Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type

We consider the quasi-linear Keller-Segel system of singular type, where the principal part Δum represents a fast diffusion like 0<m<1. We first construct a global weak solution with small initial data in the scaling invariant norm LN(q≥m)2 for all dimensions Nq≥2 and all exponents qq≥2. As for the large initial data, we show that there exists a blow-up solution in the case of N=2. In the second part, the decay property in Lr with 1<r<q≥ for 1?2Nmq≥<1 with the mass conservation is shown. On the other hand, in the case of 0<m<<;12N, the extinction phenomenon of solution is proved. It is clarified that the case of m=1<2N exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 12N.< For the borderline case of m=1<2N, our solution decays in Lr exponentially as <.