= Solution
Both <Mass conservation for the nonlinear Schrödinger equation> and <Energy conservation for the nonlinear Schrödinger equation> hold throughout the maximal lifespan. Since $\|u_0\|_2<\|Q\|_2$, the <Sharp Gagliardo-Nirenberg inequality> gives the uniform estimate
$$
\frac12\left[1-\left(\frac{\|u_0\|_2}{\|Q\|_2}\right)^{4/d}\right]
\|\nabla u(t)\|_2^2
\leq E(u(t))=E(u_0).
$$
Thus 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.
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