August  2021, 14(8): 2933-2946. doi: 10.3934/dcdss.2020393

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

1. 

LAMFA, UMR 7352 CNRS UPJV, 33 rue St Leu, 80039, Amiens Cedex, France

2. 

Equipe de recherche Analyse, Probabilités et Fractals, Av. de l'environnement, 5000 Monastir, Tunisie

* Corresponding author: Olivier Goubet

Received  October 2019 Revised  February 2020 Published  August 2021 Early access  June 2020

Fund Project: This article is dedicated to the memory of Ezzeddine Zahrouni who passed away in december 2018.

We consider here a damped forced nonlinear logarithmic Schrödinger equation in $ \mathbb{R}^N $. We prove the existence of a global attractor in a suitable energy space. We complete this article with some open issues for nonlinear logarithmic Schrödinger equations in the framework of infinite-dimensional dynamical systems.

Citation: Olivier Goubet, Ezzeddine Zahrouni. Global attractor for damped forced nonlinear logarithmic Schrödinger equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2933-2946. doi: 10.3934/dcdss.2020393
References:
[1]

M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32.  doi: 10.1016/0893-9659(93)90073-V.

[2]

M. AbounouhH. Al MoatassimeJ. P. Chehab and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems, Commun. Pure Appl. Anal., 7 (2008), 211-127.  doi: 10.3934/cpaa.2008.7.211.

[3]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.

[4]

A. H. Ardila, Logarithmic Schrödinger equation: On the orbital stability of the Gausson, Journal of Differential Equations, 335 (2016), 1-9. 

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis, 155 (2017), 52-64.  doi: 10.3934/eect.2017009.

[6]

A. H. Ardila, Stability of ground states for logarithmic Schrödinger equation with a $\delta'$-interaction, Evol. Equ. Control Theory, 6 (2017), 155-175.  doi: 10.3934/eect.2017009.

[7]

A. H. Ardila and M. Squassina, Gausson dynamic for logarithmic Schrödinger equations, Asymptot. Anal., 107 (2018), 203-226.  doi: 10.3233/ASY-171458.

[8]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348. Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[9]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[10]

W. Z. BaoR. CarlesC. M. Su and Q. L. Tang, Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.

[11]

W. Z. BaoR. CarlesC. M. Su and Q. L. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.

[12]

W. Z. Bao and D. Jacksch, An explicit unconditionaly stable numerical method for solving damped nonlinear Schrödinger equation with focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), 1406-1426.  doi: 10.1137/S0036142902413391.

[13]

P. BenilanM. G. Crandall and A. Pazy, "Bonnes solutions" d'un problème d'évolution semi-linéaire, C. R. Acad. Sci. Paris sér I Math., 306 (1988), 527-530. 

[14]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations, SIAM J. Num. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.

[15]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.

[16]

H. R. Brezis, Les opérateurs monotones, Secrétariat Mathématique, Paris, Séminaire Choquet: 1965/66, Initiation à l' Analyse, Fasc. 2, Exp. 10, (1968), 33 pp.

[17]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[18]

C. CalgaroO. Goubet and E. Zahrouni, Finite-dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation, Math. Methods Appl. Sci., 40 (2017), 5563-5574.  doi: 10.1002/mma.4409.

[19]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.

[20]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[21]

T. Cazenave and A. Haraux Équation d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 21–51. doi: 10.5802/afst.543.

[22]

P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp. doi: 10.1142/S0219199713500326.

[23]

E. EzzougO. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schrödinger equation, Differential Integral Equations, 23 (2010), 237-252. 

[24]

G. Ferriere, The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition, arXiv: 1910.09436v1.

[25]

C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), 509-538. 

[26]

P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004.

[27]

J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.

[28]

J.-M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schr${{\rm{\ddot d}}}$inger equations. Attractors, inertial manifolds and their approximation, RAIRO Modél. Math. Anal. Numér., 23 (1989), 433-443.  doi: 10.1051/m2an/1989230304331.

[29]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.  doi: 10.1080/00036819608840420.

[30]

O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schrödinger equations in $L^2(\mathbb{R})$, Nonlinear Anal., 71 (2009), 317-320.  doi: 10.1016/j.na.2008.10.078.

[31]

O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 7 (2008), 1429-1442.  doi: 10.3934/cpaa.2008.7.1429.

[32]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[33]

K. N. Lu and B. X. Wang, Global attractor for the Klein-Gordon-Schrödinger equations in unbounded domains, Journal of Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.

[34]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103-200.  doi: 10.1016/S1874-5717(08)00003-0.

