Global existence for the mass-subcritical generalized KdV equation

For the integer powers $p=2,3,4$, the Generalized Korteweg–De Vries equation is mass-subcritical. Its Gagliardo-Nirenberg interpolation inequality isThe exponent $\alpha =(p-1)/2$ is strictly below two. Since KdV mass and energy conservation fixes $\Vert u{\Vert }_2$ and $E(u)$, the gradient size $X=\Vert u_x{\Vert }_2$ 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.