Mathematische Annalen

, 345:133 | Cite as

Semilinear Schrödinger flows on hyperbolic spaces: scattering in H1

Article

Abstract

We prove global well-posedness and scattering in H1 for the defocusing nonlinear Schrödinger equations
$$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$$
on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H1 is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.University of Wisconsin–MadisonMadisonUSA
  2. 2.Massachusetts Institute of TechnologyCambridgeUSA

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