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#### Analysis of PDEsmathscidoc:1702.03040

Communications in Mathematical Physics, 323, (3), 1017–1070, 2013.11
This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m ≤ 2 − 2 / d, the L p space for both dynamic and steady solutions are detected with ${p:=\frac{d(2-m)}{2} }$ . Firstly, the global existence of the weak solution is proved for small initial data in L p . Moreover, when m > 1 − 2 / d, the weak solution preserves mass and satisfies the hyper-contractive estimates in L q for any p < q < ∞. Furthermore, for slow diffusion 1 < m ≤ 2 − 2/d, this weak solution is also a weak entropy solution which blows up at finite time provided by the initial negative free energy. For m > 2 − 2/d, the hyper-contractive estimates are also obtained. Finally, we focus on the L p norm of the steady solutions, it is shown that the energy critical exponent m = 2d/(d + 2) is the critical exponent separating finite L p norm and infinite L p norm for the steady state solutions.
@inproceedings{shen2013dynamic,
title={Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent m > 0},
author={Shen Bian, and Jian-Guo Liu},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208212528160836315},
booktitle={Communications in Mathematical Physics},
volume={323},
number={3},
pages={1017–1070},
year={2013},
}

Shen Bian, and Jian-Guo Liu. Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent m > 0. 2013. Vol. 323. In Communications in Mathematical Physics. pp.1017–1070. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208212528160836315.