Long time dynamics for the focusing inhomogeneous fractional Schr\"odinger equation

Abstract

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-\Delta)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and $1+\frac{2(2s-b)}{N}<p<1+\frac{2(2s-b)}{N-2s}$. We prove the ground state threshold of global existence and scattering versus finite time blow-up of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy ({Proc. Am. Math. Soc.} {145}: {4859--4867}, 2017). This method is based on Tao's scattering criteria and Morawetz estimates. One describes the threshold using some non-conserved quantities in the spirit of the recent paper by Dinh (Discr. Cont. Dyn. Syst. 40: 6441--6471, 2020). The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of the non-local fractional Laplacian operator. The second one is the presence of a singular weight in the non-linearity. The greater part of this paper is devoted to prove the scattering of global solutions in $H^s(\mathbb{R}^N)$.