The one-dimensional <Gagliardo-Nirenberg interpolation inequality> yields
$$
\|u\|_{p+1}^{p+1}
\leq C_p\|u_x\|_2^{(p-1)/2}\|u\|_2^{(p+3)/2}.
$$
Mass conservation fixes $\|u\|_2$. If $X=\|u_x\|_2$, energy conservation therefore gives
$$
E(u_0)
\geq\frac12X^2-C(u_0)X^{(p-1)/2}.
$$
For $p<5$, 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 $\|u(t)\|_{H^1}$. The stated blowup criterion rules out a finite endpoint, proving <Global existence for the energy-subcritical generalized KdV equation>.

Solution

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

{c}

For $p<5$, the one-dimensional <Gagliardo-Nirenberg interpolation inequality> gives
$$
\|u\|_{p+1}^{p+1}
\lesssim\|u_x\|_2^{(p-1)/2}\|u\|_2^{(p+3)/2}.
$$
The exponent of $\|u_x\|_2$ 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.

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