Existence of blowing-up solutions to some Schrödinger equations including nonlinear amplification with small initial data

Abstract

We consider the existence of blowing-up solutions to some Schroedinger equations including nonlinear amplification. The blow-up is considered in L2(R). Even though initial data are taken so small, there exist some solutions blowing-up in finite time. The theorem in this paper is an extension of Cazenave-Martel-Zhao’s result [7] from the point of making the lower bound of power of nonlinearity extended and from the point of ensuring that blowing-up solutions exist even for small initial data.