On blow-up criterion for the nonlinear Schrödinger equation

Abstract

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