• Previous Article
    Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms
  • EECT Home
  • This Issue
  • Next Article
    Approximation theorems for controllability problem governed by fractional differential equation
doi: 10.3934/eect.2020082

Blow-up criteria for linearly damped nonlinear Schrödinger equations

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  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.
Citation: Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2020082
References:
[1]

G. D. AkrivisV. A. DougalisO. 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. CollianderM. KeelG. StaffilaniH. 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. GoldmanK. 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. ÖzsariV. 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-GarciaM. 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. RasmussenO. 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. SierraA. KasimovP. 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. TaoM. 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. AkrivisV. A. DougalisO. 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. CollianderM. KeelG. StaffilaniH. 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. GoldmanK. 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. ÖzsariV. 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-GarciaM. 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. RasmussenO. 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. SierraA. KasimovP. 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. TaoM. 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

[1]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[2]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[3]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[4]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

[5]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[6]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[7]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021018

[8]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[9]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[10]

Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318

[11]

Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328

[12]

Masaru Hamano, Satoshi Masaki. A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1415-1447. doi: 10.3934/dcds.2020323

[13]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

[14]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[15]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[16]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284

[17]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[18]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316

[19]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[20]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (49)
  • HTML views (291)
  • Cited by (0)

Other articles
by authors

[Back to Top]