Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity

Qi Wang

doi:10.3934/dcdsb.2015.20.1231

Article Contents

2015, Volume 20, Issue 4: 1231-1250. Doi: 10.3934/dcdsb.2015.20.1231

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Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity

Qi Wang^1,

1.
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130

Received: April 30, 2014

Revised: October 31, 2014

Published: May 2015

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Abstract

In this paper, we study the nonconstant positive steady states of a Keller-Segel chemotaxis system over a bounded domain $\Omega\subset \mathbb{R}^N$, $N\geq 1$. The sensitivity function is chosen to be $\phi(v)=\ln (v+c)$ where $c$ is a positive constant. For the chemical diffusion rate being small, we construct positive solutions with a boundary spike supported on a platform. Moreover, this spike approaches the most curved part of the boundary of the domain as the chemical diffusion rate shrinks to zero. We also conduct extensive numerical simulations to illustrate the formation of stable boundary and interior spikes of the system. These spiky solutions can be used to model the self--organized cell aggregation phenomenon in chemotaxis.

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Mathematics Subject Classification: Primary: 92C17, 35K51; Secondary: 35K50.

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