The nonlinear Schrödinger equation on the hyperbolic space

Details The nonlinear Schrödinger equation on the hyperbolic space The nonlinear Schrödinger equation on the hyperbolic space

2004-06-03

arxiv.org/abs/math/0406058v3

Comm. P.D.E. 32 (2007), no. 10, 1643-1677

Mathematics

Analysis of PDEs

32 pages, final preprint version

Scientific paper

In this article we study some aspects of dispersive and concentration phenomena for the Schr\"odinger equation posed on hyperbolic space $\mathbb{H}^n$, 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 $\mathbb{R}^n$. 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.

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