Dynamics of Partial Differential Equations

Volume 12 (2015)

Number 1

Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space

Pages: 53 – 96

DOI: https://dx.doi.org/10.4310/DPDE.2015.v12.n1.a4


Valeria Banica (Laboratoire de Mathématiques et de Modélisation d’Évry, Département de Mathématiques, Université d’Évry, France)

Thomas Duyckaerts (Laboratoire Analyse, Géométrie et Applications, Institut Galilée, Université Paris 13, Sorbonne Paris-Cité, Villetaneuse, France)


We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schrödinger equations on the hyperbolic space. We show in particular the existence of a critical element for scattering for all energy-subcritical power nonlinearities. For mass-supercritical nonlinearity, we show a scattering vs blow-up dichotomy for radial solutions of the equation in low dimension, below natural mass and energy thresholds given by the ground states of the equation. The proofs are based on trapping by mass and energy, compactness and rigidity, and are similar to the ones on the Euclidean space, with a new argument, based on generalized Pohozaev identities, to obtain appropriate monotonicity formulas.


nonlinear Schrödinger equation, hyperbolic space, scattering, blow-up

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35B30, 35B40, 35B44, 35R01.

Published 19 March 2015