Local and global well-posedness, and Lp′-decay estimates for 1D nonlinear Schrodinger equations with Cauchy data in Lp

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 criticalL(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 theL(p)-framework including linear weighted estimates and bi-linear estimates for Duhamel type operators play a central role in proving the main results. (C) 2020 Elsevier Inc. All rights reserved.