Blowup for fractional NLS

We consider fractional NLS with focusing power-type nonlinearity
\[
i\partial_t u = (-\Delta)^s u - |u|^{2\sigma}u, (t,x) \in \mathbb{R}\times\mathbb{R}^N,
\]
where $1/2 < s < 1$ and $0 < \sigma < \infty$ for $s\ge N/2$ and $0 < \sigma \le 2s/(N-2s)$ 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 $\sigma\ge 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $\Omega\subset\mathbb{R}^N$ 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 $(-\Delta)^s$ from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.