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September  2021, 10(3): 599-617. doi: 10.3934/eect.2020082

Blow-up criteria for linearly damped nonlinear Schrödinger equations

Van Duong Dinh 1,2,,

1. 

Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France
2. 

Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

* Corresponding author: Van Duong Dinh

Received  February 2020 Revised  May 2020 Published  September 2021 Early access  July 2020

We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations
$ i\partial_t u + \Delta u + i a u = \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb R^N, $
where
$ a>0 $
 and
$ \alpha>0 $
. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up
$ H^1 $
 solutions to the focusing problem in the mass-critical and mass-supercritical cases.
Keywords: Damped nonlinear Schrödinger equation, Scattering, blow-up, localized virial estimates, radial Sobolev embedding.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q44.
Citation: Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2021, 10 (3) : 599-617. doi: 10.3934/eect.2020082
References:
[1]

G. D. Akrivis, V. A. Dougalis, O. A. Karakashian and V. R. McKinney, Numerical approximation of singular solutions of the damped nonlinear Schrödinger equation, ENUMATH, 97 (Heidelberg), World Scientific, River Edge, NJ, (1998), 117-124.   Google Scholar

[2]

M. M. Cavalcanti, W. J. Corrêa, T. Özsari, M. Sepúlveda and R. Véjar-Asem, Exponential stability for the nonlinear Schrödinger equation with locally distributed damping, Comm. Partial Differential Equations, (in press), (2020). Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.  Google Scholar

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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $ \mathbb R^3$, Annal. Math., 2008 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.  Google Scholar

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G. Chen, J. Zhang and Y. Wei, A small initial data criterion of global existence for the damped nonlinear Schrödinger equation, J. Phys. A: Math. Theor., 42 (2009), 055205. doi: 10.1088/1751-8113/42/5/055205.  Google Scholar

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M. Darwich, Blow-up for the damped $L^2$-critical nonlinear Schrödinger equation, Adv. Differential Equations, 17 (2012), 337-367.   Google Scholar

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V. D. Dinh, Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 174 (2018), 169-188.  doi: 10.1016/j.na.2018.04.024.  Google Scholar

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M. V. Goldman, K. Rypdal and B. Hafizi, Dimensionality and dissipation in Langmuir collapse, Phys. Fluids, 23 (1980), 945-955.  doi: 10.1063/1.863074.  Google Scholar

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H. Hajaiej, S. Ibrahim and N. Masmoudi, Ground state solutions of the complex Gross-Pitaevskii associated to Exciton-Polariton Bose-Einstein condensates, preprint arXiv: 1905.07660. Google Scholar

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V. K. Kalantarov and T. Özsari, Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line, J. Math. Phys., 18 (2016), 021511. doi: 10.1063/1.4941459.  Google Scholar

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C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

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F. Merle and P. Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math., 161 (2005), 157-222.  doi: 10.4007/annals.2005.161.157.  Google Scholar

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G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705.  doi: 10.1137/S0036139999362609.  Google Scholar

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G. Fibich, The nonlinear Schrödinger equations: Singular solutions and optical collapse, Applied Mathematical Sciences 192, Springer, New York, 2015. doi: 10.1007/978-3-319-12748-4.  Google Scholar

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T. Inui, Asymptotic behavior of the nonlinear damped Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 763-773.  doi: 10.1090/proc/14276.  Google Scholar

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T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

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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

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T. Özsari, V. K. Kalantarov and I. Lasiecka, Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differential Equations, 251 (2011), 1841-1863.  doi: 10.1016/j.jde.2011.04.003.  Google Scholar

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T. Özsari, Weakly-damped focusing nonlinear Schrödinger equations with Dirichlet control, J. Math. Anal. Appl., 389 (2012), 84-97.  doi: 10.1016/j.jmaa.2011.11.053.  Google Scholar

[21]

T. Özsari, Global existence and open loop exponential stabilization of weak solutions for nonlinear Schrödinger equations with localized external Neumann manipulation, Nonlinear Anal., 80 (2013), 179-193.  doi: 10.1016/j.na.2012.10.006.  Google Scholar

[22]

T. Özsari, Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities, Commun. Pure Appl. Anal., 18 (2019), 549-558.  doi: 10.3934/cpaa.2019027.  Google Scholar

[23]

V. Perez-Garcia, M. Porras and L. Vazquez, The nonlinear Schrödinger equation with dissipation and the moment method, Phys. Lett. A, 202 (1995), 176-182.  doi: 10.1016/0375-9601(95)00263-3.  Google Scholar

[24]

K. O. Rasmussen, O. Bang and P. I. Christiansen, Driving and collapse in a nonlinear Schrödinger equation, Phys. Lett. A, 184 (1994), 241-244.  doi: 10.1016/0375-9601(94)90382-4.  Google Scholar

[25]

J. Sierra, A. Kasimov, P. Markowich and R. M. Weishäupl, On the Gross-Pitaevskii equation with pumping and decay: stationary states and their stability, J. Nonlinear Sci., 25 (2015), 709-739.  doi: 10.1007/s00332-015-9239-8.  Google Scholar

[26]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[27]

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

[28]

M. Tsutsumi, On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl., 145 (1990), 328-341.  doi: 10.1016/0022-247X(90)90403-3.  Google Scholar

