Well-posedness in weighted spaces for the generalized Hartree equation with p < 2
Well-posedness in weighted spaces for the generalized Hartree equation with p < 2
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DOI：
10.1142/s0219199721500747
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发表时间：
2020-12
影响因子：
1.6
通讯作者：
A. Arora;Oscar G. Riaño;S. Roudenko
中科院分区：
文献类型：
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作者：
A. Arora;Oscar G. Riaño;S. Roudenko
我们研究了广义Hartree方程[公式：见正文]，[公式：见正文]，[公式：见正文]中的适定性，对于低的非线性幂，[公式：见正文]。我们建立了加权Sobolev空间中一类数据的局部适定性，遵循Cazenave和Naumkin的思想，Local existence，global existence，and scattering for the nonlinear Schrodinger equation，Comm. Contemp.数学19（2）（2017）1650038.这主要依赖于Riesz变换在加权Lebesgue空间中的有界性。因此，我们得到了一类全局存在的数据，而且，在正时间分散。此外，在聚焦的情况下，在[公式：见正文]-超临界设置中，我们得到了一个子集的局部适定的数据与正能量，在有限的时间爆炸。
We investigate the well-posedness in the generalized Hartree equation [Formula: see text], [Formula: see text], [Formula: see text], for low powers of nonlinearity, [Formula: see text]. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the [Formula: see text]-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.