Dynamics of Partial Differential Equations
Volume 10 (2013)
Blow up on a curve for a nonlinear Schrödinger equation on Riemannian surfaces
Pages: 99 – 155
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  and our result shows that the log log rate persists in other geometries with the assumption of a radial symmetry of the manifold.
blow up, nonlinear Schrödinger equation, Riemannian surface
2010 Mathematics Subject Classification