Blowup for fractional NLS

We consider fractional NLS with focusing power-type nonlinearity
i∂tu=(−Δ)su−|u|2σu,(t,x)∈R×RN,
where 1/2<s<11/2<s<1 and 0<σ<∞0<σ<∞ for s⩾N/2s⩾N/2 and 0<σ⩽2s/(N−2s)0<σ⩽2s/(N−2s) for s<N/2s<N/2. We prove a general criterion for blowup of radial solutions in RNRN with N⩾2N⩾2 for L2L2-supercritical and L2L2-critical powers σ⩾2s/Nσ⩾2s/N. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain Ω⊂RNΩ⊂RN in any dimension N⩾1N⩾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)u(t,x).
For the blowup proof in RNRN, we derive a localized virial estimate for fractional NLS in RNRN, which uses Balakrishnan's formula for the fractional Laplacian (−Δ)s(−Δ)s from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.