Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35B40, 35Q55, 35B41; Secondary: 76B03, 37L30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
    [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.
    [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.
    [4] A. H. ArdilaL. Cely and M. Squassina, Logarithmic Bose-Einstein condensates with harmonic potential, Asymptotic Anal., 116 (2020), 27-40.  doi: 10.3233/ASY-191538.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [14] I. D. Chueshov, Introduction to The Theory of Infinite-Dimensional Dissipative Systems, University Lectures in Contemporary Mathematics, Vol. 19, ACTA, 2002.
    [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.
    [16] E. Di NezzaG. 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.
    [17] Z. Ding and H. Hajaiej, On a fractional Schrödinger equation in the presence of harmonic potential, preprint, arXiv: math/1908.05719.
    [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.
    [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.
    [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.
    [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.
    [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. 
    [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.
    [24] L. Grafakos and S. Oh, The Kato-Ponce inequality, Commun. Partial Differ. Equ., 39 (2014), 1128-1157.  doi: 10.1080/03605302.2013.822885.
    [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.
    [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.
    [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.
    [28] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980.  doi: 10.1353/ajm.1998.0039.
    [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.
    [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.
    [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.
    [32] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), Art. 56108. doi: 10.1103/PhysRevE.66.056108.
    [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.
    [34] Q. LiuY. ZhouJ. 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.
    [35] S. Longhi, Fractional Schrödinger equation in optics, Optics Lett., 40 (2015), 1117-1120.  doi: 10.1364/OL.40.001117.
    [36] C. Martinez and M. Sanz, The Theory of Fractional Powers of Operators, North-Holland Mathematics Studies, North Holland, Vol. 187, 2001.
    [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.
    [38] H. Pollard, The mean convergence of orthogonal series â…¡, Trans. Amer. Math. Soc., 63 (1948), 355-367.  doi: 10.2307/1990435.
    [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.
    [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.
    [41] E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confl. Math., 3 (2011), 1-119.  doi: 10.1142/S1793744211000278.
    [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.
    [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.
    [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.
    [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.
    [46] Y. ZhangH. ZhongM. BeliećN. AhmedY. Zhang and M. Xiao, Diffraction free beams in fractional Schrödinger equation, Sci. Rep., 6 (2016), 1-8.  doi: 10.1038/srep23645.
  • 加载中

Article Metrics

HTML views(135) PDF downloads(290) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint