Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with an initial-boundary value problem for the chemotaxis system ut=∇·(D(u)∇u)-∇·(u∇v),x∈Ω,t>0,vt=Δv-uv,x∈Ω,t>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$\end{document} under homogeneous Neumann boundary conditions in a convex smooth bounded domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega\subset \mathbb{R}^n}$$\end{document} with n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n\geq3}$$\end{document}, where the diffusion function D(u) satisfying D(u)≥cDum-1for allu>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$\end{document}with some cD > 0 and m > 1. The main goal of this paper was to extend a previous result on global existence of solutions by Wang et al. (Z Angew Math Phys 65:1137–1152, 2014) under the condition that m>2-2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m > 2-\frac{2}{n}}$$\end{document} can be relaxed to m>2-6n+4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m > 2-\frac{6}{n+4}}$$\end{document}.