Abstract An L p -theory of local and global solutions for the one dimensional nonlinear Schrodinger equations with pure power like nonlinearities is developed. Firstly, twisted local well-posedness results in scaling subcritical L p -spaces are established for p 2 . This extends Zhou's earlier results for the gauge-invariant cubic NLS equation. Secondly, by a similar functional framework, the global well-posedness for small data in critical L p -spaces is proved, and as an immediate consequence, L p ′ - L p type decay estimates for the global solutions are derived, which are well known for the global solutions to the corresponding linear Schrodinger equation. Finally, global well-posedness results for gauge-invariant equations with large L p -data are proved, which improve earlier existence results, and from which it is shown that the global solution u has a smoothing effect in terms of spatial integrability at any large time. Various Strichartz type inequalities in the L p -framework including linear weighted estimates and bi-linear estimates for Duhamel type operators play a central role in proving the main results.

中文翻译：

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L 中柯西数据的一维非线性薛定谔方程的局部和全局适定性和 Lp'-衰减估计

摘要 发展了具有纯幂类非线性的一维非线性薛定谔方程的局部和全局解的L p 理论。首先，扭曲的局部适定性导致缩放亚临界 L p 空间为 p 2 建立。这扩展了 Zhou 早期关于规范不变三次 NLS 方程的结果。其次，通过类似的函数框架，证明了关键 L p 空间中小数据的全局适定性，并且作为直接结果，导出了全局解的 L p ′ - L p 型衰减估计，这是很好的以对应线性薛定谔方程的全局解而闻名。最后，证明了具有大 L p 数据的规范不变方程的全局适定性结果，改善了早期存在的结果，从中可以看出，全局解 u 在任何大的时间都具有空间可积分性方面的平滑效果。L p 框架中的各种 Strichartz 型不等式，包括 Duhamel 型算子的线性加权估计和双线性估计，在证明主要结果方面发挥着核心作用。