Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models

We consider the degenerate Keller-Segel system (KS)_m with 1 < m ≤ 2-2/N of Nagai type below. Our aim is to find the two types of conditions on the initial data which divide the situation of the solution (u, v) into the global existence and the finite time blow-up; one is the assumption on the size of the other one is the assumption on the size of ∂ u₀^N(2-m)/2 dx. Moreover, we discuss the critical case of m = 2-2/N (N ≥ 3). We find the upper bound and the lower bound on the size of the L¹(= L ^N(2-m)/2 )-norm of the initial data which assures the global existence and the finite time blowup, respectively. Our results cover the well-known threshold number 8π/αχ in R² for the semi-linear case, since both bounds correspond to 8π/αχ by substituting m = 1 and N = 2 formally. For our main results, we also give a proof of the mass conservation law in ℝ^N to (KS)_m.