On blow-up criterion for the nonlinear Schrödinger equation

Dapeng  Du; Yifei  Wu; Kaijun  Zhang

doi:10.3934/dcds.2016.36.3639

Article Contents

2016, Volume 36, Issue 7: 3639-3650. Doi: 10.3934/dcds.2016.36.3639

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On blow-up criterion for the nonlinear Schrödinger equation

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024
2.
Center for Applied Mathematics, Tianjin University, Tianjin 300072

Received: February 28, 2015

Revised: October 31, 2015

Published: May 2017

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Abstract

The blowup is studied for the nonlinear Schrödinger equation $iu_{t}+\Delta u+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)<0$ blows up in finite or infinite time. A new proof is also presented for the previous result in [9], in which a similar result in a case of energy-subcritical was shown.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B44.

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