Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

Chao Deng; Tong Li

doi:10.1016/j.jde.2014.05.014

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Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

Journal article

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Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

Chao Deng and Tong Li

Journal of Differential Equations, Vol.257(5), pp.1311-1332

09/01/2014

DOI: 10.1016/j.jde.2014.05.014

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Abstract

We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L2, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H1, then they remain to be small in L2×H1 for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.

Asymptotic behavior

Decay properties

Energy estimates

Keller–Segel system

Parabolic–hyperbolic system

Well-posedness

Details

Title: Subtitle: Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework
Creators: Chao Deng - Jiangsu Normal University
Tong Li - University of Iowa
Resource Type: Journal article
Publication Details: Journal of Differential Equations, Vol.257(5), pp.1311-1332
DOI: 10.1016/j.jde.2014.05.014
ISSN: 0022-0396
eISSN: 1090-2732
Publisher: Elsevier Inc
Grant note: name: NSF of Jiangsu Higher Education Institutions, award: 13KJB110008; DOI: 10.13039/501100012246, name: PAPD, award: 9212112101; name: NSF of Jiangsu Province, award: BK20130225; DOI: 10.13039/501100001809, name: NSFC, award: 11226180; DOI: 10.13039/501100001809, name: NSFC, award: 11301228; DOI: 10.13039/501100001809, name: NSFC, award: 11171357, 11271166, 11271167
Language: English
Date published: 09/01/2014
Academic Unit: Mathematics
Record Identifier: 9984241054002771

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