Ultra-contractivity for Keller-Segel model  with diffusion exponent $m>1-2/d$

Shen Bian; Jian-Guo Liu; Chen Zou

doi:10.3934/krm.2014.7.9

Article Contents

2014, Volume 7, Issue 1: 9-28. Doi: 10.3934/krm.2014.7.9

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Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$

1.
Department of Mathematics, Ocean University of China, Qingdao, 266003
2.
Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708
3.
Department of Mathematical Sciences, Peking University, Beijing, 100871

Received: September 30, 2013

Revised: September 30, 2013

Published: February 2014

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Abstract

This paper establishes the hyper-contractivity in $L^\infty(\mathbb{R}^d)$ (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent $m>1-2/d$. The results show that for the supercritical and critical case $1-2/d < m ≤ 2-2/d$, if $||U_0||_{d(2-m)/2} < C_{d,m}$ where $C_{d,m}$ is a universal constant, then for any $t>0$, $||u(\cdot,t)||_{L^\infty(\mathbb{R}^d)}$ is bounded and decays as $t$ goes to infinity. For the subcritical case $m>2-2/d$, the solution $u(\cdot,t) \in L^\infty(\mathbb{R}^d)$ with any initial data $U_0 \in L_+^1(\mathbb{R}^d)$ for any positive time.

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Mathematics Subject Classification: Primary: 35K65; Secondary: 35B45, 35J20.

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