Global Solutions And Standing Waves For Nonlinear Schr(?)dinger Equations With Potentials

Nonlinear dispersive equations are the present frontier topics in the field of mathematical physics and partial differential equations.Nonlinear Schrodinger equations are a class of typical nonlinear dispersive equations,whose general form is as follows:where i=(?)is an imaginary unit,ψ(t,x)is a complex-valued wave function,Δ=Σj=1N ?2/?xj2 stands for Laplace operator and f(x,|ψ|2)is a real-valued function defined on The research contents of(NLS)mainly include the following three aspects:firstly,the long time asymptotic behavior of global solutions of(NLS);secondly,the dynamic behavior of the blow-up solutions of(NLS);thirdly,the stability/instability of standing wave solutions(also called solitons)of(NLS).Among them,the existence of the global solutions,the existence of the blow-up solutions in finite time and the sharp conditions of global existence and blow-up in finite time are the key contents to study the above problems.This dissertation mainly considers that the sharp conditions of global existence and blow-up in finite time and the stability/instability of standing wave solutions of the nonlinear Schrodinger equations with three different nonlinear terms f(x,|ψ|2)ψ.In chapter 1,we briefly introduce the physical backgrounds and current research status of nonlinear Schrodinger equations at home and aboard.Especially summarize the global solutions and the relevant standing waves of three kinds of nonlinear Schrodinger equations.In chapter 2,we study the nonlinear Schrodinger equation with partial harmonic potentials,i.e.(NLS)with We mainly consider the three-dimensional nonlinear Schrodinger equation with two harmonic confined potentials and the N-dimensional nonlinear Schrodinger equation with one harmonic confined potential.By means of Hamilton conservation laws,scaling invariance and Virial identity of the equation,applying the cross-constrained variational method,the cross invariant manifolds of this equation are established and the sharp conditions of global existence and blow-up in finite time are obtained.In chapter 3,we consider the inhomogeneous nonlinear Schrodinger equation,i.e.(NLS)with f(x,|ψ|2)=|x|-b|ψ|p-1.Firstly,using the cross-constrained variational method,the Hamilton conservation laws,scaling invariance and Virial identity,the cross invariant manifolds of the equation are established,and then the sharp conditions of global existence and blow-up in finite time of this equation are obtained.Secondly,the existence of standing wave solutions of this equation is proved by the characteristics of this equation and the weakly sequential continuity of the functional ∫|x|-b|ψ|p+1 dx.By virtue of symmetric rearrangement of functions,the characteristics of standing wave solutions are analyzed.Finally,the strong instability of standing wave solutions of this equation is proved.In chapter 4,we study the nonlinear Schrodinger equation with inverse power,i.e.(NLS)with f(x,|ψ|2)=γ|x|-α+(Iβ*|ψ|p)|ψ|p-2.Firstly,according to the characteristics of the equation,the Brezis-Lieb lemma for the nonlocal terms of the functional ∫(Iβ*|ψ|p)|ψ|pdx and the norm equivalence theory,the existence of the standing wave solutions of this equation is showed.Secondly,based on the Hamilton conservation laws and Virial identity,the existence of blow-up solutions is demonstrated.Finally,we prove the strong instability of standing wave solutions of this equation.