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September  2020, 19(9): 4545-4573. doi: 10.3934/cpaa.2020206

Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential

Brahim Alouini

Laboratoire de Recherche: Analyse, Probabilité et Fractals, Université de Monastir, Faculté des Sciences, Avenue de l'environnement, 5019 Monastir, Tunisie

Received  January 2020 Revised  March 2020 Published  June 2020

We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.

Keywords: Schrödinger equation, Bose-Einstein equation, global attractor, fractal dimension.
Mathematics Subject Classification: Primary: 35B40, 35Q55, 35B41; Secondary: 76B03, 37L30.
Citation: Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4545-4573. doi: 10.3934/cpaa.2020206
References:
[1]

B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain, Commun. Pure Appl. Anal., 10 (2011), 1629-1643.  doi: 10.3934/cpaa.2011.10.1629.  Google Scholar

[2]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 14 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.  Google Scholar

[3]

B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete Contin. Dyn. Syst. B, 19 (2014), 651-677.  doi: 10.3934/dcdsb.2014.19.651.  Google Scholar

[4]

A. H. Ardila, L. Cely and M. Squassina, Logarithmic Bose-Einstein condensates with harmonic potential, Asymptotic Anal., 116 (2020), 27-40.  doi: 10.3233/ASY-191538.  Google Scholar

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal., 155 (2017), 52-64.  doi: 10.1016/j.na.2017.01.006.  Google Scholar

[6]

R. Askey and S. Wainger, Mean convergence of expensions in Laguerre and Hermite series, Amer. J. Math., 87 (1965), 695-708.  doi: 10.2307/2373069.  Google Scholar

[7]

Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, preprint, arXiv: math/1810.01385. Google Scholar

[8]

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

[9]

B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360.  doi: 10.1007/BF02829750.  Google Scholar

[10]

R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Ann. Henri Poincare, 3 (2002), 757-772.  doi: 10.1007/s00023-002-8635-4.  Google Scholar

[11]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[12]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phy., 53 (2012), Art. 043507. doi: 10.1063/1.3701574.  Google Scholar

[13]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Notices, 2018 (2018), 699-738.  doi: 10.1093/imrn/rnw246.  Google Scholar

[14]

I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, Vol. 19, ACTA, 2002.  Google Scholar

[15]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the American Mathematical Society, Amer. Math. Soc., Vol. 195, 2008. doi: 10.1090/memo/0912.  Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[17]

Z. Ding and H. Hajaiej, On a fractional Schrödinger equation in the presence of harmonic potential, preprint, arXiv: math/1908.05719. Google Scholar

[18]

E. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2017), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[19]

A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg type inequality and applications, Bull. Braz. Math. Soc., New Series, 48 (2017), 175-185.  doi: 10.1007/s00574-016-0017-5.  Google Scholar

[20]

G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[21]

P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Sci. de L'école Norm. Super., 43 (2010), 761-810.  doi: 10.24033/asens.2133.  Google Scholar

[22]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $ \mathbb{R}^2$, Adv. Differ. Equ., 3 (1998), 337-360.   Google Scholar

[23]

O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59. doi: 10.1007/s00030-017-0482-6.  Google Scholar

[24]

L. Grafakos and S. Oh, The Kato-Ponce inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[25]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differ. Equ., 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[26]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[27]

C. Huang and L. Dong, Beam propagation management in a fractional Schrödinger equation, Sci. Rep., 7 (2017), 5442. doi: 10.1038/s41598-017-05926-5.  Google Scholar

[28]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[29]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Physica D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.  Google Scholar

[30]

H. Koch and D. Tataru, $L^p$ Eigenfunction bounds for the Hermite operator, Duke Math. J., 128 (2005), 369-392.  doi: 10.1215/S0012-7094-04-12825-8.  Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), Art. 56108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[33]

P. Laurençot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $\mathbb{R}^N, \; N\leq 3$, NoDEA, 2 (1995), 357-369.  doi: 10.1007/BF01261181.  Google Scholar

[34]

Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487.  doi: 10.1016/j.amc.2005.11.024.  Google Scholar

[35]

S. Longhi, Fractional Schrödinger equation in optics, Optics Lett., 40 (2015), 1117-1120.  doi: 10.1364/OL.40.001117.  Google Scholar

[36]

C. Martinez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, North Holland, Vol. 187, 2001.  Google Scholar

[37]

