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

We consider the degenerate Keller-Segel system (KS)$_m$ with $1 < m \le 2-\frac{2}{N}$ 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 $R_m(u_0):=\displaystyle \frac{\int u_0 v_{0} \ dx}{\int u_0^{m} \ dx}$; the other one is the assumption on the size of $\int u_0^{\frac{N(2-m)}{2}} \ dx$. Moreover, we discuss the critical case of $m=2-\frac{2}{N} \ (N\ge3)$. We find the upper bound and the lower bound on the size of the $L^1(=L^{\frac{N(2-m)}{2}})$-norm of the initial data which assures the global existence and the finite time blow-up, respectively. Our results cover the well-known threshold number $\frac{8\pi}{\alpha \chi}$ in $\mathbb R^2$ for the semi-linear case, since both bounds correspond to $\frac{8\pi}{\alpha \chi}$ by substituting $m=1$ and $N=2$ formally. For our main results, we also give a proof of the mass conservation law in $\mathbb R^N$ to (KS)$_m$.