Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon

Xin Xu

doi:10.3934/cpaa.2020195

Article Contents

2020, Volume 19, Issue 9: 4327-4348. Doi: 10.3934/cpaa.2020195

This issue Previous Article On special regularity properties of solutions of the benjamin-ono-zakharov-kuznetsov (bo-zk) equation Next Article Asymptotic behavior of solutions for nonlinear integral equations with Hénon type on the unit Ball

Existence of monotone positive solutions of a neighbour based chemotaxis model and aggregation phenomenon

Xin Xu

Faculty of Science and Technology, University of Macau, Macau, China, co. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

Received: August 31, 2019

Revised: January 31, 2020

Published: June 2020

The author is supported by NSFC-11671190, NNSFC11571390, Macao S.A.R. (FDCT 038/ 2017/A1)

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Abstract

Using bifurcation theory, we prove the existence of spiky steady states and investigate the stability of bifurcating solutions of the one-dimensional continuous neighbour based chemotaxis model, in which the one-step jumping probability rate of cells is determined only by the chemoattractant concentration at the destination. These spiky steady states are crucial when we model cell aggregation, the most important phenomenon in chemotaxis.

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Mathematics Subject Classification: Primary: 35A01, 35B40, 35K57, 35Q92; Secondary: 92C15, 92C17.

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Figure 1. On the horizontal axis is the average mass of chemical substance, which is $ v* $. On the vertical axis is the space of $ (u, v) $. Every point on the cure is a solution of (1.7). On the "upper", branch, the solutions are decreasing in $ \Omega $ and vice versa. $ \bar{v}_1^* $ is a positive constant and will be given in Section 2

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Figure 2. $ D = \alpha = \beta = 1 $, $ u^* = 20 $, $ l = 2 $. This figure shows the graph of $ u $ and $ v $ at time $ 200 $. The $ u $ curve is red and the $ v $ curve is blue

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