Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions
Critical mass for a Patlak-Keller-Segel model with degenerate diffusion in higher dimensions
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DOI：
10.1007/s00526-008-0200-7
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发表时间：
2009-06-01
影响因子：
2.1
通讯作者：
Laurencot, Philippe
中科院分区：
文献类型：
--
作者：
Blanchet, Adrien;Carrillo, Jose A.;Laurencot, Philippe
本文致力于分析 d >= 3 的经典抛物线椭圆 Patlak-Keller-Segel 系统和多孔介质非线性扩散的非负解的推广。这里，选择非线性扩散，使其尺度与泊松项的尺度一致。我们证明解的定性行为是由系统的初始质量决定的。实际上，存在尖锐的临界质量 M-c，使得如果 M 是 (0, M-c) 的元素，则解在时间上全局存在，否则存在爆炸解。我们还证明了 M 是 (0, M-c) 的元素时存在自相似解。在表征 M = M-c 的可能的无限时间爆炸剖面时，我们观察到长时渐近在维度上比经典 Patlak-Keller-Segel 系统复杂得多二。
This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak-Keller-Segel system with d >= 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such away that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M-c such that if M is an element of (0, M-c] solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for M is an element of (0, M-c). While characterising the possible infinite time blowing-up profile for M = M-c, we observe that the long time asymptotics are much more complicated than in the classical Patlak-Keller-Segel system in dimension two.