Blowup for fractional NLS

TL;DR

This paper establishes criteria for finite-time blowup of solutions to fractional nonlinear Schrödinger equations in both unbounded and bounded domains, using virial and Pohozaev estimates for fractional Laplacians.

Contribution

It introduces new blowup criteria for fractional NLS with general nonlinearities, applicable to radial and non-radial solutions in various domain settings.

Findings

Blowup criteria for radial solutions in D and higher dimensions.

General blowup results for solutions on star-shaped bounded domains.

Development of localized virial and Pohozaev estimates for fractional Laplacians.

Abstract

We consider fractional NLS with focusing power-type nonlinearity $$i \partial_t u = (-\Delta)^s u - |u|^{2 \sigma} u, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^N,$$ where $1/2< s < 1$ and $0 < \sigma < \infty$ 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 $N \geq 2$ for $L^2$-supercritical and $L^2$-critical powers $\sigma \geq 2s/N$. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain $\Omega \subset \mathbb{R}^N$ in any dimension $N \geq 1$ 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.