On the 4D Nonlinear Schr\"odinger equation with combined terms under the energy threshold

TL;DR

This paper studies the long-term behavior of solutions to a 4D nonlinear Schrödinger equation with combined focusing and defocusing terms, extending previous results to non-radial cases and proving scattering below the energy threshold.

Contribution

It extends the analysis of 4D energy-critical NLS with combined terms to non-radial solutions and establishes scattering using interaction Morawetz estimates.

Findings

Proves scattering for solutions below the ground state energy threshold.

Extends previous results to four dimensions without radial symmetry.

Overcomes logarithmic difficulties in the double Duhamel argument.

Abstract

In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schr\"odinger equation with a defocusing energy-subcritical 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 265-298, 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 Schr\"oinger 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.