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August  2021, 14(8): 2655-2670. doi: 10.3934/dcdss.2020410

Global attractor for a one dimensional weakly damped half-wave equation

Brahim Alouini

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

Received  February 2020 Revised  April 2020 Published  August 2021 Early access  July 2020

We discuss the asymptotic behavior of the solutions for the fractional nonlinear Schrödinger equation that reads
$ u_t-iD u+ig(|u|^2)u+\gamma u = f\, . $
We prove that this behavior is characterized by the existence of a compact global attractor in the appropriate energy space.
Keywords: Schrödinger Equation, Half-Wave Equation, Global Attractor.
Mathematics Subject Classification: Primary: 35B40, 35Q55; Secondary: 76B03, 37L30.
Citation: Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410
References:
[1]

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

[2]

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

[3]

A. Babin and M. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

[4]

Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, (2018), 985–989. arXiv: math/1810.01385 Google Scholar

[5]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: three prespectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[6]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[7]

H. Brezis and S. Wainger, A note on limiting cases of sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[8]

D. Cai, A. Majda, D. McLaughlin and E. Tabak, Spectrat bifurcation in dispersive wave turbulence, PNAS, 96 (1999), 14216-14221.  doi: 10.1073/pnas.96.25.14216.  Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems - A, 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.  Google Scholar

[12]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Communications on Pure and Applied Analysis, 10 (2011), 1121-1128.  doi: 10.3934/cpaa.2011.10.1121.  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. Not., 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, 19, ACTA, 1999. Available from: http://www.emis.de/monographs/Chueshov/book.pdf  Google Scholar

[15]

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

[16]

V. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces, arXiv: math/1609.06181v3. Google Scholar

[17]

V. Dinh, On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Communications on Pure and Applied Analysis, 18 (2019), 689-708.  doi: 10.3934/cpaa.2019034.  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), 171-185.  doi: 10.1007/s00574-016-0017-5.  Google Scholar

[20]

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

[21]

P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Analysis and PDE, 5 (2012), 1139-1155.  doi: 10.2140/apde.2012.5.1139.  Google Scholar

[22]

P. Gérard and S. Grellier, The cubic Szegö equation and Hankel operators, Société Mathématiques de France Astérisques, 389 (2017), vi+112 pp, Available from: https://hal.archives-ouvertes.fr/hal-01187657  Google Scholar

[23]

O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59-74.  doi: 10.1007/s00030-017-0482-6.  Google Scholar

[24]

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

[25]

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

[26]

N. Karachalios and N. M. Stavrakakis, Global attractor for the weakly damped driven Schrödinger equation in $H^2(\mathbb{R})$, NoDEA, 9 (2002), 347-360.  doi: 10.1007/s00030-002-8132-y.  Google Scholar

[27]

J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.  Google Scholar

[28]

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

[29]

N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002), 56108, 7pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[30]

S. Lula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations, 29 (2016), 455-492.   Google Scholar

[31]

A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.  doi: 10.1007/BF02679124.  Google Scholar

[32]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Analysis, 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[33]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.  Google Scholar

[34]

A. Ouled Elmounir and F. Simondon, Attracteurs compacts pour des problèmes d'évolutions sans unicité, Annales de la Faculté des Sciences de Toulouse, 9 (2000), 631-654.  doi: 10.5802/afst.975.  Google Scholar

[35]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.  Google Scholar

[36]

O. Pocovnicu, First and second order approximations for a nonlinear wave equation, J. Dyn. Diff. Equa., 25 (2013), 305-333.  doi: 10.1007/s10884-013-9286-5.  Google Scholar

[37]

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.  Google Scholar

[38]

J. C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.  Google Scholar

[39]

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

[40]

F. Takahashi, Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$, Advances in Nonlinear Analysis, 8 (2019), 868-884.  doi: 10.1515/anona-2017-0116.  Google Scholar

[41]

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

[42]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. doi: 10.1115/1.3424338.  Google Scholar

[43]

M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR, 275 (1984), 780-783.   Google Scholar

[44]

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

[45]

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, 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

[2]

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

[3]

A. Babin and M. Vishik, Attractors of partial differential evolution equations in an unbounded domain, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 116 (1990), 221-243.  doi: 10.1017/S0308210500031498.  Google Scholar

[4]

Y. Bahri, S. Ibrahim and H. Kikuchi, Remarks on solitary waves and Cauchy problem for a half-wave Schrödinger equations, (2018), 985–989. arXiv: math/1810.01385 Google Scholar

[5]

