Blowup for fractional NLS

We consider fractional NLS with focusing power-type nonlinearity $$i{\mathrm{\partial }}_{t}u=(-\mathrm{\Delta }{)}^{s}u-\mathrm{\mid }u{\mathrm{\mid }}^{2\sigma }u,(t,x)\in \mathbb{R}\times {\mathbb{R}}^N,$$ where $1\mathrm{/}2<s<1$ and $0<\sigma <\mathrm{\infty }$ for $s \geq N/2$and$0 < \sigma \leq 2s/(N-2s)$for$s < N/2$. We prove a general criterion for blowup of radial solutions in ${\mathbb{R}}^N$ with $N\ge 2$ for $L^2$-supercritical and $L^2$-critical powers $\sigma \ge 2s\mathrm{/}N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ in any dimension $N\ge 1$ and subject to exterior Dirichlet conditions. In this setting, we prove a general blowup result without imposing any symmetry assumption on $u(t,x)$. For the blowup proof in ${\mathbb{R}}^N$, we derive a localized virial estimate for fractional NLS in ${\mathbb{R}}^N$, which uses Balakrishnan’s formula for the fractional Laplacian $(-\mathrm{\Delta }{)}^s$ from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.