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PARABOLIC ELLIPTIC TYPE KELLER-SEGEL SYSTEM ON THE WHOLE SPACE CASE

机译：抛物线椭圆型Keller-Segel系统在整个空间外壳上

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摘要

This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass 8π. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent m = 2 — 2 which comes from the scaling invariance of the mass, and the exponent m = 2n/(n + 2) which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.
机译：本说明讨论了在整个空间壳体上对抛物线椭圆型Patlak-Keller-Segel系统的解决方案的存在和爆发。两个维度的问题与对数硬性小屋-SoboLev不等式密切相关，这直接引入了临界质量8π。虽然在较高的尺寸案例中，它与哈迪特小木-Sobolev不等式有关。因此，已经引入了多孔介质型非线性扩散以平衡聚集。我们将审查文献中引入的批判性指数，即指数M = 2 - 2 / N，来自质量的缩放不变性，以及来自的指数M = 2n /（n + 2）熵的共形不变性。最后，将给出具有普遍潜力的模型的新结果，从哈欠Littlewood-Sobolev-Sobolev不等式中发出。

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