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

This paper investigates infinite-time spreading and finite-time blow-up for the Keller-Segel system. For 0 < m a parts per thousand currency sign 2 - 2 / d, the L (p) space for both dynamic and steady solutions are detected with . 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 < a. Furthermore, for slow diffusion 1 < m a parts per thousand currency sign 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.