F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann and J. Baumberg, Fractional quantum mechanics in polariton condensates with velocity dependent mass, Phys. Rev. B, 92 (2015), Art. 195310. doi: 10.1103/PhysRevB.92.195310.  Google Scholar

[38]

H. Pollard, The mean convergence of orthogonal series â…¡, Trans. Amer. Math. Soc., 63 (1948), 355-367.  doi: 10.2307/1990435.  Google Scholar

[39]

G. Raugel, Global Attractors in Partial Differential Equations, Handbook of dynamical systems, North-Holland, Vol. 2,885?82, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[40]

J. C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and The Theorie of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1115/1.1579456.  Google Scholar

[41]

E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confl. Math., 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.  Google Scholar

[42]

B. A. Stickler, Potential condensed-matter realisation of space-fractional quantum mechanics: the one dimensional Lévy crystal, Phys. Rev. E, 88 (2013), Art. 012120. doi: 10.1103/PhysRevE.88.012120.  Google Scholar

[43]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer applied mathmatical sciences, Vol. 68, Springer-Verlag, 2$^nd$ Edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[44]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

[45]

H. Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z., 286 (2017), 443-489.  doi: 10.1007/s00209-016-1768-9.  Google Scholar

[46]

Y. Zhang, H. Zhong, M. Belieć, N. Ahmed, Y. Zhang and M. Xiao, Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.  doi: 10.1038/srep23645.  Google Scholar

show all references

References:
[1]

B. Alouini, Long-time behavior of a Bose-Einstein equation in a two dimensional thin domain, Commun. Pure Appl. Anal., 10 (2011), 1629-1643.  doi: 10.3934/cpaa.2011.10.1629.  Google Scholar

[2]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 14 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.  Google Scholar

[3]

B. Alouini and O. Goubet, Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Discrete Contin. Dyn. Syst. B, 19 (2014), 651-677.  doi: 10.3934/dcdsb.2014.19.651.  Google Scholar

[4]

A. H. Ardila, L. Cely and M. Squassina, Logarithmic Bose-Einstein condensates with harmonic potential, Asymptotic Anal., 116 (2020), 27-40.  doi: 10.3233/ASY-191538.  Google Scholar

[5]

A. H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal., 155 (2017), 52-64.  doi: 10.1016/j.na.2017.01.006.  Google Scholar

[6]

R. Askey and S. Wainger, Mean convergence of expensions in Laguerre and Hermite series, Amer. J. Math., 87 (1965), 695-708.  doi: 10.2307/2373069.  Google Scholar

[7]

Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, preprint, arXiv: math/1810.01385. Google Scholar

[8]

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

[9]

B. Bongioanni and J. L. Torrea, Sobolev spaces associated to the harmonic oscillator, Proc. Indian. Acad. Sci. (Math. Sci.), 116 (2006), 337-360.  doi: 10.1007/BF02829750.  Google Scholar

[10]

R. Carles, Remarks on nonlinear Schrödinger equation with harmonic potential, Ann. Henri Poincare, 3 (2002), 757-772.  doi: 10.1007/s00023-002-8635-4.  Google Scholar

[11]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/010.  Google Scholar

[12]

M. Cheng, Bound state for the fractional Schrödinger equation with unbounded potential, J. Math. Phy., 53 (2012), Art. 043507. doi: 10.1063/1.3701574.  Google Scholar

[13]

A. Choffrut and O. Pocovnicu, Ill-posedness of the cubic nonlinear half-wave equation and other fractional NLS on the real line, Int. Math. Res. Notices, 2018 (2018), 699-738.  doi: 10.1093/imrn/rnw246.  Google Scholar

[14]

I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, Vol. 19, ACTA, 2002.  Google Scholar

[15]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the American Mathematical Society, Amer. Math. Soc., Vol. 195, 2008. doi: 10.1090/memo/0912.  Google Scholar

[16]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[17]

Z. Ding and H. Hajaiej, On a fractional Schrödinger equation in the presence of harmonic potential, preprint, arXiv: math/1908.05719. Google Scholar

[18]

E. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2017), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[19]

A. Esfahani and A. Pastor, Sharp constant of an anisotropic Gagliardo-Nirenberg type inequality and applications, Bull. Braz. Math. Soc., New Series, 48 (2017), 175-185.  doi: 10.1007/s00574-016-0017-5.  Google Scholar

