= Localized virial identity
For $iu_t+\Delta u+|u|^{p-1}u=0$, a smooth real weight $\chi$ defines the localized mass $I_\chi(t)=\int\chi|u|^2$. For a sufficiently regular decaying solution and a weight with the required integrability,
$$
I_\chi'(t)=2\operatorname{Im}\int\nabla\chi\cdot\nabla u\,\overline u,
$$
and
$$
I_\chi''(t)=4\sum_{j,k}\int\partial_{jk}\chi\,\operatorname{Re}(\partial_j u\,\overline{\partial_k u})-\int\Delta^2\chi\,|u|^2-\frac{2(p-1)}{p+1}\int\Delta\chi\,|u|^{p+1}.
$$
These formulas follow by substituting the <nonlinear Schrödinger equation> and <integration by parts>. The <Hessian> controls the gradient contribution; the fourth <derivative> term records variation of the cutoff.
Choose $\chi_R(x)=R^2\chi(x/R)$, with $\chi(x)=|x|^2$ near zero and constant outside a larger ball. The interior terms reproduce the <virial identity>, while the remaining terms are supported in the transition region or controlled by spatial tails. A bounded weight makes $I_{\chi_R}$ meaningful for finite <mass> without requiring finite variance. A blowup proof still needs estimates that control those tail errors.
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