Blowup for fractional NLS

On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities

“…For the local nonlinearity |u| 2p u, the well-posedness and illposedness in the Sobolev space H s have been investigated in [7,19]. In [1], Boulenger et al have obtained a general criterion for blow-up of radial solution of (1.1) with p ≥ 2s N in R N with N ≥ 2. Although a general existence theorem for blow-up solutions of this problem has remained an open problem, it has been strongly supported by numerical evidence [20].…”

“…The existence of blow-up solutions for the fractional nonlinear Schrödinger equation (1.1) with the local nonlinearity |u| 2p u has been investigated in [1]. The dynamical properties of blow-up solutions for the L 2 -critical nonlinear Schrödinger equation (1.4) have been discussed in [18].…”

Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one

“…Precisely, the paper shows that there exist (a wide class of) compactly supported potentials V , with α " ş R V pxq dx, such that the sequence of self-adjoint operators DpH ε s q :" H 2s pRq and H ε s ψ :"`p´∆q s`1 ε V`¨ε˘˘ψ converges in the norm resolvent sense to H α s . Therefore, H α s defined by (8) and (9) provides a rigorous version of the informal expression p´∆q s`δ .…”

“…Main results: nonlinear delta perturbations. Let us introduce the nonlinear analogue of (10) by posing α " αpψq " β|ψp0q| 2σ , σ ą 0, β P R, in (8), that is…”

“…where ψ (henceforth) is the solution of (16) provided by (19) and (21) (see for instance [8] and the references therein). We notice that the fractional momentum operator defined in the appendix in [20] bears some similarities with (49).…”