A study on blowup solutions to the focusing L2-supercritical nonlinear fractional Schrödinger equation

Abstract

We study the dynamical properties of blowup solutions to the focusing L2-supercritical nonlinear fractional Schrödinger equation i∂tu − (−Δ)su = −|u|αu on [0,+∞)×Rd, where d≥2,d2d−1≤s<1, 4sd<α<4sd−2s, and the initial data u(0)=u0∈Ḣsc∩Ḣs is radial with the critical Sobolev exponent sc. To this end, we establish a compactness lemma related to the equation by means of the profile decomposition for bounded sequences in Ḣsc∩Ḣs. As a result, we obtain the Ḣsc-concentration of blowup solutions with bounded Ḣsc-norm and the limiting profile of blowup solutions with critical Ḣsc-norm.

On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

“…We again refer the reader to [20] for the proof of this result. In this case, Strichartz estimates have no loss of derivatives.…”

“…We refer the reader to [20] (or [19]) for the proof of this result. Note that in the case of non-radial H s initial data, Strichartz estimates have a loss of derivatives.…”

“…The fractional differential operator (−∆) s is defined by (−∆) s u = F −1 [|ξ| 2s F(u)], where F and F −1 are the Fourier transform and inverse Fourier transform, respectively. Recently, equation (1.1) has attracted more and more attentions in both the physics and mathematics fields, see [2,3,7,9,12,13,14,15,16,19,20,21,26,23,24,25,30,32,35,47,48,49]. For the Hartree-type nonlinearity f (u) = ±(|x| −γ * |u| 2 )u, Cho et al in [8] proved existence and uniqueness of local and global solutions of (1.1).…”

“…Dynamics of blow-up solutions were studied recently by the first author in [20,21]. The sharp threshold of blow-up and scattering in the mass-supercritical and energy-subcritical case was established in [43,33].…”

Local well-posedness of semilinear space-time fractional Schrödinger equation

“…in the more recent paper [6]. Other works about fractional Schrödinger equation include [10], [19], [20]references therein. Recently, Laskin published a book [18] which gives a systematic introduction about both space and time fractional Schrödinger equation.…”