Communications in Mathematical Sciences

Volume 19 (2021)

Number 4

Global dynamics below the ground states for NLS under partial harmonic confinement

Pages: 993 – 1032



Alex H. Ardila (Department of Mathematics, Instituto de Ciências Exata, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

Rémi Carles (CNRS, IRMAR-UMR 6625,Université de Rennes, France)


We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schrödinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the variational characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles.


nonlinear Schrödinger equation, ground states, global existence, blow-up, scattering

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35P25, 37K45.

R. Carles is supported by Rennes Métropole through its AIS program.

Received 28 October 2020

Accepted 23 November 2020

Published 18 June 2021