Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model

Wenting Cong; Jian-Guo Liu

doi:10.3934/dcdsb.2017015

Article Contents

2017, Volume 22, Issue 2: 307-338. Doi: 10.3934/dcdsb.2017015

This issue Previous Article Optimality conditions for a controlled sweeping process with applications to the crowd motion model Next Article Analysis of a nonlocal-in-time parabolic equation

Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model

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

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

* Corresponding author: Wenting Cong
* Corresponding author: Wenting Cong

Received: January 31, 2016

Revised: September 30, 2016

Published: December 2017

The first author is supported by NSFC grant 11271154.

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Abstract

This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$ . We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

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Mathematics Subject Classification: Primary:35K65, 35K59, 92C17.

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