# American Institute of Mathematical Sciences

March  2017, 22(2): 669-686. doi: 10.3934/dcdsb.2017032

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

 1 School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China 2 School of Mathematics and Statistics, Beijing, Institute of Technology, Beijing 100081, China

* Corresponding author

Received  February 2016 Revised  August 2016 Published  December 2016

The Neumann boundary value problem for the chemotaxis system generalizing the prototype
 $\left\{ \begin{array}{l}{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (u\nabla v),\;\;\;\;x \in \Omega ,t < 0,\\{v_t} = \Delta v - uv,\;\;\;\;\;x \in \Omega ,t < 0,\end{array} \right. \tag{KS}\label{KS}$
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.
Citation: Jiashan Zheng, Yifu Wang. A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 669-686. doi: 10.3934/dcdsb.2017032
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