带部分调和势的非齐次非线性Schrödinger方程的爆破解  

On the Blow-Up Solutions of Inhomogeneous Nonlinear Schrödinger Equation with a Partial Confinement

作　　者：简慧 龚敏 王莉 Jian Hui;Gong Min;Wang Li(School of Science,East China Jiaotong University,Nanchang 330013)

机构地区：[1]华东交通大学理学院,南昌330013

出　　处：《数学物理学报（A辑）》2023年第5期1350-1372,共23页Acta Mathematica Scientia

基　　金：国家自然科学基金(11761032,12161038);江西省自然科学基金(20212BAB211006,20202BABL211004);江西省教育厅科技项目(GJJ212204)。

摘　　要：该文致力于研究带部分调和势的非齐次非线性Schrödinger方程的Cauchy问题.该方程是玻色-爱因斯坦凝聚中的一个重要模型.结合非线性椭圆方程基态解的变分特征及质量和能量守恒,首先得到了该问题整体解的存在性,并利用尺度变换技巧证明了该方程在一些特殊初值情形下存在爆破解.其次讨论了爆破解的L^(2)集中现象.最后利用与上述基态解相关的变分结论研究了L^(2)最小质量爆破解的动力学性质,即具有最小质量的爆破解的极限profile、精细质量集中和爆破速率.该文将Zhang^([35])的全局存在性和爆破结果推广到带非齐次非线性项的情形,并将Pan和Zhang^([24])的部分结果改进到空间维数N≥2且非线性项为非齐次的情形.This paper is devoted to the Cauchy problem of inhomogeneous nonlinear Schrödinger equation in the presence of a partial confinement,which is an important model in Bose-Einstein condensates.Combining the variational characterization of the ground state of a nonlinear elliptic equation and the conservations of mass and energy,we first obtain a global solution and show the existence of blow-up solutions for some special initial data by scaling techniques.Then,we study the L^(2)-concentration phenomenon for the blow-up solutions.Finally,we apply the variational arguments connected to the above ground state to investigate the dynamics of L^(2)-minimal blow-up solutions,i.e.,the limiting profile,mass-concentration and blow-up rate of the blow-up solutions with minimal mass.We extend the global existence and blow-up results of Zhang^([35])to the case of inhomogeneous nonlinearities and improve partial results of Pan and Zhang^([24])to space dimensions N≥2 in the inhomogeneous case.

关 键 词：非齐次非线性 Schrödinger 方程 部分调和势 爆破 质量集中 极限 profile 

分 类 号：O175.23[理学—数学]