# American Institute of Mathematical Sciences

December  2019, 24(12): 6837-6854. doi: 10.3934/dcdsb.2019169

## On global existence and blow-up for damped stochastic nonlinear Schrödinger equation

 1 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 School of Mathematical Science, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author: Jianbo Cui

Received  October 2018 Published  July 2019

Fund Project: This work was supported by National Natural Science Foundation of China (No. 91630312, No. 91530118, No. 11021101 and No. 11290142).

In this paper, we consider the well-posedness of the weakly damped stochastic nonlinear Schrödinger(NLS) equation driven by multiplicative noise. First, we show the global existence of the unique solution for the damped stochastic NLS equation in critical case. Meanwhile, the exponential integrability of the solution is proved, which implies the continuous dependence on the initial data. Then, we analyze the effect of the damped term and noise on the blow-up phenomenon. By modifying the associated energy, momentum and variance identity, we deduce a sharp blow-up condition for damped stochastic NLS equation in supercritical case. Moreover, we show that when the damped effect is large enough, the damped effect can prevent the blow-up of the solution with high probability.

Citation: Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
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