Nonlinear Schr?dinger Equation （NLS） With Square Potential And Inverse-square Potential

Schr?dinger equation is a basic mathematical model of quantum mechanics.According to quantum mechanics,all particles in the universe are described by wave functions.The quantum wave function is described by Schr?dinger equation.At the same time,the nonlinear Schr?dinger equation is an important mathematical model to describe various physical phenomena.Different nonlinear Schr?dinger equations have different physical backgrounds.The thesis studies the dynamic properties of the blow-up solutions of the Cauchy problem of nonlinear Schr?dinger equation with square potential and inverse-square potential.For the subcritical nonlinear index,the solutions exist globally and the standing wave solutions are orbitally stable.For the critical nonlinear index,when the initial mass is less than the critical mass,the solution of the equation exists globally and the standing wave solution is orbital stable.When the initial mass is not less than the critical mass,there exists finite time blow-up solutions.In this paper,the existence of blow-up solution with minimum mass is proved and the mass concentration property of the solution is discussed.For the supercritical nonlinear index,the sharp conditions for the solution of cauchy problem to blow up or exist globally in finite time are given by constructing invariant manifolds.