On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model

Tian Xiang

doi:10.3934/dcds.2014.34.4911

Article Contents

2014, Volume 34, Issue 11: 4911-4946. Doi: 10.3934/dcds.2014.34.4911

This issue Previous Article The existence of strong solutions to the $3D$ Zakharov-Kuznestov equation in a bounded domain Next Article Liouville type theorem for nonlinear elliptic equation with general nonlinearity

On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model

Tian Xiang^1,

1.
Department of Mathematics, Tulane University, New Orleans, LA 70118

Received: August 31, 2013

Revised: January 31, 2014

Published: October 2014

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Abstract

In this paper, we continue to study a general nonlocal gradient Patlak-Keller-Segel chemotaxis model in a one dimensional spatial domain. By utilizing the properties of the nonlocal gradient, we first apply the well-known Moser-Alikakos iteration technique plus the heat semigroup theory to obtain the boundedness and hence the global existence of its solution. Then we study the asymptotic behavior of the time-dependent solution, and obtain the limiting equations when the sampling radius $\rho\rightarrow 0$ as well as convergence results when time $t\rightarrow \infty$. Along this way, a ``global" stability issue of the spiky stationary solution for the minimal model is formulated. Finally and importantly, we study the stability of the nonconstant bifurcating solutions. Interestingly, the small size of the cells enhances the occurrence of pattern formation, the stability results are independent of the net creation rate of the chemical, and the stability is closely related to the cell radius $\rho$. Typically, when the cell (net) degradation rate lies below a threshold (stabilizing) value, the cell is stable. Surprisingly, this threshold value is an increasing function of the cell radius. The large cells can compensate their degradation of the chemical signal, and become stable; however, for small cells to be stable, their degradation rate must be less than a threshold value.

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Mathematics Subject Classification: Primary: 35K57, 35B40, 92C17, 35B32; Secondary: 35K45, 35J67, 35B41, 35B36.

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