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

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 . The global weak solution obtained in this article does not need any smallness assumption on the initial data. Furthermore, a uniform in time L estimate of the weak solutions is obtained via the Moser iteration, where the constant in Lp estimate for the gradient of the chemical concentration has been exactly formulated in order to complete the iteration process.