A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

The Neumann boundary value problem for the chemotaxis system generalizing the prototype KS $\left\{ \begin{array}{l}{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (u\nabla v),\;\;\;\;x \in \Omega ,t < 0,\\{v_t} = \Delta v - uv,\;\;\;\;\;x \in \Omega ,t < 0,\end{array} \right. \tag{KS}\label{KS} $ is considered in a smooth bounded convex domain $Ω\subset \mathbb{R}^N(N≥2)$, where $D(u)≥ C_D(u+1)^{m-1}~~ \mbox{for all}~~ u≥0~~\mbox{with some}~~ m > 1~~\mbox{and}~~ C_D>0.$ If $m >\frac{3N}{2N+2}$ and suitable regularity assumptions on the initial data are given, the corresponding initial-boundary problem possesses a global classical solution. Our paper extends the results of Wang et al. ([24]), who showed the global existence of solutions in the cases $m>2-\frac{6}{N+4}$ ($N≥3$). If the flow of fluid is ignored, our result is consistent with and improves the result of Tao, Winkler ([15]) and Tao, Winkler ([17]), who proved the possibility of global boundedness, in the case that $N=2,m>1$ and $N= 3$, $m > \frac{8}{7}$, respectively.