Global attractor for damped forced nonlinear logarithmic Schrödinger equations

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

Global attractor for damped forced nonlinear logarithmic Schrödinger equations

Olivier Goubet ^1,, and Ezzeddine Zahrouni ^2,

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.

Keywords: Global attractor, damped forced nonlinear logarithmic Schrodinger equations.

Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B41.

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. 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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