Local and global well-posedness, and [formula omitted]-decay estimates for 1D nonlinear Schrödinger equations with Cauchy data in Lp

Abstract

An Lp-theory of local and global solutions for the one dimensional nonlinear Schrödinger equations with pure power like nonlinearities is developed. Firstly, twisted local well-posedness results in scaling subcritical Lp-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 Lp-spaces is proved, and as an immediate consequence, Lp′-Lp type decay estimates for the global solutions are derived, which are well known for the global solutions to the corresponding linear Schrödinger equation. Finally, global well-posedness results for gauge-invariant equations with large Lp-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 Lp-framework including linear weighted estimates and bi-linear estimates for Duhamel type operators play a central role in proving the main results.