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

Shen Bian Ocean University of China Jian-Guo Liu Duke University Chen Zou Duke University

Analysis of PDEs mathscidoc:1702.03038

Kinetic & Related Models, 7, (1), 9 - 28, 2014.3
This paper establishes the hyper-contractivity in L∞ (ℝd) (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the dif;fusion exponent m > 1 -2/d. The results show that for the supercritical and critical case 1 - 2/d < m ≤ 2 - 2/d, if || U0||d(2-m)/2) < Cd, m where Cd, m is a universal constant, then for any t > 0, || u (⋅, t)|| ... is bounded and decays as t goes to infinity, For the subcritical case m > 2 - 2/d, the solution u (⋅, t) ∈ L∞ (ℝd) with any initial data U0 ∈ L1+ (ℝd) for any positive time.
Hyper-contractive, ultra-contractive, chemotaxis, nonlocal aggregation, degenerate diffusion.
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@inproceedings{shen2014ultra-contractivity,
  title={Ultra-contractivity for Keller-Segel model with diffusion exponent m>1-2/d},
  author={Shen Bian, Jian-Guo Liu, and Chen Zou},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208211558641210313},
  booktitle={Kinetic & Related Models},
  volume={7},
  number={1},
  pages={9 - 28},
  year={2014},
}
Shen Bian, Jian-Guo Liu, and Chen Zou. Ultra-contractivity for Keller-Segel model with diffusion exponent m>1-2/d. 2014. Vol. 7. In Kinetic & Related Models. pp.9 - 28. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208211558641210313.
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