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. 10171070, ISSN 00103616 [doi]
(last updated on 2017/12/14)
Abstract: This paper investigates infinitetime spreading and finitetime blowup for the KellerSegel system. For 0 < m ≤ 2  2/d, the L p space for both dynamic and steady solutions are detected with (Formula presented.). 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 hypercontractive 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 hypercontractive 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. © 2013 SpringerVerlag Berlin Heidelberg.


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