Set
$$
I_\chi(t)=\int\chi|u|^2,
\qquad
V_\chi(t)=\operatorname{Im}\int\nabla\chi\mathbin{\cdot}\nabla u\,\overline u.
$$
Substituting $u_t=i(\Delta u+|u|^{p-1}u)$ and integrating by parts gives the <localized virial identity>
$$
\frac12I_\chi'(t)=V_\chi(t).
$$
Differentiating once more, integrating the Laplacian terms twice, and observing that the gauge-invariant nonlinearity contributes only through $\Delta\chi$, gives
$$
\frac12V_\chi'(t)
=\int \operatorname{Hess}\chi(\nabla u,\nabla\overline u)
-\frac14\int\Delta^2\chi|u|^2
-\left(\frac12-\frac1{p+1}\right)
\int\Delta\chi|u|^{p+1}.
$$
These are the required formulas, with $\chi''|\nabla u|^2$ interpreted as the Hessian quadratic form for a radial weight.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-154/2/2/3/solution