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Both Mass conservation for the nonlinear Schrödinger equation and Energy conservation for the nonlinear Schrödinger equation hold throughout the maximal lifespan. Since $\Vert u_0{\Vert }_2<\Vert Q{\Vert }_2$, the Sharp Gagliardo-Nirenberg inequality gives the uniform estimateThus the gradient norm stays bounded. The Blowup alternative for the nonlinear Schrödinger equation rules out a finite endpoint of the lifespan in either time direction, so the solution is global.

Under the mass-critical spatial scalingthe $L^2$ norm is invariant and the gradient norm is multiplied by ${\lambda }_n$. Therefore chooseFinite-time blowup and the Blowup alternative for the nonlinear Schrödinger equation imply $\Vert \nabla u(t_n){\Vert }_2\to \infty$, so ${\lambda }_n\to 0$.

The energy has scaling degree two:where the second equality uses Energy conservation for the nonlinear Schrödinger equation.

Interpolate between $L^2$ and ${\dot{H}}^1$ and then use the preceding Sobolev embedding theorem:Mass conservation for the nonlinear Schrödinger equation fixes $\Vert u(t){\Vert }_2=\Vert u_0{\Vert }_2$, while Energy conservation for the nonlinear Schrödinger equation and $E(u_0)\le 0$ implySince $\sigma (p+1)>2$, rearrangement givesuniformly on the maximal lifespan.