Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos
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December 2016, 9(4): 687-714. doi: 10.3934/krm.2016012
A degenerate $p$-Laplacian Keller-Segel model
Wenting Cong 1, and Jian-Guo Liu 2,
School of Mathematics, Jilin University, Changchun 130012, China
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708
Received February 2016 Revised April 2016 Published September 2016
This paper investigates the existence of a uniform in time $L^{\infty}$ bounded weak solution for the $p$-Laplacian Keller-Segel system with the supercritical diffusion exponent $1 < p < \frac{3d}{d+1}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(3-p)}{p}}$ norm of initial data is smaller than a universal constant. We also prove the local existence of weak solutions and a blow-up criterion for general $L^1\cap L^{\infty}$ initial data.
Keywords: fast diffusion, monotone operator, global existence, Chemotaxis, critical space, non-Newtonian filtration..
Mathematics Subject Classification: Primary: 35K65, 35K92, 92C1.
Citation: Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
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Wenting Cong Jian-Guo Liu