Abstract and Applied Analysis

A Time-Oscillating Hartree-Type Schrödinger Equation

Xu Chen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the time-oscillating Hartree-type Schrödinger equation i u t + Δ u + θ ω t x - γ * u 2 u = 0 , where θ is a periodic function. For the mean value I ( θ ) of θ , we show that the solution u ω converges to the solution of i U t + Δ U + I θ x - γ * U 2 U = 0 for their local well-posedness and global well-posedness.

Article information

Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 950132, 7 pages.

First available in Project Euclid: 2 October 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Chen, Xu. A Time-Oscillating Hartree-Type Schrödinger Equation. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 950132, 7 pages. doi:10.1155/2014/950132. https://projecteuclid.org/euclid.aaa/1412278110

Export citation


  • J. Fröhlich and E. Lenzmann, “Mean-field limit of quantum Bose gases and nonlinear Hartree equation,” in Séminaire: Équations aux Dérivées Partielles (2003-2004), Exp. No. XIX, p. 26, École Polytechnique, Palaiseau, France, 2004.
  • Y. B. Gaididei, K. ${\text{\O}}$. Rasmussen, and P. L. Christiansen, “Nonlinear excitations in two-dimensional molecular structures with impurities,” Physical Review E, vol. 52, no. 3, pp. 2951–2962, 1995.
  • L. P. Pitaevskii, “Vortex lines in an imperfect Bose gas,” Soviet Physics–-Journal of Experimental and Theoretical Physics, vol. 13, pp. 451–454, 1961.
  • T. Cazenave, Semilinear Schrödinger Equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York, NY, USA; American Mathematical Society, Providence, RI, USA, 2003.
  • T. Cazenave and F. B. Weissler, “The Cauchy problem for the nonlinear Schrödinger equation in ${H}^{1}$,” Manuscripta Mathematica, vol. 61, no. 4, pp. 477–494, 1988.
  • C. X. Miao, G. X. Xu, and L. F. Zhao, “Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data,” Journal of Functional Analysis, vol. 253, no. 2, pp. 605–627, 2007.
  • C. X. Miao, G. X. Xu, and L. F. Zhao, “Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case,” Colloquium Mathematicum, vol. 114, no. 2, pp. 213–236, 2009.
  • C. X. Miao, G. X. Xu, and L. F. Zhao, “The Cauchy problem of the Hartree equation,” Journal of Partial Differential Equations, vol. 21, no. 1, pp. 22–44, 2008.
  • T. Cazenave and M. Scialom, “A Schrödinger equation with time-oscillating nonlinearity,” Revista Matemática Complutense, vol. 23, no. 2, pp. 321–339, 2010.
  • D. Fang and Z. Han, “A Schrödinger equation with time-oscillating critical nonlinearity,” Nonlinear Analysis: Theory, Methods & Applications A: Theory and Methods, vol. 74, no. 14, pp. 4698–4708, 2011.
  • R. S. Strichartz, “Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations,” Duke Mathematical Journal, vol. 44, no. 3, pp. 705–714, 1977.
  • J. Ginibre and G. Velo, “The global Cauchy problem for the nonlinear Schrödinger equation revisited,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 2, no. 4, pp. 309–327, 1985.
  • K. Yajima, “Existence of solutions for Schrödinger evolution equations,” Communications in Mathematical Physics, vol. 110, no. 3, pp. 415–426, 1987.
  • M. Keel and T. Tao, “Endpoint Strichartz estimates,” The American Journal of Mathematics, vol. 120, no. 5, pp. 955–980, 1998.
  • T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 2nd edition, 1980. \endinput