# American Institute of Mathematical Sciences

April  2017, 37(4): 2077-2102. doi: 10.3934/dcds.2017089

## Almost global existence for cubic nonlinear Schrödinger equations in one space dimension

 1 Department of Mathematics, University of California, 970 Evans Hall, Berkeley, CA 94720-3840, USA 2 Department of Mathematics, Princeton University, Fine Hall, 304 Washington Rd, Princeton, NJ 08544, USA

* Corresponding author: Jason Murphy

Received  May 2016 Revised  November 2016 Published  December 2016

We consider non-gauge-invariant cubic nonlinear Schrödinger equations in one space dimension.We show that initial data of size $\varepsilon$ in a weighted Sobolev space lead to solutions with sharp $L_x^∞$ decay up to time $\exp(C\varepsilon^{-2})$. We also exhibit norm growth beyond this time for a specific choice of nonlinearity.

Citation: Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089
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