Choose a smooth radial cutoff $\theta_R$ that vanishes on $|x|\leq R$, equals one on $|x|\geq2R$, and satisfies $|\nabla\theta_R|\leq C/R$. The first <localized virial identity> and the estimate from part 2, applied to $w=u(t)$ and $\psi=\theta_R$, give
$$
\left|\frac d{dt}\int\theta_R|u(t)|^2\right|
\leq\frac{C(u_0)}R,
$$
because the mass and energy are conserved.

For fixed $R$, the strong profile convergence from part 5 and $\lambda_n\to0$ imply
$$
\int\theta_R(x)|u(t_n,x)|^2,dx
=\int\theta_R(\lambda_n y)|v_n(y)|^2,dy\longrightarrow0.
$$
Integrating the flux estimate from $t$ to $t_n$ and then letting $n\to\infty$ yields
$$
\int_{|x|\geq2R}|u(t,x)|^2,dx
\leq\frac{C(u_0)T}{R}
$$
uniformly for $0\leq t<T$. Taking $R$ sufficiently large proves the claim.

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-154/3/3/6/solution