阻尼对带逆平方势的非线性Schrdinger方程整体解的影响

Effect of Damping for Global Solution of Nonlinear Schrdinger Equation with Inverse Square Potential

本文研究一类带阻尼和逆平方势的非线性Schr¨odinger方程.旨在关心阻尼对系统整体解的影响.对于系统的次临界情形,通过引入一个特殊的变换和运用反证法,证明阻尼并不影响系统的整体解:系统的解对于任意初值都整体存在.对于系统的临界和超临界情形,分析阻尼对系统的影响,运用变分法构建一类适用于任何阻尼强度的系统整体解存在的准则.

This work is concerned with the damped nonlinear SchrSdinger equation with inverse square potential. The main goal is to address that the damping affects the global existence of the solutions. For the subcritical case, a special transform is proposed and the contradiction method is applied to obtain that the damping can not disturb the global existence： the solutions globally exist with all initial data. For the critical and supercritical cases, the variational method is used to analyze the effect of damping, and to construct a criterion of global existence which is adapted to the system with arbitrary damping intensity.