[29]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

show all references

References:
[1]

G. D. Akrivis, V. A. Dougalis, O. A. Karakashian and V. R. McKinney, Numerical approximation of singular solutions of the damped nonlinear Schrödinger equation, ENUMATH, 97 (Heidelberg), World Scientific, River Edge, NJ, (1998), 117-124.   Google Scholar

[2]

M. M. Cavalcanti, W. J. Corrêa, T. Özsari, M. Sepúlveda and R. Véjar-Asem, Exponential stability for the nonlinear Schrödinger equation with locally distributed damping, Comm. Partial Differential Equations, (in press), (2020). Google Scholar

[3]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.  Google Scholar

[4]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $ \mathbb R^3$, Annal. Math., 2008 (2008), 767-865.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[5]

G. Chen, J. Zhang and Y. Wei, A small initial data criterion of global existence for the damped nonlinear Schrödinger equation, J. Phys. A: Math. Theor., 42 (2009), 055205. doi: 10.1088/1751-8113/42/5/055205.  Google Scholar

[6]

M. Darwich, Blow-up for the damped $L^2$-critical nonlinear Schrödinger equation, Adv. Differential Equations, 17 (2012), 337-367.   Google Scholar

[7]

V. D. Dinh, Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal., 174 (2018), 169-188.  doi: 10.1016/j.na.2018.04.024.  Google Scholar

[8]

R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491.  Google Scholar

[9]

M. V. Goldman, K. Rypdal and B. Hafizi, Dimensionality and dissipation in Langmuir collapse, Phys. Fluids, 23 (1980), 945-955.  doi: 10.1063/1.863074.  Google Scholar

[10]

H. Hajaiej, S. Ibrahim and N. Masmoudi, Ground state solutions of the complex Gross-Pitaevskii associated to Exciton-Polariton Bose-Einstein condensates, preprint arXiv: 1905.07660. Google Scholar

[11]

V. K. Kalantarov and T. Özsari, Qualitative properties of solutions for nonlinear Schrödinger equations with nonlinear boundary conditions on the half-line, J. Math. Phys., 18 (2016), 021511. doi: 10.1063/1.4941459.  Google Scholar

[12]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy critical, focusing, nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[13]

F. Merle and P. Raphael, Blow up dynamic and upper bound on the blow up rate for critical nonlinear Schrödinger equation, Ann. Math., 161 (2005), 157-222.  doi: 10.4007/annals.2005.161.157.  Google Scholar

[14]

G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705.  doi: 10.1137/S0036139999362609.  Google Scholar

[15]

G. Fibich, The nonlinear Schrödinger equations: Singular solutions and optical collapse, Applied Mathematical Sciences 192, Springer, New York, 2015. doi: 10.1007/978-3-319-12748-4.  Google Scholar

[16]

T. Inui, Asymptotic behavior of the nonlinear damped Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 763-773.  doi: 10.1090/proc/14276.  Google Scholar

[17]

T. Ogawa and Y. Tsutsumi, Blow-up of $H^1$ solutions for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[18]

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

[19]

T. Özsari, V. K. Kalantarov and I. Lasiecka, Uniform decay rates for the energy of weakly damped defocusing semilinear Schrödinger equations with inhomogeneous Dirichlet boundary control, J. Differential Equations, 251 (2011), 1841-1863.  doi: 10.1016/j.jde.2011.04.003.  Google Scholar

[20]

T. Özsari, Weakly-damped focusing nonlinear Schrödinger equations with Dirichlet control, J. Math. Anal. Appl., 389 (2012), 84-97.  doi: 10.1016/j.jmaa.2011.11.053.  Google Scholar

[21]

T. Özsari, Global existence and open loop exponential stabilization of weak solutions for nonlinear Schrödinger equations with localized external Neumann manipulation, Nonlinear Anal., 80 (2013), 179-193.  doi: 10.1016/j.na.2012.10.006.  Google Scholar

[22]

T. Özsari, Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities, Commun. Pure Appl. Anal., 18 (2019), 549-558.  doi: 10.3934/cpaa.2019027.  Google Scholar

[23]

V. Perez-Garcia, M. Porras and L. Vazquez, The nonlinear Schrödinger equation with dissipation and the moment method, Phys. Lett. A, 202 (1995), 176-182.  doi: 10.1016/0375-9601(95)00263-3.  Google Scholar

[24]

K. O. Rasmussen, O. Bang and P. I. Christiansen, Driving and collapse in a nonlinear Schrödinger equation, Phys. Lett. A, 184 (1994), 241-244.  doi: 10.1016/0375-9601(94)90382-4.  Google Scholar

[25]

J. Sierra, A. Kasimov, P. Markowich and R. M. Weishäupl, On the Gross-Pitaevskii equation with pumping and decay: stationary states and their stability, J. Nonlinear Sci., 25 (2015), 709-739.  doi: 10.1007/s00332-015-9239-8.  Google Scholar

[26]

T. Tao, M. Visan and X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.  doi: 10.1080/03605300701588805.  Google Scholar

[27]

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

[28]

M. Tsutsumi, On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl., 145 (1990), 328-341.  doi: 10.1016/0022-247X(90)90403-3.  Google Scholar

[29]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.   Google Scholar

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