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September  2020, 9(3): 721-732. doi: 10.3934/eect.2020030

Remarks on the damped nonlinear Schrödinger equation

Tarek Saanouni

Departement of Mathematics, College of Sciences and Arts of Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia

Received  June 2019 Revised  August 2019 Published  September 2020 Early access  December 2019

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.

Keywords: Nonlinear Schrödinger equation, damping, global existence, scattering.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 37K10.
Citation: Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030
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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