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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  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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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