四维具有组合项且在能量阈值下的非线性 Schrödinger 方程

本文考虑四维空间中，在基态能量阈值下，具有散焦能量次临界扰动项的聚焦能量临界 Schrödinger 方程解的长时间动力学。这将 Miao 等人（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）的结果推广到四维且无需径向假设，其散射证明基于 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）中发展的相互作用 Morawetz 估计，其主要关键在于我们需要克服四维中双重 Duhamel 论证的对数失效。