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

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.

Citation: Brahim Alouini. Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential. Communications on Pure and 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.

[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.

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.

[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.

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