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

TL;DR

This paper establishes conditions for global existence, scattering, and blow-up of focusing nonlinear Schrödinger equations on hyperbolic space, extending Euclidean results with new identities.

Contribution

It introduces a novel approach using generalized Pohozaev identities to analyze NLS on hyperbolic space, including a scattering vs blow-up dichotomy for radial solutions.

Findings

Proves global well-posedness and scattering for energy-subcritical cases.

Establishes blow-up results under certain conditions.

Identifies a critical element for scattering across all energy-subcritical nonlinearities.

Abstract

We prove global well-posedness, scattering and blow-up results for energy-subcritical focusing nonlinear Schr\"odinger 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.