Global boundedness of  solutions to a Keller-Segel system with nonlinear sensitivity

Hao  Yu; Wei  Wang; Sining  Zheng

doi:10.3934/dcdsb.2016.21.1317

Article Contents

2016, Volume 21, Issue 4: 1317-1327. Doi: 10.3934/dcdsb.2016.21.1317

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Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024

Received: July 2015

Revised: November 2015

Published: June 2016

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Abstract

This paper considers the parabolic-parabolic Keller-Segel system with nonlinear sensitivity $u_t=\Delta u-\nabla (u^{\alpha}\nabla v)$, $v_t=\Delta v-v+u$, subject to homogeneous Neumann boundary conditions with smooth and bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq1$. It is proved that if $\alpha\geq\max\{1,\frac{2}{n}\}$, then the solutions are globally bounded, and both the components $u$ and $v$ decay to the same constant steady state $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u_0(x) dx$ exponentially in the $L^\infty$-norm provided both $\|u_0\|_{L^{q^{\ast}}(\Omega)}$ and $\|\nabla v_0\|_{L^{p^{\ast}}(\Omega)}$ small enough with $q^{\ast}=\frac{n\alpha K}{n+K}$, $p^{\ast}=\frac{n\alpha K}{n+K-n\alpha}$, $K\in[n,2n\alpha-n]\cap ((\alpha-1)n,\infty)$.

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

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