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Global solutions to a chemotaxis model with consumption of chemoattractant

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Abstract

This paper is devoted to the following chemotaxis system

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

under homogeneous Neumann boundary conditions in a smooth bounded domain \({\Omega\subset \mathbb{R}^n}\) (\({n\geq2}\)), not necessarily being convex. There are some constants \({c_D > 0}\), \({c_S > 0}\), \({m\in\mathbb{R}}\) and \({q\in\mathbb{R}}\) such that

$$D(u) \geq c_D(u+1)^{m-1} \quad\text{and} \quad S(u)\leq c_S(u+1)^{q-1}\quad for all \,\,\,u\geq0.$$

If \({q < m+\frac{n+2}{2n}}\), it is shown that the model possesses a unique global classical solution which is uniformly bounded; if \({q < \frac{m}{2}+\frac{n+2}{2n}}\), the global existence of solution is established.

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Authors and Affiliations

  1. College of Sciences, Chongqing University of Posts and Telecommunications, Chongqing, 400065, China

    Liangchen Wang & Xuegang Hu

  2. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Chunlai Mu

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  1. Liangchen Wang
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Correspondence to Liangchen Wang.

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Wang, L., Mu, C. & Hu, X. Global solutions to a chemotaxis model with consumption of chemoattractant. Z. Angew. Math. Phys. 67, 96 (2016). https://doi.org/10.1007/s00033-016-0693-4

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  • DOI: https://doi.org/10.1007/s00033-016-0693-4

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