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

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Abstract

This paper deals with an initial-boundary value problem for the chemotaxis system

$$\left\{\begin{array}{ll} u_t = \nabla \cdot (D (u) \nabla u)- \nabla \cdot (u \nabla v), \quad & x\in \Omega, \quad t > 0, \\ v_t= \Delta v-uv, \quad & x \in \Omega, \quad t > 0, \end{array}\right.$$

under homogeneous Neumann boundary conditions in a convex smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) with \({n\geq3}\), where the diffusion function D(u) satisfying

$$\begin{array}{ll}D(u)\geq c_Du^{m-1}\quad\text{for all}\,\,u > 0 \end{array}$$

with some c D  > 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 \({m > 2-\frac{2}{n}}\) can be relaxed to \({m > 2-\frac{6}{n+4}}\).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, People’s Republic of China

    Liangchen Wang, Chunlai Mu, Ke Lin & Jie Zhao

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  1. Liangchen Wang
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Correspondence to Liangchen Wang.

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Wang, L., Mu, C., Lin, K. et al. Global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant. Z. Angew. Math. Phys. 66, 1633–1648 (2015). https://doi.org/10.1007/s00033-014-0491-9

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  • DOI: https://doi.org/10.1007/s00033-014-0491-9

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