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For the mass-critical focusing nonlinear Schrödinger equation, let $u\in C([0,T_{+});H^1({\mathbb{R}}^d))$ be the maximal forward solution in the standard local $H^1$ theory. Either $T_{+}=\infty$, orBecause Mass conservation for the nonlinear Schrödinger equation fixes $\Vert u(t){\Vert }_2$, this is equivalent to divergence of $\Vert \nabla u(t){\Vert }_2$.

The proof uses a local existence time bounded below for data in a bounded H1 norm ball. If a sequence $t_n\uparrow T_{+}$ had uniformly bounded H1 norms, restart the equation at $t_n$ with that common time interval. Uniqueness joins the restarted solution to the old one and extends it beyond $T_{+}$, contradicting maximality.

The dependence of lifespan on the norm is a hypothesis of this argument. Other critical solution theories may use a spacetime control norm instead, so the stated H1 norm conclusion is attached here to the mass-critical equation and its $H^1$-subcritical local theory.^[ref]

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.

For a sufficiently regular solution of the mass-critical focusing nonlinear Schrödinger equation with finite variance, the variance and its first derivative areA second differentiation and integration by parts give the virial identityThese constants use Laplacian coefficient one in the evolution equation. Energy conservation makes the second derivative constant. If the energy is negative, the resulting quadratic expression for $I(t)$ eventually becomes negative, contradicting its definition as a nonnegative integral; this proves negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation under the finite-variance hypotheses.

The weighted moment condition is additional to an H1 norm bound. A localized virial identity replaces the unbounded weight by a smooth cutoff when only tail control is available.