带奇异势的非线性Schrödinger方程解的渐近行为研究

本项目拟研究偶极玻色-爱因斯坦凝聚方程和玻色星体方程及其相关带奇异势的非线性Schrödinger方程，其爆破机制、基态解的动力学行为等方面与经典的不带奇异势非线性Schrödinger方程有显著差异。利用Profile分解理论、集中紧-刚性引理等深入挖掘关联孤立子的变分特征，尤其是由奇异势导出的新变分结构。研究方程整体解的渐近行为，包括：解整体存在的最佳门槛质量及其大初值散射性质、门槛质量附近驻波的渐近动力学行为等。研究方程爆破解的渐近行为，包括在临界-次临界非线性幂下方程爆破解的存在性、爆破速率、爆破点的极限图景等。最后，对以上性质进行数值模拟与计算，力争对带奇异势的非线性Schrödinger方程解的渐近行为有全景刻画。

In this project, we study models for the dipolar Bose-Einstein condensations and the Boson star, and the related nonlinear Schrödinger equations with singular potentials, whose blow-up mechanics, dynamics of ground states are quite different from that for the nonlinear Schrödinger equation without any singular potential. Firstly, we apply the modern analysis tools including the profile decomposition theory and the compactness-rigidity method etc, to explore variational characteristics for related solitons. Particularly, we aim to find out new variational characteristics generated from the singular potentials. Secondly, we investigate asymptotic behavior of global solutions for the above two nonlinear Schrödinger equations with singular potentials, including sharp threshold mass for global solutions, scattering of global solutions with big initial data and asymptotic dynamical properties of standing waves near the sharp threshold mass etc. Thirdly, we investigate asymptotic behavior of blow-up solutions for the above nonlinear Schrödinger equations with singular potentials, including existence of blow-up solutions under the critical and subcritical nonlinearities, blow-up rate, limiting profile of blow-up points etc. Finally, we consider numerical simulations and computations for the above dynamical properties, and pursue a panorama of the asymptotic behavior for the nonlinear Schrödinger equations with singular potentials.