A degenerate $p$-Laplacian Keller-Segel model

Wenting  Cong; Jian-Guo Liu

doi:10.3934/krm.2016012

Article Contents

2016, Volume 9, Issue 4: 687-714. Doi: 10.3934/krm.2016012

This issue Previous Article A Vlasov-Poisson plasma with unbounded mass and velocities confined in a cylinder by a magnetic mirror Next Article Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos

A degenerate $p$-Laplacian Keller-Segel model

Wenting Cong^1, and
Jian-Guo Liu^2,

1.
School of Mathematics, Jilin University, Changchun 130012
2.
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708

Received: February 2016

Revised: April 2016

Published: December 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35K65, 35K92, 92C17.

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