Exact criterion for global existence and blow up to a degenerate Keller-Segel system
Exact criterion for global existence and blow up to a degenerate Keller-Segel system
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DOI：
10.4171/dm/441
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发表时间：
2013-11
影响因子：
0.9
通讯作者：
Li Chen;Jinhuan Wang
中科院分区：
文献类型：
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作者：
Li Chen;Jinhuan Wang
研究了多维扩散指数 m 为 2n n+2 < m < 2 − 2 n 的简并 Keller-Segel 系统。获得了全局存在和解爆炸的精确准则。对解的 L 2n n+2 范数的估计在我们的分析中发挥着重要作用。这些估计与 Hardy-Littlewood-Sobolev 不等式中的最优常数密切相关。在初始自由能小于通用常数（取决于总质量的倒数）的情况下，存在一个常数，如果初始数据的 L 2n n+2 范数小于该常数，则全局存在弱解；如果初始数据的 L 2n n+2 范数大于同一常数，则解必须在有限时间内爆炸。我们的结果表明，在二维情况下起确定性作用的总质量可能不是多维中存在和爆炸讨论的适当标准，而初始数据的 L 2n n+2 范数以及初始自由能和初始质量之间的关系更为重要。
A degenerate Keller-Segel system with diffusion exponent m with 2n n+2 < m < 2 − 2 n in multi dimension is studied. An exact criterion for global existence and blow up of solution is obtained. The estimates on L 2n n+2 norm of the solution play important roles in our analysis. These estimates are closely related to the optimal constant in the Hardy- Littlewood- Sobolev inequality. In the case of initial free energy less than a universal constant which depends on the inverse of total mass, there exists a constant such that if the L 2n n+2 norm of initial data is less than this constant, then the weak solution exists globally; if the L 2n n+2 norm of initial data is larger than the same constant, then the solution must blow-up in finite time. Our result shows that the total mass, which plays the deterministic role in two dimension case, might not be an appropriate criterion for existence and blow up discussion in multi-dimension, while the L 2n n+2 norm of the initial data and the relation between initial free energy and initial mass are more important.