# 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
##### References:
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##### References:
 [1] V. Barbu, M. Röckner and D. Zhang, Stochastic nonlinear Schrödinger equations, Nonlinear Anal., 136 (2016), 168-194. doi: 10.1016/j.na.2016.02.010. Google Scholar [2] V. Barbu, M. Röckner and D. Zhang, Stochastic nonlinear Schrödinger equations: No blow-up in the non-conservative case, J. Differential Equations, 263 (2017), 7919-7940. doi: 10.1016/j.jde.2017.08.030. Google Scholar [3] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, volume 46 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1999. doi: 10.1090/coll/046. Google Scholar [4] C. E. Bréhier, J. Cui, and J. Hong, Strong convergence rates of semi-discrete splitting approximations for stochastic Allen–Cahn equation, IMA J. Numer. Anal., dry052, https://doi.org/10.1093/imanum/dry052, 2018.Google Scholar [5] Z. Brzeźniak and A. Millet, On the stochastic Strichartz estimates and the stochastic nonlinear Schrödinger equation on a compact Riemannian manifold, Potential Anal., 41 (2014), 269-315. doi: 10.1007/s11118-013-9369-2. Google Scholar [6] T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010. Google Scholar [7] S. Cox, M. Hutzenthaler and A. Jentzen, Local lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations, arXiv: 1309.5595.Google Scholar [8] J. Cui and J. Hong, Analysis of a splitting scheme for damped stochastic nonlinear Schrödinger equation with multiplicative noise, SIAM J. Numer. Anal., 56 (2018), 2045-2069. doi: 10.1137/17M1154904. Google Scholar [9] J. Cui, J. Hong and Z. Liu, Strong convergence rate of finite difference approximations for stochastic cubic Schrödinger equations, J. Differential Equations, 263 (2017), 3687-3713. doi: 10.1016/j.jde.2017.05.002. Google Scholar [10] J. Cui, J. Hong, Z. Liu and W. Zhou, Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equations, J. Differential Equations, 266 (2019), 5625-5663. doi: 10.1016/j.jde.2018.10.034. Google Scholar [11] A. de Bouard and A. Debussche, A stochastic nonlinear Schrödinger equation with multiplicative noise, Comm. Math. Phys., 205 (1999), 161-181. doi: 10.1007/s002200050672. Google Scholar [12] A. de Bouard and A. Debussche, On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation, Probab. Theory Related Fields, 123 (2002), 76-96. doi: 10.1007/s004400100183. Google Scholar [13] A. de Bouard and A. Debussche, The stochastic nonlinear Schrödinger equation in H1, Stochastic Anal. Appl., 21 (2003), 97-126. doi: 10.1081/SAP-120017534. Google Scholar [14] A. de Bouard and A. Debussche, Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise, Ann. Probab., 33 (2005), 1078-1110. doi: 10.1214/009117904000000964. Google Scholar [15] F. Hornung, The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates, J. Evol. Equ., 18 (2018), 1085-1114. doi: 10.1007/s00028-018-0433-7. Google Scholar [16] M. Hutzenthaler and A. Jentzen, On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients, arXiv: 1401.0295.Google Scholar [17] M. Ohta and G. Todorova, Remarks on global existence and blowup for damped nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 23 (2009), 1313-1325. doi: 10.3934/dcds.2009.23.1313. Google Scholar [18] C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation, volume 139 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. Google Scholar [19] M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal., 15 (1984), 357-366. doi: 10.1137/0515028. Google Scholar [20] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576. Google Scholar
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