The Sharp Condition For Blow-up And Global Existence Of The Solution Of Nonlinear Schr(o|¨)dinger Equations With Harmonic Potential

Abstract:In this paper, we shall research some types of known nonlinear Schrodinger equations with harmonic potential. Our main thought is based on the well-posedness of the Cauchy problem, We analyze the structure of the equation, set several appropriate functions, pose cross-constrained variational problem and construct so-called invariant manifolds of the evolution flow of these equation. Then we derive a sharp criterion for blow-up and global existence of the solutions. This paper contains four chapters. The main contents are as follows:In Chapter1, we put forward the physical background and the latest research on these equations. In addition, main work of this paper is also given.In Chapter2, we considered the nonlinear Schrodinger equation with harmonic potential and critical exponent. We are concerned with the relations between the global existence of the Cauchy problem and the ground state solution, which is the positive solution of the nonlinear equation-△φ+φ-|φ|4/Nφ=0. And by using scaling argument, we derive a sharp criterion for blow-up and global existence of the solution. Further-more the question that how small the initial data are, the global solutions exist is answered.In Chapter3, we discussed the inhomogeneous nonlinear Schrodinger equation with harmonic potential. By establishing a variational structure, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solution. In addition, the question that how small the initial data are, the global solutions exist is answered. What’s more, we can prove that when is1+4/N<p<p*, we can take the mass of initial data as we want such that the solution of Cauchy equation exist globally in time. In Chapter4, we studied the coupled nonlinear Schrodinger equation with harmonic potential. By establishing variational structure of the ground state solution, constructing a cross-constrained variational problem and so called invariant manifolds of the evolution flow, and applying the potential well argument and the concavity method, we derive a sharp criterion for blow-up and global existence of the solutions. And the question that how small the initial data are, the global solutions exist is answered. The thesis has sixty references.