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The Nonlinear Schrödinger Equation on Hyperbolic Space
Abstract
In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space n , in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties: we prove that the dispersion inequality is valid, in a stronger form than the one on
n . However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.
Keywords
- Blow-up,
- Dispersion estimates,
- Hyperbolic space,
- Nonlinear Schrödinger equations on manifolds,
- Representation of solutions
Mathematics Subject Classification
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Details
- Citation information:
- Available online: 23 Oct 2007
Author affiliations
- a Laboratoire d'Analyse et Probabilité, Département de Mathématiques, Université d'Evry, Evry, France
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