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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:257

Article

Deng, Chao¹  Li, Tong² 

[1] Jiangsu Normal Univ, Sch Math & Stat, Xuzhou 221116, Peoples R China

[2] Univ Iowa, Dept Math, Iowa City, IA 52242 USA

关键词: Keller-Segel system;    Parabolic hyperbolic system;    Energy estimates;    Well-posedness;    Asymptotic behavior;    Decay properties;   

DOI  :  10.1016/j.jde.2014.05.014

来源: Elsevier

【 摘 要 】

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 L-2, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H-1, then they remain to be small in L-2 x H-1 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. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free