Blowup for fractional NLS

We consider fractional NLS with focusing power-type nonlinearity i partial derivative(t)u = (-Delta)(3)u - vertical bar u vertical bar(2 sigma)u, (t, x) epsilon R x R-N, where 1/2 < s < 1 and 0 < sigma < infinity for s N/2 and 0 < sigma <= 2s/(N - 2s) for s < N/2. We prove a general criterion for blowup of radial solutions in R-N with N >= 2 for L-2-supercritical and L-2-critical powers sigma >= 2s/N. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain Omega subset of R-N in any dimension N >= 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 R-N, we derive a localized virial estimate for fractional NLS in 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 Pohozaevtype estimate for the fractional Laplacian to prove blowup. (C) 2016 Elsevier Inc. All rights reserved.