Interpolate between $L^2$ and $\dot H^1$ and then use the preceding <Sobolev embedding theorem>:
$$
\|u\|_{p+1}^{p+1}
\leq C\|u\|_2^{(1-\sigma)(p+1)}
\|\nabla u\|_2^{\sigma(p+1)}.
$$
<Mass conservation for the nonlinear Schrödinger equation> fixes $\|u(t)\|_2=\|u_0\|_2$, while <Energy conservation for the nonlinear Schrödinger equation> and $E(u_0)\leq0$ imply
$$
\frac12\|\nabla u(t)\|_2^2
\leq\frac1{p+1}\|u(t)\|_{p+1}^{p+1}.
$$
Since $\sigma(p+1)>2$, rearrangement gives
$$
\|\nabla u(t)\|_2\geq c(u_0)>0
$$
uniformly on the maximal lifespan.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 2 / 2 / Solution

{c}

For the focusing power equation $iu_t+\Delta u+|u|^{p-1}u=0$, the conserved energy is
$$
E(u)=\frac12\int|\nabla u|^2-
\frac1{p+1}\int|u|^{p+1}.
$$


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