Dynamics of spike in a Keller-Segel's minimal chemotaxis model

Yajing Zhang; Xinfu Chen; Jianghao Hao; Xin Lai; Cong Qin

doi:10.3934/dcds.2017046

Article Contents

2017, Volume 37, Issue 2: 1109-1127. Doi: 10.3934/dcds.2017046

This issue Previous Article Computer-assisted equilibrium validation for the diblock copolymer model Next Article Preface

Dynamics of spike in a Keller-Segel's minimal chemotaxis model

1.
School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China
2.
Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
3.
Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China
4.
Center for Financial Engineering, Soochow University, Suzhou, Jiangsu 215006, China

* Corresponding author: Yajing Zhang
* Corresponding author: Yajing Zhang

Received: September 30, 2014

Revised: February 29, 2016

Published: May 2017

This work is partially supported by NSF DMS-1008905, NNSFC (No. 61374089), China Scholarship Council, NSF of Shanxi Province(No. 2014011005-2), Hundred Talent Program of Shanxi and International Cooperation Projects of Shanxi Province (No. 2014081026).

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Abstract

The dynamics are studied for the Keller-Segel's minimal chemotaxis model

$τ u_t=(u_x-kuv_x)_x, \ \ \ \ v_t=v_{xx}-v+u$

on a bounded interval with homogeneous Neumann boundary conditions, where $\tau\geqslant 0$ and $k\gg 1$ are parameters and the total mass of $u$ is scaled to be one. In general, the dynamics can be divided into three stages: the first stage is very short in which $u$ quickly becomes a delta like function with mass concentrated near the point of global maximum of $v$; in the second stage, the point of the global maximum of $v$ drifts towards the boundary of the domain and reaches it at the end of the second stage; in the third stage, the profile of the solution evolves to a steady state profile. This paper considers a special case in which the relaxation parameter $\tau$ is set to be zero, so the first stage takes no time. A free boundary problem describing the second stage is presented. Rigorous asymptotic behavior is proven for the third stage evolution.

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Mathematics Subject Classification: Primary:35B40, 92C17; Secondary: 92D15.

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Figure 1. A numerical solution of (1) with $\tau=0$, $k=200$, $\ell=1$, and mesh sizes $\Delta x=1/400$, $\Delta t=3\times 10^{-6}$. Left: snapshots of $v(k,\cdot,t)$ with constant frequency; middle: snapshots of $u$; right: location of the point of maximum of $v(k,\cdot,t)$, which reaches the boundary at $T\approx 0.13$.

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Figure 2. First Row: a numerical solution of (6) with $k=100$, $\ell=4$, and $v_0(x)=\frac{\ell}{\pi}\big|\cos\frac{\pi(x-2.5)}{\ell}\big|$; left is location of maximum of $v$; middle is snapshots of $v(k,\cdot,t)$ with non-uniform time elapses; right is snapshots of $v_x(k,\cdot,t)$. Second Row: the solution of the free boundary problem (7) for $t\in[0,T^-]$ ($T\approx 5.2$), combined with the solution of the boundary value problem (8) for $t\geqslant T$; left is the location of free boundary; middle is snapshots of $w$; right is snapshots of $w_x$.

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