Blow-up criteria for linearly damped nonlinear Schrödinger equations

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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 and 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.
[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).
[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.
[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.
[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.
[6]	M. Darwich, Blow-up for the damped $L^2$-critical nonlinear Schrödinger equation, Adv. Differential Equations, 17 (2012), 337-367.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705. doi: 10.1137/S0036139999362609.
[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.
[16]	T. Inui, Asymptotic behavior of the nonlinear damped Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 763-773. doi: 10.1090/proc/14276.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

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.
[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).
[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.
[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.
[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.
[6]	M. Darwich, Blow-up for the damped $L^2$-critical nonlinear Schrödinger equation, Adv. Differential Equations, 17 (2012), 337-367.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[14]	G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705. doi: 10.1137/S0036139999362609.
[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.
[16]	T. Inui, Asymptotic behavior of the nonlinear damped Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 763-773. doi: 10.1090/proc/14276.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[29]	Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), 115-125.

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