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

Wenting Cong Jilin University Jian-Guo Liu Duke University

Analysis of PDEs mathscidoc:1702.03017

Discrete and Continuous Dynamical Systems - Series B, 22, (2), 307-338, 2017.2
This paper investigates the existence of a uniform in time $L^{\infty}$ 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 $L^1\cap L^{\infty}$ initial data.
Chemotaxis, fast diffusion, critical space, semi-group theory, global existence
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@inproceedings{wenting2017uniform,
  title={Uniform $L^{\infty}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model},
  author={Wenting Cong, and Jian-Guo Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207181701136774258},
  booktitle={Discrete and Continuous Dynamical Systems - Series B},
  volume={22},
  number={2},
  pages={307-338},
  year={2017},
}
Wenting Cong, and Jian-Guo Liu. Uniform $L^{\infty}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. 2017. Vol. 22. In Discrete and Continuous Dynamical Systems - Series B. pp.307-338. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170207181701136774258.
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