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For $p<5$, the one-dimensional Gagliardo-Nirenberg interpolation inequality givesThe exponent of $\Vert u_x{\Vert }_2$ is below two, so the conserved mass and energy control the $H^1$ norm. The blowup alternative then extends every local $H^1$ solution globally.

In three dimensions, the Hardy–Littlewood–Sobolev inequality and Gagliardo-Nirenberg interpolation inequality giveConserved mass and energy therefore bound the $H^1$ norm by Young inequality, and the blowup alternative gives global existence.

The one-dimensional Gagliardo-Nirenberg interpolation inequality yieldsMass conservation fixes $\Vert u{\Vert }_2$. If $X=\Vert u_x{\Vert }_2$, energy conservation therefore givesFor $p<5$, the second exponent is strictly smaller than two. The right side tends to infinity with $X$, so this inequality bounds $X$ uniformly throughout the lifespan. The conserved $L^2$ norm then bounds $\Vert u(t){\Vert }_{H^1}$. The stated blowup criterion rules out a finite endpoint, proving Global existence for the energy-subcritical generalized KdV equation.

In three dimensions, the Hardy–Littlewood–Sobolev inequality applied to the Newtonian kernel givesThe Gagliardo-Nirenberg interpolation inequality then givesWriting $X=\Vert \nabla u{\Vert }_2$ and using the conserved mass, conservation of energy impliesThus $X$ remains bounded. Together with the conserved $L^2$ norm this bounds $\Vert u(t){\Vert }_{H^1}$, and the supplied blowup criterion proves Global H1 solutions of the three-dimensional gravitational Hartree equation.

Set $p=2+4/d$. For $u_{a,\lambda }(x)=au(\lambda x)$, the change of variables $y=\lambda x$ givesandThe factors cancel, so the Weinstein functional satisfies $J(u_{a,\lambda })=J(u)$.

The Gagliardo-Nirenberg interpolation inequality givesConsequently $J(u)\ge C_d^{-1}$ for every nonzero $u\in H^1$, and therefore $I>0$.