The $L^2$-supercritical gBO blow-up and global-existence dichotomy

Let $m>3$, let $u_0\in X\cap H^{1/2}(\mathbb R)$ be sufficiently localized, where $X$ is the corresponding local well-posedness space for the given $m$, and let $Q$ be the ground state solution of the equation defining $Q$ for that $m$. Define

$$D[u]=\| (\mathcal H\partial_x)^{1/2}u\|_{L^2}^2$$

and let $u(x,t)$ be the gBO time evolution of $u_0(x)$. Put $s=\frac12-\frac1{m-1}$, and write $M[u]$ and $E[u]$ for the mass and energy. The $L^2$-supercritical gBO dichotomy. The solution satisfies the following assertions:

1. $u(x,t)$ blows up in finite time if $E[u_0]<0$;
2. if

$$M[u_0]^{\frac12-s}E[u_0]^s<M[Q]^{\frac12-s}E[Q]^s,\qquad E[u_0]>0,$$

then $u(x,t)$ exists globally in time when

$$M[u_0]^{\frac12-s}D[u_0]^s<M[Q]^{\frac12-s}D[Q]^s,$$

and blows up in finite time when

$$M[u_0]^{\frac12-s}D[u_0]^s>M[Q]^{\frac12-s}D[Q]^s.$$

These claims concern the poorly understood $L^2$-supercritical regime of generalized Benjamin–Ono equations. The source presents them as conjectures motivated by numerical investigations; the negative-energy criterion and the threshold dichotomy describe the expected alternatives for global existence and stable blow-up.