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Remarks on the damped nonlinear Schrödinger equation
Departement of Mathematics, College of Sciences and Arts of Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia |
It is the purpose of this note to investigate the initial value problem for a focusing semi-linear damped Schrödinger equation. Indeed, in the energy sub-critical regime, one obtains global well-posedness and scattering in the energy space, depending on the order of the fractional dissipation.
References:
[1] |
G. D. Akrivis, V. A. Dougalis, O. A. Karakashian and W. R Mckinney, Numerical approximation of singular solution of the damped nonlinear Schrödinger equation, ENUMATH 97 (Heidelberg), World Scientific River Edge, NJ, (1998), 117–124. |
[2] |
I. V. Barashenkov, N. V. Alexeeva and E. V. Zemlianaya, Two and three dimensional oscillons in nonlinear Faraday resonance, Phys. Rev. Lett., 89 (2002), 104101.
doi: 10.1103/PhysRevLett.89.104101. |
[3] |
M. M. Cavalcanti, W. J. Correa, V. N. Domingos Cavalcanti and L. Tebou,
Well-posedness and energy decay estimates in the Cauchy problem for the damped defocusing Schrödinger equation, J. Differential Equations, 262 (2017), 2521-2539.
doi: 10.1016/j.jde.2016.11.002. |
[4] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and F. Natali,
Exponential stability for the 2-D defocusing Schrödinger equation with locally distributed damping, Differential Integral Equations, 22 (2009), 617-636.
|
[5] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano and F. Natali,
Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization, J. Differential Equations, 248 (2010), 2955-2971.
doi: 10.1016/j.jde.2010.03.023. |
[6] |
T. Cazenave, Semilinear Schrödinger Equations, Vol. 10, Lecture Notes in Mathematics, New York University Courant Institute of Mathematical sciences, New York, 2003.
doi: 10.1090/cln/010. |
[7] |
M. Darwich,
Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation, Annali Dell Universita Di Ferrara, 64 (2018), 323-334.
doi: 10.1007/s11565-018-0307-5. |
[8] |
T. Duyckaerts and F. Merle,
Dynamic of threshold solutions for energy-critical NlS, Geometric and Functional Analysis, 18 (2009), 1787-1840.
doi: 10.1007/s00039-009-0707-x. |
[9] |
G. Fibich,
Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705.
doi: 10.1137/S0036139999362609. |
[10] |
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. |
[11] |
C. D. Levermore and M. Oliver,
The complex Ginzburg-Landau equation as a model problem, Lectures in Appl. Math., 31 (1996), 141-189.
|
[12] |
P. L. Lions,
Symetrie et compacité dans les espaces de Sobolev, J. F. A., 49 (1982), 315-334.
doi: 10.1016/0022-1236(82)90072-6. |
[13] |
C. Morosi and L. Pizzocchero,
On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities, Expositiones Mathematicae, 36 (2018), 32-77.
doi: 10.1016/j.exmath.2017.08.007. |
[14] |
F. Natali,
Exponential stabilization for the nonlinear Schrödinger equation with localized damping, J. Dyn. Control Syst., 21 (2015), 461-474.
doi: 10.1007/s10883-015-9270-y. |
[15] |
F. Natali,
A note on the exponential decay for the nonlinear Schrödinger equation, Osaka J. Math., 53 (2016), 717-729.
|
[16] |
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. |
[17] |
T. Passot, C. Sulem and P. L. Sulem,
Linear versus nonlinear dissipation for critical NLS equation, Physica D, 203 (2005), 167-184.
doi: 10.1016/j.physd.2005.03.011. |
[18] |
T. Saanouni,
Global well-posedness of a damped Schrödinger equation in two space dimensions, Math. Meth. Appl. Sci., 37 (2014), 488-495.
doi: 10.1002/mma.2804. |
[19] |
T. Saanouni, Remarks on damped fractional Schrödinger equation with pure power nonlinearity, Journal of Mathematical Physics, 56 (2015), 061502, 14pp.
