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

The homogeneous Sobolev embedding theorem givessoNow $s_c>0$ is equivalent to $p-1>4/d$, which gives $\sigma >2/(p+1)$. Similarly $s_c<1$ is equivalent to $(d-2)(p-1)<4$, which gives $\sigma <1$.

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Interpolate between $L^2$ and ${\dot{H}}^1$ and then use the preceding Sobolev embedding theorem:Mass conservation for the nonlinear Schrödinger equation fixes $\Vert u(t){\Vert }_2=\Vert u_0{\Vert }_2$, while Energy conservation for the nonlinear Schrödinger equation and $E(u_0)\le 0$ implySince $\sigma (p+1)>2$, rearrangement givesuniformly on the maximal lifespan.

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SetSubstituting $u_t=i(\Delta u+|u{|}^{p-1}u)$ and integrating by parts gives the localized virial identityDifferentiating once more, integrating the Laplacian terms twice, and observing that the gauge-invariant nonlinearity contributes only through $\Delta \chi$, givesThese are the required formulas, with ${\chi }^{\prime \prime }|\nabla u{|}^2$ interpreted as the Hessian quadratic form for a radial weight.

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For a smooth radial function that decays at infinity,The Cauchy-Schwarz inequality therefore yields the Radial Sobolev inequalityup to the harmless common normalization of surface measure. Taking the supremum over $r\ge R$ gives the claimed estimate; density extends it from smooth radial functions to every $u\in H_r^1$.

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For ${\psi }_R(x)=R^2\psi (x/R)$, the assumptions imply $\mathrm{Hess}{\psi }_R\le I$, while on $|x|\le 2R$ one hasApply the localized virial identity and compare its interior terms withThe coefficient $d(p-1)/4$ is strictly greater than one because $s_c>0$. The resulting negative multiple of $\Vert \nabla u{\Vert }_2^2$ can be moved to the left. All errors in the nonlinear term are supported on $|x|\ge 2R$, and ${\Delta }^2{\psi }_R=O(R^{-2})$ is supported on $2R\le |x|\le 10R$. Thus

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Suppose that $T=\infty$. Since $s_c<1$ and $d\ge 3$,The Radial Sobolev inequality and Mass conservation for the nonlinear Schrödinger equation giveBecause $(p-1)/2<2$, Young inequality and a sufficiently large fixed $R$ absorb this term into the left side of the estimate from part 5. The annular term is at most $C(u_0)R^{-2}$. Using the uniform lower bound $\Vert \nabla u(t){\Vert }_2\ge c(u_0)$ from part 2 and enlarging $R$ once more yields

Since $I_{{\psi }_R}^{\prime }=2V_{{\psi }_R}$, two integrations giveThe right-hand side is negative for large $t$, contradicting $I_{{\psi }_R}(t)=\int {\psi }_R|u{|}^2\ge 0$. Hence the maximal forward lifespan is finite: $T<\infty$.

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