Opuscula Math. 41, no. 4 (2021), 465-488
https://doi.org/10.7494/OpMath.2021.41.4.465

 
Opuscula Mathematica

Remarks on damped Schrödinger equation of Choquard type

Lassaad Chergui

Abstract. This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient.

Keywords: damped Choquard equation, global existence, scattering, invariant sets.

Mathematics Subject Classification: 35Q55.

Full text (pdf)

  1. G.D. Akrivis, V.A. Dougalis, O.A. Karakashian, W.R. Mckinney, Numerical approximation of singular solution of the damped nonlinear Schrödinger equation, [in:] ENUMATH, vol. 97, World Scientific, River Edge, NJ, 1998, 117-124.
  2. I.V. Barashenkov, N.V. Alexeeva, E.V. Zemlianaya, Two- and three-dimensional oscillons in nonlinear Faraday resonance, Phys. Rev. Lett. 89 (2002), 104101.
  3. P. Bégout, J.I. Díaz, Finite time extinction for the strongly damped nonlinear Schrödinger equation in bounded domains, J. Differ. Equ. 268 (2020), no. 7, 4029-4058.
  4. C. Bonanno, P. d'Avenia, M. Ghimenti, M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl. 417 (2014), 180-199.
  5. R. Carles, P. Antonelli, C. Sparber, On nonlinear Schrödinger type equations with nonlinear damping, Int. Math. Res. Not. 3 (2013), 740-762.
  6. T. Cazenave, Semilinear Schrödinger Equations, Lecture Notes in Mathematics, Courant Institute of Mathematical Sciences, New York, 2003.
  7. J. Chen, B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica D 227 (2007), 142-148.
  8. M. Darwich, On the Cauchy problem for the nonlinear Schrödinger equation including fractional dissipation with variable coefficient, Math. Methods Appl. Sci. 41 (2018), 2930-2938.
  9. M. Darwich, L. Molinet, Some remarks on the nonlinear Schrödinger equation with fractional dissipation, J. of Math. Phys. 57 (2015), 101502.
  10. B. Feng, X. Yuan, On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory 4 (2015), no. 4, 431-445.
  11. G. Fibich, Self-focusing in the damped nonlinear Schrödinger equation, SIAM J. Appl. Math. 61 (2001), no. 5, 1680-1705.
  12. H. Genev, G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S 5 (2012), 903-923.
  13. M.V. Goldman, K. Rypdal, B. Hafizi, Dimensionality and dissipation in Langmuir collapse, Phys. Fluids 23 (1980), 945-955.
  14. E.P. Gross, E. Meeron, Physics of Many-particle Systems, vol. 1, Gordon Breach, New York, 1966, 231-406.
  15. M. Lewin, N. Rougerie, Derivation of Pekar's polarons from a microscopic model of quantum crystal, SIAM J. Math. Anal. 45 (2013), 1267-1301.
  16. E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Amer. Math. Soc., Providence RI 14, 2001.
  17. P.L. Lions, The Choquard equation and related questions, Nonlinear Anal. 4 (1980), 1063-1073.
  18. P.L. Lions, Symetrie et compacité dans les espaces de Sobolev, J. Funct. Anal. 49 (1982), 315-334.
  19. V. Moroz, J.V. Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics, J. Funct. Anal. 265 (2013), 153-184.
  20. V. Moroz, J.V. Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl. 19 (2017), 773-813.
  21. M. Ohta, G. Todorova, Remarks on global existence and blowup for damped non-linear Schrödinger equations, Discret. Contin. Dyn. Syst. 23 (2009), 1313-1325.
  22. L.E. Payne, D.H. Sattinger, Saddle points and instability of non-linear hyperbolic equations, Israel J. Math. 22 (1976), 273-303.
  23. R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. Roy. Soc. (Lond.) A 356 (1998), 1927-1939.
  24. T. Saanouni, Remarks on damped fractional Schrödinger equation with pure power nonlinearity, J. Math. Phys. 56 (2015), 061502.
  25. T. Saanouni, Damped non-linear coupled Schrödinger equation, Complex Anal. Oper. Theory 13 (2019), 1093-1110.
  26. T. Saanouni, Scattering threshold for the focusing Choquard equation, Nonlinear Differ. Equ. Appl. 26 (2019), Article no. 41.
  27. T. Saanouni, Sharp threshold of global well-posedness vs finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514.
  28. T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, vol. 106, CBMS Regional Conference Series in Mathematics, 2006.
  29. M. Tsutsumi, Nonexistence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, SIAM J. Math. Anal. 15 (1984), 357-366.
  30. M. Tsutsumi, On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations, J. Math. Anal. Appl. 145 (1990), 328-341.
  • Lassaad Chergui
  • Qassim University, Department of Mathematics, College of Science and Arts in Uglat Asugour, Kingdom of Saudia Arabia
  • University Tunis El-Manar, Department of Mathematics, Preparatory Institute for Engineering Studies, Elmanar, Campus, BP 244 CP 2092, Elmanar 2, Tunis, Tunisia
  • Communicated by J.I. Díaz.
  • Received: 2020-08-03.
  • Revised: 2021-03-08.
  • Accepted: 2021-04-01.
  • Published online: 2021-07-09.
Opuscula Mathematica - cover

Cite this article as:
Lassaad Chergui, Remarks on damped Schrödinger equation of Choquard type, Opuscula Math. 41, no. 4 (2021), 465-488, https://doi.org/10.7494/OpMath.2021.41.4.465

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise that this website uses cookies to help us understand how the site is used. All data is anonymized. Recent versions of popular browsers provide users with control over cookies, allowing them to set their preferences to accept or reject all cookies or specific ones.