具有临界非线性项的薛定谔方程解的渐近行为

项目名称： 具有临界非线性项的薛定谔方程解的渐近行为

项目编号： No.11461074

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 李春花

作者单位： 延边大学

项目金额： 37万元

中文摘要： 具有临界非线性项的薛定谔方程是一种典型的非线性色散方程，在量子物理、光学和流体力学等领域中有着广泛的应用。由于临界非线性项的长程作用，使得具有临界非线性项的薛定谔方程解的性质非常复杂。基于此，具有临界非线性项的薛定谔方程解的长时间渐近行为一直是研究的热门课题。另外，在实际应用中，往往涉及具有临界非线项的薛定谔方程组问题。本项目拟研究如下问题：不满足gauge 条件的、低维具有临界非线性项的薛定谔方程的初值问题；在粒子质量共振或非共振的条件下，探讨低维具有临界非线性项的薛定谔方程组的初值和终值问题。针对上述问题我们讨论整体解的存在性、解的时间衰减估计、解的长时间渐近行为和修正波动算子的存在性。

中文关键词： 解的渐近行为；临界非线性项；质量共振；非线性薛定谔方程

英文摘要： The nonlinear Schr?dinger equation with critical nonlinearities is a kind of typical nonlinear dispersive equation, and it is studied widely in quantum physics, optics, fluid mechanics, and etc.. Since the critical nonlinearities are long range interactions, the properties of solutions to the nonlinear Schr?dinger equation with critical nonlinearities become more complex. Therefore the asymptotic behavior of solutions to nonlinear Schr?dinger equations with critical nonlinearities is a popular research topic. Moreover, the nonlinear Schr?dinger systems are always concerned in practical application. The project aims to investigate some important problems as follows. We study the initial problem of nonlinear Schr?dinger equations with critical nonlinearities which do not satisfy the gauge condition in low space dimensions. We also investigate the initial and final problem of nonlinear Schr?dinger systems with critical nonlinearities under the mass resonance condition or mass non resonance condition in low space dimensions. For all the problems mentioned above, we discuss global existence of solutions, time decay estimates of solutions, large time behavior of solutions and existence of modified wave operators.

英文关键词： Asymptotic behavior of solutions;Critical nonlinearity;Mass resonance;Nonlinear Schrodinger equation