doi: 10.1063/1.4922114. |
[20] |
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. |
[21] |
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. |
show all references
References:
[1] |
G. D. Akrivis, V. A. Dougalis, O. A. Karakashian and W. R Mckinney, Numerical approximation of singular solution of the damped nonlinear Schrödinger equation, ENUMATH 97 (Heidelberg), World Scientific River Edge, NJ, (1998), 117–124. |
[2] |
I. V. Barashenkov, N. V. Alexeeva and E. V. Zemlianaya, Two and three dimensional oscillons in nonlinear Faraday resonance, Phys. Rev. Lett., 89 (2002), 104101.
doi: 10.1103/PhysRevLett.89.104101. |
[3] |
M. M. Cavalcanti, W. J. Correa, V. N. Domingos Cavalcanti and L. Tebou,
Well-posedness and energy decay estimates in the Cauchy problem for the damped defocusing Schrödinger equation, J. Differential Equations, 262 (2017), 2521-2539.
doi: 10.1016/j.jde.2016.11.002. |
[4] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and F. Natali,
Exponential stability for the 2-D defocusing Schrödinger equation with locally distributed damping, Differential Integral Equations, 22 (2009), 617-636.
|
[5] |
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano and F. Natali,
Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization, J. Differential Equations, 248 (2010), 2955-2971.
doi: 10.1016/j.jde.2010.03.023. |
[6] |
T. Cazenave, Semilinear Schrödinger Equations, Vol. 10, Lecture Notes in Mathematics, New York University Courant Institute of Mathematical sciences, New York, 2003.
doi: 10.1090/cln/010. |
[7] |
M. Darwich,
Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation, Annali Dell Universita Di Ferrara, 64 (2018), 323-334.
doi: 10.1007/s11565-018-0307-5. |
[8] |
T. Duyckaerts and F. Merle,
Dynamic of threshold solutions for energy-critical NlS, Geometric and Functional Analysis, 18 (2009), 1787-1840.
doi: 10.1007/s00039-009-0707-x. |
[9] |
G. Fibich,
Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math., 61 (2001), 1680-1705.
doi: 10.1137/S0036139999362609. |
[10] |
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. |
[11] |
C. D. Levermore and M. Oliver,
The complex Ginzburg-Landau equation as a model problem, Lectures in Appl. Math., 31 (1996), 141-189.
|
[12] |
P. L. Lions,
Symetrie et compacité dans les espaces de Sobolev, J. F. A., 49 (1982), 315-334.
doi: 10.1016/0022-1236(82)90072-6. |
[13] |
C. Morosi and L. Pizzocchero,
On the constants for some fractional Gagliardo–Nirenberg and Sobolev inequalities, Expositiones Mathematicae, 36 (2018), 32-77.
doi: 10.1016/j.exmath.2017.08.007. |
[14] |
F. Natali,
Exponential stabilization for the nonlinear Schrödinger equation with localized damping, J. Dyn. Control Syst., 21 (2015), 461-474.
doi: 10.1007/s10883-015-9270-y. |
[15] |
F. Natali,
A note on the exponential decay for the nonlinear Schrödinger equation, Osaka J. Math., 53 (2016), 717-729.
|
[16] |
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. |
[17] |
T. Passot, C. Sulem and P. L. Sulem,
Linear versus nonlinear dissipation for critical NLS equation, Physica D, 203 (2005), 167-184.
doi: 10.1016/j.physd.2005.03.011. |
[18] |
T. Saanouni,
Global well-posedness of a damped Schrödinger equation in two space dimensions, Math. Meth. Appl. Sci., 37 (2014), 488-495.
doi: 10.1002/mma.2804. |
[19] |
T. Saanouni, Remarks on damped fractional Schrödinger equation with pure power nonlinearity, Journal of Mathematical Physics, 56 (2015), 061502, 14pp.
doi: 10.1063/1.4922114. |
[20] |
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. |
[21] |
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. |
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