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

Supercritical degenerate parabolic-parabolic Keller-Segel system -- existence criterion given by the best constant in Sobolev's inequality