A universal bound on the blow-up rate for the focusing mass-critical nonlinear Schrödinger equation

Abstract

In this paper, we investigate a universal blow-up bound for the focusing mass-critical nonlinear Schrödinger equation for general initial data in $L^2({\mathbb{R}}^d)$, extending previous knowledge for mass near the ground-state threshold due to Merle and Raphaël. The main results are twofold. First, we show the nonexistence of self-similar rate blow-up solutions. Second, under radial symmetry, we establish the sharp log--log correction to the self-similar bound on the blow-up rate. The proofs are based on a new analysis of general blow-up solutions, which does not rely on any ansatz or variational structure.

Strengthened the consequences of the main result (Corollary 1.4 and Theorem 1.5) and added the corresponding proofs. Simplified some proofs in the paper