Sharp threshold of blow-up and scattering for the fractional Hartree equation
This article presents the fractional Hartree equation \[ iu_t-(-\Delta)^su+(\frac{1}{| x|^\gamma}\ast| u|^2)u=0, \] where \(s\in(0,1)\), \(\gamma\in(2s,\min\{N,4s\})\) and \(x\in\mathbb R^N\). Hence, the authors consider the supercritical case, in the \(L^2\) sense; moreover, the equation possesses scale invariance. The results obtained are for the case of radial data. Here, a sharp blow-up/scattering dichotomy result is formulated. The fractional Laplacian present here is based on the fractional Schrödinger equation [\textit{N. Laskin}, ``Fractional Schrödinger equation'', Phys. Rev. E (3) 66, No. 5, Article ID 056108, 7 p. (2002; \url{doi:10.1103/PhysRevE.66.056108})].
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