Geometric and Functional Analysis

, Volume 18, Issue 2, pp 367-399

First online:

Scattering Theory for Radial Nonlinear Schrödinger Equations on Hyperbolic Space

  • Valeria BanicaAffiliated withDépartement de Mathématiques, Université d’Evry Email author 
  • , Rémi CarlesAffiliated withCNRS & Université Montpellier 2, Mathématiques, CC 051
  • , Gigliola StaffilaniAffiliated withMIT

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We study the long-time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short-range and long-range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz-type inequalities. The latter are established without symmetry assumptions.

Keywords and phrases:

Nonlinear Schrödinger equations on manifolds asymptotic behavior Strichartz estimates Morawetz estimates

AMS Mathematics Subject Classification:

35B40 35P25 35Q55 58J37