On the blow up phenomenon of the critical nonlinear Schrödinger equation

Abstract

In this paper we consider the blow up phenomenon of critical nonlinear Schrödinger equations in dimension 1D and 2D. We define the minimal mass as the $L^2$ norm necessary to ignite a wave collapse and we stress its role in the blow up mechanism. Asymptotic compactness properties and $L^2$-concentration are proved. The proof relies on linear and nonlinear profile decompositions.