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Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P.R. China

    Shen Bian

  2. Department of Physics and Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Shen Bian & Jian-Guo Liu

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  1. Shen Bian
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  2. Jian-Guo Liu
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Correspondence to Jian-Guo Liu.

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Communicated by P. Constantin

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Bian, S., Liu, JG. Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0. Commun. Math. Phys. 323, 1017–1070 (2013). https://doi.org/10.1007/s00220-013-1777-z

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