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arXiv · 1801.03475

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

Jinhuan Wang·Yue Li·Li Chen
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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.

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Jinhuan Wang, Yue Li, Li Chen. 2018-01-10. Supercritical degenerate parabolic-parabolic Keller-Segel system -- existence criterion given by the best constant in Sobolev's inequality. https://doi.org/10.1007/s00033-019-1115-1

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