Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data
Chenmin Sun Hua Wang Xiaohua Yao Jiqiang Zheng

The aim of this paper is to adapt the strategy in [8] [ See, B. Dodson, J. Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS. The main ingredient is to apply the fractional virial identity proved in [3] [ See, T. Boulenger, D. Himmelsbach, E. Lenzmann, Blow up for fractional NLS, J. Func. Anal, 271(2016), 2569-2603 ] to exclude the concentration of mass near the origin.

keywords: Fractional Schrödinger equation scattering Morawetz estimate
The energy-critical NLS with inverse-square potential
Rowan Killip Changxing Miao Monica Visan Junyong Zhang Jiqiang Zheng

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

keywords: Nonlinear Schröodinger equation scattering inverse-square potential concentration compactness
Scattering theory for energy-supercritical Klein-Gordon equation
Changxing Miao Jiqiang Zheng
In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{t t}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\mathbb{R}^d)\times H_x^{s_c-1}(\mathbb{R}^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.
keywords: concentration compactness. scattering theory Klein-Gordon equation Strichartz estimate Energy supercritical

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