\begin{document}$${\gamma\geq1}$$\end{document} for all u≥0\documentclass[12pt]{minimal}
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\begin{document}$${u\geq0}$$\end{document}. It is shown that if
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\begin{document}$$\alpha + 2\beta < \left\{\begin{array}{lll}\gamma - 1 + \frac{2}{n}, &\quad {\rm for} \, 1 \leq \gamma < 2,\\ \gamma - 1 + \frac{4}{n + 2}, &\quad {\rm for} \, \gamma \geq 2,\end{array}\right.$$\end{document}then for sufficiently smooth initial data, the problem possesses a unique global classical solution which is uniformly bounded.
