Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$ Read More: http://epubs.siam.org/doi/abs/10.1137/110839102

Li Chen Tsinghua University Jian-Guo Liu Duke University Jinhuan Wang Liaoning University

Analysis of PDEs mathscidoc:1702.03048

Siam Journal on Mathematical Analysis, 44, (2), 1077–1102, 2012.2
This paper deals with a degenerate diffusion Patlak–Keller–Segel system in $n\geq 3$ dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent $m=2n/(n+2)$, which is smaller than the usual exponent $m^{*}=2-2/n$ used in other studies. With the exponent $m=2n/(n+2)$, the associated free energy is conformal invariant, and there is a family of stationary solutions $U_{\lambda,x_0}(x)=C(\frac{\lambda} {\lambda^2+|x-x_0|^2})^{\frac{n+2}{2}}$ $\forall \lambda>0$, $x_0\in {\mathbb R}^n$. For radially symmetric solutions, we prove that if the initial data are strictly below $U_{\lambda,0}(x)$ for some $\lambda$, then the solution vanishes in $L^1_{loc}$ as $t\to\infty$; if the initial data are strictly above $U_{\lambda,0}(x)$ for some $\lambda$, then the solution either blows up at a finite time or has a mass concentration at $r=0$ as time goes to infinity. For general initial data, we prove that there is a global weak solution provided that the $L^m$ norm of initial density is less than a universal constant, and the weak solution vanishes as time goes to infinity. We also prove a finite time blow-up of the solution if the $L^m$ norm for initial data is larger than the $L^m$ norm of $U_{\lambda,x_0}(x)$, which is constant independent of $\lambda$ and $x_0$, and the free energy of initial data is smaller than that of $U_{\lambda,x_0}(x)$.
chemotaxis, critical diffusion exponent, nonlocal aggregation, critical stationary solution, global existence, mass concentration, radially symmetric solution
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@inproceedings{li2012multidimensional,
  title={Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$   Read More: http://epubs.siam.org/doi/abs/10.1137/110839102},
  author={Li Chen, Jian-Guo Liu, and Jinhuan Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208224536466908327},
  booktitle={Siam Journal on Mathematical Analysis},
  volume={44},
  number={2},
  pages={1077–1102},
  year={2012},
}
Li Chen, Jian-Guo Liu, and Jinhuan Wang. Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$ Read More: http://epubs.siam.org/doi/abs/10.1137/110839102. 2012. Vol. 44. In Siam Journal on Mathematical Analysis. pp.1077–1102. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170208224536466908327.
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