研究了一類帶調和勢的Schr?dinger方程的解，運用能量守衡定律和質量守衡定律以及利用矢量分析的知識，引入積分不等式和解微分不等式的方法，得到了初值滿足一定條件的柯西問題的解會在有限的時間里發生爆破的結論。由于所討論方程更具有一般性，從而推廣了已有的結論，所得到結論也可以對能量和質量的集中現象作進一步解釋。
關 鍵 詞 調和勢; 非線性Schr?dinger方程; 爆破; 能量守衡

Abstract In the paper, a class of Schr?dinger equation with harmonic Potential, which concerns Bose-Einstin condensates, is investigated. Now that the equation of Bose-einstin condensates describes lot of phenomena, that we research it has special significance. With the help of using energy conservative law and quality conservative law as well as knowledge of vector analysis, integral and differential inequality, we prove that the solution to the Cauchy’s problem will blow up in finite time in case initial value satisfy with some conditions. As the equation in the paper is more general, we have got extensive conclusion, by means of which we may deeply understand the aggregative phenomena on energy and quality.
Key words nonlinear Schr?dinger equation; harmonic potential; blow up; energy