薛定谔(Schrodinger)方程是非相对论量子力学的基本方程，具有深刻的物理背景。Hartree方程是一类非局部的非线性Schrodinger方程，它起源于量子场论中对多粒子系统的研究，如激光束，Bose-Einstein凝聚等等。本项目我们将集中于研究解的爆破理论以及轨道稳定性理论，主要研究目标包括：（1）研究质量临界非齐次Hartree方程的极小爆破解的性质，证明当非齐次项存在非退化最大值点时极小爆破解的存在性和唯一性，并给出极小爆破解存在的充分必要条件；（2）研究半经典非齐次Hartree方程的驻波解的性质，讨论当非齐次项满足一定条件时驻波解的轨道稳定性和不稳定性。我们将分别利用Modulation方法以及Lyapunov-Schmidt约化方法来研究爆破解和驻波解的性质。本项目拟研究的问题是偏微分方程的热点问题，将促使我们更好地理解非线性Schrodinger方程的动力学性质。

Schrodinger equation is the fundamental equation of the non-relativistic quantum mechanics, while the Hartree equation is one of nonlocal nonlinear Schrodinger equations. It arises in the study of many-body systems in quantum field theory, such as the laser beams and the Bose-Einstein condensation. In this project, we will focus on the blow-up theory and orbital stability theory of inhomogeneous Hartree equations. There are two main targets: (1) to prove the existence and uniqueness of the minimal blow-up solutions of inhomogeneous mass-critical Hartree equations, and to depict the sharp condition for the existence of the minimal blow-up solutions; (2) to study the orbital stability of the standing-wave solutions of semi-classical inhomogeneous mass-critical Hartree equations. We will apply the Modulation method and Lyapunov-Schmidt reduction method to study the properties of blow-up solutions and standing-wave solutions respectively. The project we propose is fundamental and important in the theory of partial differential equation, and will help us to understand the dynamics of nonlinear Schrodinger equations better.