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

作者：Deng Chao^*; Li Tong

来源：Journal of Differential Equations, 2014, 257(5): 1311-1332.

DOI：10.1016/j.jde.2014.05.014

• Keller-Segel system
• Parabolic hyperbolic system
• Energy estimates
• Well-posedness
• Asymptotic behavior
• Decay properties

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.

• 出版日期2014-9-1