[20]

G. B. Folland, Fourier Analysis and Its Applications, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1992.  Google Scholar

[21]

P. Gérard and S. Grellier, The cubic Szegö equation, Ann. Sci. de L'école Norm. Super., 43 (2010), 761-810.  doi: 10.24033/asens.2133.  Google Scholar

[22]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $ \mathbb{R}^2$, Adv. Differ. Equ., 3 (1998), 337-360.   Google Scholar

[23]

O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59. doi: 10.1007/s00030-017-0482-6.  Google Scholar

[24]

L. Grafakos and S. Oh, The Kato-Ponce inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.  Google Scholar

[25]

B. Guo and Z. Huo, Global well-posedness for the fractional nonlinear Schrödinger equation, Commun. Partial Differ. Equ., 36 (2011), 247-255.  doi: 10.1080/03605302.2010.503769.  Google Scholar

[26]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[27]

C. Huang and L. Dong, Beam propagation management in a fractional Schrödinger equation, Sci. Rep., 7 (2017), 5442. doi: 10.1038/s41598-017-05926-5.  Google Scholar

[28]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[29]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Physica D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.  Google Scholar

[30]

H. Koch and D. Tataru, $L^p$ Eigenfunction bounds for the Hermite operator, Duke Math. J., 128 (2005), 369-392.  doi: 10.1215/S0012-7094-04-12825-8.  Google Scholar

[31]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[32]

N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), Art. 56108. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[33]

P. Laurençot, Long-time behavior for weakly damped driven nonlinear Schrödinger equations in $\mathbb{R}^N, \; N\leq 3$, NoDEA, 2 (1995), 357-369.  doi: 10.1007/BF01261181.  Google Scholar

[34]

Q. Liu, Y. Zhou, J. Zhang and W. Zhang, Sharp condition of global existence for nonlinear Schrödinger equation with a harmonic potential, Appl. Math. Comput., 177 (2006), 482-487.  doi: 10.1016/j.amc.2005.11.024.  Google Scholar

[35]

S. Longhi, Fractional Schrödinger equation in optics, Optics Lett., 40 (2015), 1117-1120.  doi: 10.1364/OL.40.001117.  Google Scholar

[36]

C. Martinez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, North Holland, Vol. 187, 2001.  Google Scholar

[37]

F. Pinsker, W. Bao, Y. Zhang, H. Ohadi, A. Dreismann and J. Baumberg, Fractional quantum mechanics in polariton condensates with velocity dependent mass, Phys. Rev. B, 92 (2015), Art. 195310. doi: 10.1103/PhysRevB.92.195310.  Google Scholar

[38]

H. Pollard, The mean convergence of orthogonal series â…¡, Trans. Amer. Math. Soc., 63 (1948), 355-367.  doi: 10.2307/1990435.  Google Scholar

[39]

G. Raugel, Global Attractors in Partial Differential Equations, Handbook of dynamical systems, North-Holland, Vol. 2,885?82, North-Holland, Amsterdam, 2002. doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[40]

J. C. Robinson, Infinite Dimensional Dynamical Systems, An Introduction to Dissipative Parabolic PDEs and The Theorie of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001. doi: 10.1115/1.1579456.  Google Scholar

[41]

E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confl. Math., 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.  Google Scholar

[42]

B. A. Stickler, Potential condensed-matter realisation of space-fractional quantum mechanics: the one dimensional Lévy crystal, Phys. Rev. E, 88 (2013), Art. 012120. doi: 10.1103/PhysRevE.88.012120.  Google Scholar

[43]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer applied mathmatical sciences, Vol. 68, Springer-Verlag, 2$^nd$ Edition, 1997. doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[44]

X. Wang, An energy equation for the weakly damped driven nonlinear Schrödinger equations and its application to their attractors, Physica D, 88 (1995), 167-175.  doi: 10.1016/0167-2789(95)00196-B.  Google Scholar

[45]

H. Xu, Unbounded Sobolev trajectories and modified scattering theory for a wave guide nonlinear Schrödinger equation, Math. Z., 286 (2017), 443-489.  doi: 10.1007/s00209-016-1768-9.  Google Scholar

[46]

Y. Zhang, H. Zhong, M. Belieć, N. Ahmed, Y. Zhang and M. Xiao, Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.  doi: 10.1038/srep23645.  Google Scholar

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