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

TL;DR

This paper establishes a criterion for the global existence of solutions to a degenerate Keller-Segel system based on the best Sobolev constant, without requiring small initial data, and provides uniform boundedness results.

Contribution

It links the sharp Sobolev constant to initial conditions ensuring global solutions for a degenerate Keller-Segel system, extending previous results without smallness assumptions.

Findings

Global weak solutions exist without small initial data.

A uniform in time $L^{ abla}$ estimate is achieved.

The Sobolev constant determines the initial criterion for existence.

Abstract

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.