[35]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885-982.  doi: 10.1016/S1874-575X(02)80038-8.

[36]

J. M. Sanz-Serna and J. G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. of Num. Anal., 6 (1986), 25-42.  doi: 10.1093/imanum/6.1.25.

[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.

show all references

References:
[1]

M. Abounouh, Asymptotic behaviour for a weakly damped Schrödinger equation in dimension two, Appl. Math. Lett., 6 (1993), 29-32.  doi: 10.1016/0893-9659(93)90073-V.

[2]

M. AbounouhH. Al MoatassimeJ. P. Chehab and O. Goubet, Discrete Schrödinger equations and dissipative dynamical systems, Commun. Pure Appl. Anal., 7 (2008), 211-127.  doi: 10.3934/cpaa.2008.7.211.

[3]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.

[4]

A. H. Ardila, Logarithmic Schrödinger equation: On the orbital stability of the Gausson, Journal of Differential Equations, 335 (2016), 1-9. 

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Analysis, 155 (2017), 52-64.  doi: 10.3934/eect.2017009.

[6]

A. H. Ardila, Stability of ground states for logarithmic Schrödinger equation with a $\delta'$-interaction, Evol. Equ. Control Theory, 6 (2017), 155-175.  doi: 10.3934/eect.2017009.

[7]

A. H. Ardila and M. Squassina, Gausson dynamic for logarithmic Schrödinger equations, Asymptot. Anal., 107 (2018), 203-226.  doi: 10.3233/ASY-171458.

[8]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, 348. Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[9]

J. M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52.  doi: 10.3934/dcds.2004.10.31.

[10]

W. Z. BaoR. CarlesC. M. Su and Q. L. Tang, Error estimates of a regularized finite difference method for the logarithmic Schrödinger equation, SIAM J. Numer. Anal., 57 (2019), 657-680.  doi: 10.1137/18M1177445.

[11]

W. Z. BaoR. CarlesC. M. Su and Q. L. Tang, Regularized numerical methods for the logarithmic Schrödinger equation, Numer. Math., 143 (2019), 461-487.  doi: 10.1007/s00211-019-01058-2.

[12]

W. Z. Bao and D. Jacksch, An explicit unconditionaly stable numerical method for solving damped nonlinear Schrödinger equation with focusing nonlinearity, SIAM J. Numer. Anal., 41 (2003), 1406-1426.  doi: 10.1137/S0036142902413391.

[13]

P. BenilanM. G. Crandall and A. Pazy, "Bonnes solutions" d'un problème d'évolution semi-linéaire, C. R. Acad. Sci. Paris sér I Math., 306 (1988), 527-530. 

[14]

C. Besse, A relaxation scheme for nonlinear Schrödinger equations, SIAM J. Num. Anal., 42 (2004), 934-952.  doi: 10.1137/S0036142901396521.

[15]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.

[16]

H. R. Brezis, Les opérateurs monotones, Secrétariat Mathématique, Paris, Séminaire Choquet: 1965/66, Initiation à l' Analyse, Fasc. 2, Exp. 10, (1968), 33 pp.

[17]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic Schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.

[18]

C. CalgaroO. Goubet and E. Zahrouni, Finite-dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation, Math. Methods Appl. Sci., 40 (2017), 5563-5574.  doi: 10.1002/mma.4409.

[19]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.

[20]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.

[21]

T. Cazenave and A. Haraux Équation d'évolution avec non linéarité logarithmique, Ann. Fac. Sci. Toulouse Math. (5), 2 (1980), 21–51. doi: 10.5802/afst.543.

[22]

P. d'Avenia, E. Montefusco and M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 1350032, 15 pp. doi: 10.1142/S0219199713500326.

[23]

E. EzzougO. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schrödinger equation, Differential Integral Equations, 23 (2010), 237-252. 

[24]

G. Ferriere, The focusing logarithmic Schrödinger equation: Analysis of breathers and nonlinear superposition, arXiv: 1910.09436v1.

[25]

C. Gallo, Schrödinger group on Zhidkov spaces, Adv. Differential Equations, 9 (2004), 509-538. 

[26]

P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 765-779.  doi: 10.1016/j.anihpc.2005.09.004.

[27]

J.-M. Ghidaglia, Finite dimensional behavior for the weakly damped driven Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 365-405.  doi: 10.1016/S0294-1449(16)30343-2.

[28]

J.-M. Ghidaglia, Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schr${{\rm{\ddot d}}}$inger equations. Attractors, inertial manifolds and their approximation, RAIRO Modél. Math. Anal. Numér., 23 (1989), 433-443.  doi: 10.1051/m2an/1989230304331.

