Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition
Finite time blow up and non-uniform bound for solutions to a degenerate drift-diffusion equation with the mass critical exponent under non-weight condition
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DOI：
10.1007/s00229-019-01108-x
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发表时间：
2019-02
影响因子：
0.6
通讯作者：
T. Ogawa;H. Wakui
中科院分区：
文献类型：
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作者：
T. Ogawa;H. Wakui
考虑一类具有质量临界指数的退化漂移扩散方程组的Cauchy问题的时间整体解的不存在性和非一致有界性。如果初始数据具有负自由能，则方程的相应弱解在时间上不全局存在，或者时间全局解在能量空间中不保持有界。我们强调，我们的结果不需要对初始数据的任何重量假设，因此，解决方案可能有一个无限的二阶矩。证明是基于修改的维里定律和守恒定律，我们表明，修改后的功能在负能量条件下的有限时间为零。对于径向对称的情形，解在有限时间内爆破，出现质量集中现象，且存在一个与Hardy-Littlewood-Sobolev不等式最佳常数相关的下界.
We consider the non-existence and the non-uniform boundedness of a time global solution to the Cauchy problem of a degenerate drift-diffusion system with the mass critical exponent. If the initial data has negative free energy, then either the corresponding weak solution to the equation does not exist globally in time, or the time global solution does not remain bounded in the energy space. We emphasize that our result does not require any weight assumption on the initial data, and hence, a solution may have an infinite second moment. The proof is based upon the modified virial law and conservation laws and we show that the modified moment functional vanishes for a finite time under the negative energy condition. For a radially symmetric case, the solution blows up in finite time and the mass concentration phenomenon occurs with a sharp lower bound related to the best constant for the Hardy–Littlewood–Sobolev inequality.