# American Institute of Mathematical Sciences

November  2009, 24(4): 1113-1127. doi: 10.3934/dcds.2009.24.1113

## On scattering for NLS: From Euclidean to hyperbolic space

 1 Département de Mathématiques, Univ. Evry, Bd. F. Mitterrand, 91025 Evry, France 2 Université Montpellier 2, Mathématiques, CC051, 34095 Montpellier, CNRS, UMR 5149, 34095 Montpellier 3 Département de Mathématiques, Univ. Cergy-Pontoise, CNRS UMR 8088, 2 avenue Adolphe Chauvin, BP 222, Pontoise, 95302 Cergy-Pontoise cedex, France

Received  March 2008 Revised  January 2009 Published  May 2009

We prove asymptotic completeness in the energy space for the nonlinear Schrödinger equation posed on hyperbolic space $\mathbb H^n$ in the radial case, for $n\ge 4$, and any energy-subcritical, defocusing, power nonlinearity. The proof is based on simple Morawetz estimates and weighted Strichartz estimates. We investigate the same question on spaces which sort of interpolate between Euclidean space and hyperbolic space, showing that the family of short range nonlinearities becomes larger and larger as the space approaches the hyperbolic space. Finally, we describe the large time behavior of radial solutions to the free dynamics.
Citation: Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113
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