\[

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.

First, we shall deduce sufficient criteria for blowup of radial solutions of the focusing problem in the mass-supercritical and mass-critical cases. The conditions are given in terms of inequalities between a combination of the (kinetic) energy and mass of the initial datum, and that of the ground state for the corresponding elliptic equation. Using a new method to deal with the nonlocality of the fractional Laplacian, a localized virial argument enables us to conclude blowup in finite and infinite time, respectively.

Second, we consider a special class of nondispersive solutions of the focusing fractional NLS: the traveling solitary waves. Introducing an appropriate variational problem, we establish the existence of their stationary profiles (boosted ground states). In order to deal with the lack of compactness, we use the technique of compactness modulo translations adapted to the fractional Sobolev spaces. In the case of algebraic (even integer-order) nonlinearities, we derive symmetries of boosted ground states with respect to the boost direction, relying on symmetric decreasing rearrangements in Fourier space. Moreover, we show a non-optimal spatial decay of these profiles at infinity.

Third and finally, we concentrate on the asymptotics of global solutions of the defocusing problem. To have a full range of Strichartz estimates available, we restrict to the radially symmetric case. We construct the wave operator on the radial subclass of the energy space, and show asymptotic completeness. Thus we infer that any radial solution scatters to a linear solution in infinite time. Similarly to the blowup theory, this is done in the spirit of monotonicity formulae: taking a suitable virial weight and using the favourable sign of the defocusing nonlinearity, we develop a lower bound for the Morawetz action. The resulting decay estimates permit us to build a satisfactory scattering theory in the radial case.

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.