  This article presents a relationship between the sharp constant of the
Sobolev inequality and the initial criterion to the global existence of
degenerate parabolic-parabolic Keller-Segel system with the diffusion exponent
$\frac{2n}{2+n}<m<2-\frac{2}{n}$. The global weak solution obtained in this
article does not need any smallness assumption on the initial data.
Furthermore, a uniform in time $L^{\infty}$ estimate of the weak solutions is
obtained via the Moser iteration, where the constant in $L^p$ estimate for the
gradient of the chemical concentration has been exactly formulated in order to
complete the iteration process.
