On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold

Abstract

Abstract. In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energysubcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767-808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 2015) to four dimension without radial assumption and the proof of scattering is based on the interaction Morawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schröinger problem in dimension d = 4 for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.

Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations

“…These kind of equations have been studied intensively in the past decade. In [1,2,11,30,39,40,38,49,54,55,56], the authors study the well-posedness and scattering of these kind of equations.…”

“…We also refer to [55] for further discussion about 1 + 4 d < p 1 < p 2 = 1 + 4 d−2 , d = 3. The radial assumption was removed in dimensions four and higher in [39,40] by using [18,32]. For the case 1 + 4 d = p 1 < p 2 ≤ 1 + 4 d−2 , 1 ≤ d ≤ 4, X. Cheng, C. Miao, and L. Zhao [11] proved global wellposedness and scattering in H 1 in the radial case where they gave the exact value of the threshold which is related to the ground state.…”

Energy thresholds of blow‐up for the Hartree equation with a focusing subcritical perturbation

“…in Ref. 10 to the energy threshold case. Feng 11,12 investigated the sharp threshold mass of blow‐up solutions for the focusing nonlinear Schrödinger equation with a defocusing subcritical perturbation and some dynamical properties of blow‐up solutions in the L 2 supercritical case.…”

A new proof of scattering theory for the 3D radial focusing energy-critical NLS with combined terms

“…Next, we recall the scattering and blow-up result of (1.1), which established by Miao-Xu-Zhao in [10]. We define some quantities and some variation results(refers to [12], [10], [11] for details). For ϕ ∈ H 1 , let…”