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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 |
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.
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. Abounouh, H. Al Moatassime, J. 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. Bao, R. Carles, C. 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. Bao, R. Carles, C. 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. Benilan, M. 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. Calgaro, O. 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. Ezzoug, O. 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. Abounouh, H. Al Moatassime, J. 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. Bao, R. Carles, C. 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. Bao, R. Carles, C. 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. Benilan, M. 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. Calgaro, O. 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. Ezzoug, O. 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. |
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