Math @ Duke

Publications [#246866] of JianGuo Liu
Papers Published
 Bian, S; Liu, JG, Dynamic and Steady States for MultiDimensional KellerSegel Model with Diffusion Exponent m > 0,
Communications in Mathematical Physics, vol. 323 no. 3
(2013),
pp. 154, ISSN 00103616 [doi]
(last updated on 2018/08/19)
Abstract: This paper investigates infinitetime spreading and finitetime blowup for the KellerSegel system. For 0 < m ≤ 2  2 / d, the Lp space for both dynamic and steady solutions are detected with {Mathematical expression} . Firstly, the global existence of the weak solution is proved for small initial data in Lp. Moreover, when m > 1  2 / d, the weak solution preserves mass and satisfies the hypercontractive estimates in Lq 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 hypercontractive estimates are also obtained. Finally, we focus on the Lp norm of the steady solutions, it is shown that the energy critical exponent m = 2 d/(d + 2) is the critical exponent separating finite Lp norm and infinite Lp norm for the steady state solutions. © 2013 SpringerVerlag Berlin Heidelberg.


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