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Motion of interfaces for a damped hyperbolic Allen–Cahn equation
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 |
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.
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. Ardila, L. 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 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. |
[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. 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. |
[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. 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. |
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. Ardila, L. 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 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. |
[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. 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. |
[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. 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. |
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