Remarks on the blow-up for the Schr\"odinger equation with critical mass on a plane domain

Abstract

In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schr\"odinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than $(T-t)^{-1}$, the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.