Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity

Wei Wang; Yan Li; Hao Yu

doi:10.3934/dcdsb.2017147

Article Contents

2017, Volume 22, Issue 10: 3663-3669. Doi: 10.3934/dcdsb.2017147

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Global boundedness in higher dimensions for a fully parabolic chemotaxis system with singular sensitivity

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author
* Corresponding author

Received: December 2016

Revised: December 2016

Published: September 2017

Supported by the National Natural Science Foundation of China (11101060,11671066) and by the Fundamental Research Funds for the Central Universities (DUT16LK24).

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Abstract

In this paper we study the global boundedness of solutions to the fully parabolic chemotaxis system with singular sensitivity:$u_t=\Delta u-\chi\nabla·(\frac{u}{v}\nabla v)$, $v_t=k\Delta v-v+u$, subject to homogeneous Neumann boundary conditions in a bounded and smooth domain $\Omega\subset\mathbb{R}^{n}$ ($n\ge 2$), where $\chi, \, k>0$. It is shown that the solution is globally bounded provided $0<\chi<\frac{-(k-1)+\sqrt{(k-1)^2+\frac{8k}{n}}}{2}$. This result removes the additional restriction of $n \le 8 $ in Zhao, Zheng [15] for the global boundedness of solutions.

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Mathematics Subject Classification: Primary:35B35, 35B40, 35K55;Secondary:92C17.

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