= Global existence for the mass-subcritical generalized KdV equation
= Global existence for the energy-subcritical generalized KdV equation
{synonym}
For the integer powers $p=2,3,4$, the <Generalized Korteweg–De Vries equation> is mass-subcritical. Its <Gagliardo-Nirenberg interpolation inequality> is
$$
\|u\|_{p+1}^{p+1}\le C\|u_x\|_2^{(p-1)/2}\|u\|_2^{(p+3)/2}.
$$
The exponent $\alpha=(p-1)/2$ is strictly below two. Since <KdV mass and energy conservation> fixes $\|u\|_2$ and $E(u)$, the gradient size $X=\|u_x\|_2$ obeys $X^2/2\le |E(u_0)|+C_{M(u_0)}X^\alpha$. <Young's inequality for products> absorbs the lower power into $X^2/4$ and leaves a uniform bound for $X$.
Under the standard local $H^1(\mathbb R)$ solution theory with a continuation time controlled by the <H1 norm>, this bound extends the solution globally. At $p=5$, the nonlinear estimate has gradient exponent two, so this absorption argument requires additional information such as a small-mass threshold. The scaling identifies this as an <L2 norm> threshold.
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