Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$
Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$
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DOI：
10.3934/krm.2014.7.9
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发表时间：
2013-12
影响因子：
1
通讯作者：
S. Bian;Jian‐Guo Liu;Chenlong Zou
中科院分区：
文献类型：
--
作者：
S. Bian;Jian‐Guo Liu;Chenlong Zou
本文建立了扩散指数m>1-2/d的多维Keller-Segel系统在L^\infty（\mathbb{R}^d）中的超压缩性（称为超压缩性）.结果表明，对于超临界和临界情况$1-2/d 0$，$||u（\cdot，t）||_{L^\infty（\mathbb{R}^d）}$是有界的，并且随着t$趋于无穷大而衰减。对于亚临界情形$m>2-2/d$，对于任意正时刻，解$u（\cdot，t）\in L^\infty（\mathbb{R}^d）$具有任意初始数据$U_0 \in L_+^1（\mathbb{R}^d）$。
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 $1-2/d 0$, $||u(\cdot,t)||_{L^\infty(\mathbb{R}^d)}$ 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.