Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 154 / 3 / 2

For real $h\in \mathcal{D}({\mathbb{R}}^d)$, differentiation under the integral givesThe ground-state equation $\Delta Q-Q+Q^{1+4/d}=0$ and integration by parts reduce this to

In particular, along the amplitude direction $h=Q$ the derivative is $-\Vert Q{\Vert }_2^2<0$. Hence $u_0=(1+\delta )Q$ has negative energy for every sufficiently small $\delta >0$, whilewhen $\delta$ is chosen small enough. The ground state has finite variance, so negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation shows that the corresponding solution blows up in finite time.