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

Brahim Alouini

doi:10.3934/dcdss.2020410

Article Contents

2021, Volume 14, Issue 8: 2655-2670. Doi: 10.3934/dcdss.2020410

This issue Previous Article Non-standard boundary conditions for the linearized Korteweg-de Vries equation Next Article Lipschitz stability in determination of coefficients in a two-dimensional Boussinesq system by arbitrary boundary observation

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: January 31, 2020

Revised: March 31, 2020

Early access: July 2020

Published: August 2021

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Abstract

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.

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

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[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.
[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.
[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.
[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
[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.
[6]	H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Analysis, 4 (1980), 677-681. doi: 10.1016/0362-546X(80)90068-1.
[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.
[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.
[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.
[10]	T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10, American Mathematical Society, New York, 2003. doi: 10.1090/cln/010.
[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.
[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.
[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.
[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
[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.
[16]	V. Dinh, Well-posedness of nolinear fractional Schrödinger and wave equations in Sobolev spaces, arXiv: math/1609.06181v3.
[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.
[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), 171-185. doi: 10.1007/s00574-016-0017-5.
[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.
[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.
[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
[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.
[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.
[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.
[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.
[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.
[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.
[29]	N. Laskin, Fractional Schrödinger equation,, Phys. Rev. E, 66 (2002), 56108, 7pp. doi: 10.1103/PhysRevE.66.056108.
[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.
[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.
[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.
[33]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, Rhode Island, 2001. doi: 10.1090/gsm/014.
[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.
[35]	T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012.
[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.
[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.
[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.
[39]	E. Russ, Racine carrées d'opérateurs elliptiques et espaces de Hardy, Confluente Mathematici, 3 (2011), 1-119. doi: 10.1142/S1793744211000278.
[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.
[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.
[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.
[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.
[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.
[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.