Dynamics of Partial Differential Equations

Volume 20 (2023)

Number 3

Blow-up or Grow-up for the threshold solutions to the nonlinear Schrödinger equation

Pages: 213 – 225

DOI: https://dx.doi.org/10.4310/DPDE.2023.v20.n3.a3


Stephen Gustafson (University of British Columbia, Vancouver, B.C., Canada)

Takahisa Inui (Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, Japan; and University of British Columbia, Vancouver, B.C., Canada)


We consider the nonlinear Schrödinger equation with $L^2$-supercritical and $H^1$-subcritical power type nonlinearity. Duyckaerts and Roudenko [8] and Campos, Farah, and Roudenko [3] studied the global dynamics of the solutions with same mass and energy as that of the ground state. In these papers, finite variance is assumed to show the finite time blow-up. In the present paper, we remove the finite-variance assumption and prove a blow-up or grow-up result.


nonlinear Schrödinger equation, blow-up, grow-up, threshold

2010 Mathematics Subject Classification

Primary 35Q55. Secondary 35B44.

Research of the first author is partially supported by an NSERC Discovery Grant. The second author is supported by JSPS Overseas Research Fellowship and KAKENHI Grant-in-Aid for Early-Career Scientists No. JP18K13444.

Received 11 January 2023

Published 19 May 2023