Bian, S; Liu, J-G, *Dynamic and Steady States for Multi-Dimensional Keller-Segel Model with Diffusion Exponent m > 0*,
Communications in Mathematical Physics, vol. 323 no. 3
(2013),
pp. 1017-1070 [doi] .
**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 (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 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. © 2013 Springer-Verlag Berlin Heidelberg.*