Blow up of solutions to the nonlinear Schrödinger equations on manifolds

Abstract

In this paper, we partially settle down the long-standing open problem of the finite time blow-up property about the nonlinear Schrödinger equations on some Riemannian manifolds such as the standard 2-sphere S2 and the hyperbolic 2-space H2(−1). Using the similar idea, we establish such blow-up results on higher dimensional standard sphere and hyperbolic n-space. Extensions to n-dimensional Riemannian warped product manifolds with n⩾2 are also given.