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

Abstract

The Neumann boundary value problem for the chemotaxis system generalizing the prototype KS \begin{document}$\left\{ \begin{array}{l}{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (u\nabla v),\;\;\;\;x \in \Omega ,t is considered in a smooth bounded convex domain \begin{document}$Ω\subset \mathbb{R}^N(N≥2)$\end{document} , where \begin{document}$D(u)≥ C_D(u+1)^{m-1}~~ \mbox{for all}~~ u≥0~~\mbox{with some}~~ m > 1~~\mbox{and}~~ C_D>0.$ \end{document} If \begin{document}$m >\frac{3N}{2N+2}$\end{document} 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 \begin{document}$m>2-\frac{6}{N+4}$\end{document} ( \begin{document}$N≥3$\end{document} ). 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 \begin{document}$N=2,m>1$\end{document} and \begin{document}$N= 3$\end{document} , \begin{document}$m > \frac{8}{7}$\end{document} , respectively.