Local well-posedness and blow-up conjecture for the Schrödinger equation with concentrated nonlinearity

Let $s>\frac{n}{2}$ and let $u_0\in D(\mathcal{L}_c)$. There exists $T>0$ such that the equation in (20) is locally well-posed and the conservation laws in (conserve) remain valid up to an eventual blow-up time.

Well-posedness and blow-up conjecture. If $0<\sigma<\frac{2s}{n}-1$, the solutions are global, whereas if $\sigma\geq\frac{2s}{n}-1$, finite-time blow-up is possible for some initial data.

The authors state this as the higher-dimensional analogue of the corresponding one-dimensional result. The claim is made under key assumptions of continuous dependence on initial data near the standing wave and global existence, conservation of mass, and conservation of the Hamiltonian for sufficiently close initial data; its resolution is not specified here.