TY - JOUR
T1 - On global existence and blow-up for damped stochastic nonlinear Schrödinger equation
AU - Cui, Jianbo
AU - Hong, Jialin
AU - Sun, Liying
N1 - Funding Information:
2010 Mathematics Subject Classification. Primary: 60H15; Secondary: 35Q55, 35R60. Key words and phrases. Stochastic nonlinear Schrödinger equation, multiplicative noise, global existence, blow-up, exponential integrability. This work was supported by National Natural Science Foundation of China (No. 91630312, No. 91530118, No. 11021101 and No. 11290142). ∗ Corresponding author: Jianbo Cui.
Publisher Copyright:
© 2019 American Institute of Mathematical Sciences. All rights reserved.
PY - 2019/12
Y1 - 2019/12
N2 - 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.
AB - 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.
KW - Blow-up
KW - Exponential integrability
KW - Global existence
KW - Multiplicative noise
KW - Stochastic nonlinear Schrödinger equation
UR - http://www.scopus.com/inward/record.url?scp=85072571507&partnerID=8YFLogxK
U2 - 10.3934/dcdsb.2019169
DO - 10.3934/dcdsb.2019169
M3 - Journal article
AN - SCOPUS:85072571507
SN - 1531-3492
VL - 24
SP - 6837
EP - 6854
JO - Discrete and Continuous Dynamical Systems - Series B
JF - Discrete and Continuous Dynamical Systems - Series B
IS - 12
ER -