5 problems

Supercritical blow-up conjecture. The evolution $u(t)$ blows up at finite time $t^*$ and finite location $(x^*,y^*)$, with

Critical-case conjecture. If $\|u_0\|_2<\|\mathcal Q\|_2$, then $u(t)$ is dispersed. If $u_0$ is sufficiently localized and $\|u_0\|_2>\|\mathcal Q\|_2$, then $u(t)$ blows up at fi…

Main conjecture II. For $p=3,4$, the line solitons are strongly unstable: perturbations with $c<c^*(p)$ are dispersed, whereas perturbations with $c>c^*(p)$ lead to blow-up in fini…

Main conjecture I. The line solitons $Q_c$ for $p=2$ and $c>c^*$ are unstable against the formation of lumps.

Let $U(t)$ denote the linear evolution operator and let $u_0$ be initial data, with the mixed-norm estimate … Here $d$ is the spatial dimension, $p$ is the spatial integrability ex…