Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger Equation in One Dimension

Abstract

We consider the $L^2$-critical quintic focusing nonlinear Schrödinger equation (NLS) on $\mathbf{R}$. It is well known that $H^1$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^1$ spherically symmetric blow-up solutions of the $L^2$-critical focusing NLS, there is a minimal amount of concentration of the $L^2$-norm (the mass of the ground state) at the origin. In this paper we prove the existence of a similar phenomenon for the one-dimensional case and rougher initial data, $(u_0\in H^s,s<1)$, without any additional assumption.

23 pages

References in corpus (1)

• Global well-posedness for Schrödinger equations with derivative