F. Balibrea, T. Caraballo, P. E. Kloeden and J. Valero, Recent developments in dynamical systems: three prespectives, International Journal of Bifurcation and Chaos, 20 (2010), 2591-2636.  doi: 10.1142/S0218127410027246.  Google Scholar

[6]

H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[7]

H. Brezis and S. Wainger, A note on limiting cases of sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[8]

D. Cai, A. Majda, D. McLaughlin and E. Tabak, Spectrat bifurcation in dispersive wave turbulence, PNAS, 96 (1999), 14216-14221.  doi: 10.1073/pnas.96.25.14216.  Google Scholar

[9]

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.  Google Scholar

[10]

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

[11]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems - A, 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.  Google Scholar

[12]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Communications on Pure and Applied Analysis, 10 (2011), 1121-1128.  doi: 10.3934/cpaa.2011.10.1121.  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. Not., 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, 19, ACTA, 1999. Available from: http://www.emis.de/monographs/Chueshov/book.pdf  Google Scholar

[15]

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

[16]

V. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces, arXiv: math/1609.06181v3. Google Scholar

[17]

V. Dinh, On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Communications on Pure and Applied Analysis, 18 (2019), 689-708.  doi: 10.3934/cpaa.2019034.  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), 171-185.  doi: 10.1007/s00574-016-0017-5.  Google Scholar

[20]

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

[21]

P. Gérard and S. Grellier, Effective integrable dynamics for a certain nonlinear wave equation, Analysis and PDE, 5 (2012), 1139-1155.  doi: 10.2140/apde.2012.5.1139.  Google Scholar

[22]

P. Gérard and S. Grellier, The cubic Szegö equation and Hankel operators, Société Mathématiques de France Astérisques, 389 (2017), vi+112 pp, Available from: https://hal.archives-ouvertes.fr/hal-01187657  Google Scholar

[23]

O. Goubet and E. Zahrouni, Finite dimensional global attractor for a fractional nonlinear Schrödinger equation, NoDEA, 24 (2017), 59-74.  doi: 10.1007/s00030-017-0482-6.  Google Scholar

[24]

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

[25]

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

[26]

N. Karachalios and N. M. Stavrakakis, Global attractor for the weakly damped driven Schrödinger equation in $H^2(\mathbb{R})$, NoDEA, 9 (2002), 347-360.  doi: 10.1007/s00030-002-8132-y.  Google Scholar

[27]

J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $L^2$-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.  Google Scholar

[28]

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

[29]

N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002), 56108, 7pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[30]

S. Lula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical points, Differential Integral Equations, 29 (2016), 455-492.   Google Scholar

[31]

A. Majda, D. McLaughlin and E. Tabak, A one-dimensional model for dispersive wave turbulence, J. Nonlinear Sci., 7 (1997), 9-44.  doi: 10.1007/BF02679124.  Google Scholar

[32]

V. S. Melnik and J. Valero, On attractors of multivalued semi-flows and differential inclusions, Set-Valued Analysis, 6 (1998), 83-111.  doi: 10.1023/A:1008608431399.  Google Scholar

[33]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.  Google Scholar

[34]

A. Ouled Elmounir and F. Simondon, Attracteurs compacts pour des problèmes d'évolutions sans unicité, Annales de la Faculté des Sciences de Toulouse, 9 (2000), 631-654.  doi: 10.5802/afst.975.  Google Scholar

[35]

T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.  doi: 10.1006/jfan.1995.1012.  Google Scholar

[36]

O. Pocovnicu, First and second order approximations for a nonlinear wave equation, J. Dyn. Diff. Equa., 25 (2013), 305-333.  doi: 10.1007/s10884-013-9286-5.  Google Scholar

[37]

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.  Google Scholar

[38]

J. C. Robinson, Infinite-Dimensionel Dynamical Systems, An Introduction To Dissipative Parabolic PDEs And The Theorie Of Global Attractors, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2001.  Google Scholar

[39]

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

[40]

F. Takahashi, Critical and subcritical fractional Trudinger-Moser type inequalities on $\mathbb{R}$, Advances in Nonlinear Analysis, 8 (2019), 868-884.  doi: 10.1515/anona-2017-0116.  Google Scholar

[41]

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

[42]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. doi: 10.1115/1.3424338.  Google Scholar

[43]

M. V. Vladimirov, On the solvability of mixed problem for a nonlinear equation of Schrödinger type, Dokl. Akad. Nauk SSSR, 275 (1984), 780-783.   Google Scholar

[44]

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

[45]

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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