Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1 / 3 / Solution

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.