Dynamics of Partial Differential Equations

Volume 10 (2013)

Number 2

Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces

Pages: 99 – 155

DOI: http://dx.doi.org/10.4310/DPDE.2013.v10.n2.a1

Author

Nicolas Godet (Department of Mathematics, CNRS, UMR 8088, University of Cergy-Pontoise, France)

Abstract

We consider the focusing quintic nonlinear Schrödinger equation posed on a rotationally symmetric surface, typically the sphere $S^2$ or the two dimensional hyperbolic space $H^2$. We prove the existence and the stability of solutions blowing up on a suitable curve with the log log speed. The Euclidean case is handled in [25] and our result shows that the log log rate persists in other geometries with the assumption of a radial symmetry of the manifold.

Keywords

blow up, nonlinear Schrödinger equation, Riemannian surface

2010 Mathematics Subject Classification

35-xx

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