Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system
Looking for critical nonlinearity in the one-dimensional quasilinear Smoluchowski-Poisson system
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寻找一维拟线性 Smoluchowski-Poisson 系统中的临界非线性
DOI：
10.3934/dcds.2010.26.417
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
Philippe Laurencçot
中科院分区：
文献类型：
--
作者：
Tomasz Cie'slak;Philippe Laurencçot
一维拟线性Smoluchowski-Poisson方程组具非线性扩散项$a(U)=(1+u)^{-p}$当$p>；1$时经典解可能在有限时间内爆破，当$p<；1$时整体存在经典解。因此，情况$p=1$似乎是关键的，但事实证明，在这种情况下，所有解也是全局的。本文实际上识别了两类扩散系数，一类是相应的拟线性Smoluchowski-Poisson系统的所有解都是整体的，另一类是导致足够集中的初值的有限时间爆破的。证明的基石是Smoluchowski-Poisson系统的另一种表述，它依赖于变量的新变化和维里恒等式。
It is known that classical solutions to the one-dimensional quasilinear Smoluchowski-Poisson system with nonlinear diffusion $a(u)=(1+u)^{-p}$ may blow up in finite time if $p>1$ and exist globally if $p<1$. The case $p=1$ thus appears to be critical but it turns out that all solutions are global also in that case. Two classes of diffusion coefficients are actually identified in this paper, one for which all solutions to the corresponding quasilinear Smoluchowski-Poisson system are global and the other one leading to finite time blow-up for sufficiently concentrated initial data. The cornerstone of the proof are an alternative formulation of the Smoluchowski-Poisson system which relies on a novel change of variables and a virial identity.
影响因子：
0.9
作者：
G. Burton
通讯作者：
G. Burton
DOI：
10.1007/s00526-008-0200-7
发表时间：
2009-06-01
影响因子：
2.1
作者：
Blanchet, Adrien;Carrillo, Jose A.;Laurencot, Philippe
通讯作者：
Laurencot, Philippe