[29]

O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119.  doi: 10.1080/00036819608840420.

[30]

O. Goubet and L. Molinet, Global attractor for weakly damped nonlinear Schrödinger equations in $L^2(\mathbb{R})$, Nonlinear Anal., 71 (2009), 317-320.  doi: 10.1016/j.na.2008.10.078.

[31]

O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Commun. Pure Appl. Anal., 7 (2008), 1429-1442.  doi: 10.3934/cpaa.2008.7.1429.

[32]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988.

[33]

K. N. Lu and B. X. Wang, Global attractor for the Klein-Gordon-Schrödinger equations in unbounded domains, Journal of Differential Equations, 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.

[34]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, Handbook of Differential Equations: Evolutionary Equations, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 4 (2008), 103-200.  doi: 10.1016/S1874-5717(08)00003-0.

[35]

G. Raugel, Global attractors in partial differential equations, Handbook of Dynamical Systems, North-Holland, Amsterdam, 2 (2002), 885-982.  doi: 10.1016/S1874-575X(02)80038-8.

[36]

J. M. Sanz-Serna and J. G. Verwer, Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation, IMA J. of Num. Anal., 6 (1986), 25-42.  doi: 10.1093/imanum/6.1.25.

[37]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[38]

K. Tsugawa, Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index, Commun. Pure Appl. Anal., 3 (2004), 301-318.  doi: 10.3934/cpaa.2004.3.301.

[1]

Boling Guo, Zhaohui Huo. The global attractor of the damped, forced generalized Korteweg de Vries-Benjamin-Ono equation in $L^2$. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 121-136. doi: 10.3934/dcds.2006.16.121

[2]

Kotaro Tsugawa. Existence of the global attractor for weakly damped, forced KdV equation on Sobolev spaces of negative index. Communications on Pure and Applied Analysis, 2004, 3 (2) : 301-318. doi: 10.3934/cpaa.2004.3.301

[3]

Ming Wang. Global attractor for weakly damped gKdV equations in higher sobolev spaces. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3799-3825. doi: 10.3934/dcds.2015.35.3799

[4]

Marilena N. Poulou, Nikolaos M. Stavrakakis. Global attractor for a Klein-Gordon-Schrodinger type system. Conference Publications, 2007, 2007 (Special) : 844-854. doi: 10.3934/proc.2007.2007.844

[5]

Takahiro Hashimoto. Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains. Conference Publications, 2007, 2007 (Special) : 487-494. doi: 10.3934/proc.2007.2007.487

[6]

Yi Cheng, Ying Chu. A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms. Electronic Research Archive, 2021, 29 (6) : 3867-3887. doi: 10.3934/era.2021066

[7]

Zhiming Liu, Zhijian Yang. Global attractor of multi-valued operators with applications to a strongly damped nonlinear wave equation without uniqueness. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 223-240. doi: 10.3934/dcdsb.2019179

[8]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345

[9]

Antonio Segatti. Global attractor for a class of doubly nonlinear abstract evolution equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 801-820. doi: 10.3934/dcds.2006.14.801

[10]

Francesca Bucci, Igor Chueshov, Irena Lasiecka. Global attractor for a composite system of nonlinear wave and plate equations. Communications on Pure and Applied Analysis, 2007, 6 (1) : 113-140. doi: 10.3934/cpaa.2007.6.113

[11]

Hiroshi Takeda. Global existence of solutions for higher order nonlinear damped wave equations. Conference Publications, 2011, 2011 (Special) : 1358-1367. doi: 10.3934/proc.2011.2011.1358

[12]

Masahoto Ohta, Grozdena Todorova. Remarks on global existence and blowup for damped nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1313-1325. doi: 10.3934/dcds.2009.23.1313

[13]

Sandra Lucente. Global existence for equivalent nonlinear special scale invariant damped wave equations. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021159

[14]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations and Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645

[15]

Olivier Goubet, Ezzeddine Zahrouni. On a time discretization of a weakly damped forced nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1429-1442. doi: 10.3934/cpaa.2008.7.1429

[16]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824

[17]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094

[18]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410

[19]

Brahim Alouini. Finite dimensional global attractor for a class of two-coupled nonlinear fractional Schrödinger equations. Evolution Equations and Control Theory, 2022, 11 (2) : 559-581. doi: 10.3934/eect.2021013

[20]

Varga K. Kalantarov, Edriss S. Titi. Global stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1325-1345. doi: 10.3934/dcdsb.2018153

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (270)
  • HTML views (447)
  • Cited by (0)

Other articles
by authors

[Back to Top]