# 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  December 2019 Early access  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 and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.

show all references

##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [20] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
 [1] Jinmyong An, Roesong Jang, Jinmyong Kim. Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022111 [2] Zaihui Gan, Jian Zhang. Blow-up, global existence and standing waves for the magnetic nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 827-846. doi: 10.3934/dcds.2012.32.827 [3] Dapeng Du, Yifei Wu, Kaijun Zhang. On blow-up criterion for the nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3639-3650. doi: 10.3934/dcds.2016.36.3639 [4] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264 [5] Xiaobing Feng, Shu Ma. Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 687-711. doi: 10.3934/dcdss.2021071 [6] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control and Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [7] Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034 [8] Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259 [9] Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085 [10] Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure and Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 [11] Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations and Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082 [12] Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022106 [13] Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 [14] Yajing Li, Yejuan Wang. The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2665-2697. doi: 10.3934/dcdsb.2020027 [15] Nguyen Thanh Long, Hoang Hai Ha, Le Thi Phuong Ngoc, Nguyen Anh Triet. Existence, blow-up and exponential decay estimates for a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (1) : 455-492. doi: 10.3934/cpaa.2020023 [16] Wenjun Liu, Jiangyong Yu, Gang Li. Global existence, exponential decay and blow-up of solutions for a class of fractional pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4337-4366. doi: 10.3934/dcdss.2021121 [17] Cristophe Besse, Rémi Carles, Norbert J. Mauser, Hans Peter Stimming. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence. Discrete and Continuous Dynamical Systems - B, 2008, 9 (1) : 11-36. doi: 10.3934/dcdsb.2008.9.11 [18] Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 [19] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [20] Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic and Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043

2021 Impact Factor: 1.497