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On finite Morse index solutions of higher order fractional elliptic equations
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 |
$ i\partial_t u + \Delta u + i a u = \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb R^N, $ |
$ a>0 $ |
$ \alpha>0 $ |
$ H^1 $ |
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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