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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 273959, 11 pages
http://dx.doi.org/10.1155/2013/273959
Research Article

Final State Problem for the Dirac-Klein-Gordon Equations in Two Space Dimensions

Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560 0043, Japan

Received 25 April 2013; Accepted 15 July 2013

Academic Editor: Daniel C. Biles

Copyright © 2013 Masahiro Ikeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. N. Bournaveas, “Low regularity solutions of the Dirac Klein-Gordon equations in two space dimensions,” Communications in Partial Differential Equations, vol. 26, no. 7-8, pp. 1345–1366, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. P. D'Ancona, D. Foschi, and S. Selberg, “Local well-posedness below the charge norm for the Dirac-Klein-Gordon system in two space dimensions,” Journal of Hyperbolic Differential Equations, vol. 4, no. 2, pp. 295–330, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Grünrock and H. Pecher, “Global solutions for the Dirac-Klein-Gordon system in two space dimensions,” Communications in Partial Differential Equations, vol. 35, no. 1, pp. 89–112, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. H. Pecher, “Unconditional well-posedness for the Dirac-Klein-Gordon system in two space dimensions,” http://arxiv.org/abs/1001.3065.
  5. S. Selberg and A. Tesfahun, “Unconditional uniqueness in the charge class for the Dirac-Klein-Gordon equations in two space dimensions,” Nonlinear Differential Equations and Applications, vol. 20, no. 3, pp. 1055–1063, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. Bachelot, “Problème de Cauchy global pour des systèmes de Dirac-Klein-Gordon,” Annales de l'Institut Henri Poincaré, vol. 48, no. 4, pp. 387–422, 1988. View at Zentralblatt MATH · View at MathSciNet
  7. R. B. E. Wibowo, “Scattering problem for a system of nonlinear Klein-Gordon equations related to Dirac-Klein-Gordon equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 881–890, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. Sunagawa, “On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension,” Journal of Differential Equations, vol. 192, no. 2, pp. 308–325, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Kawahara and H. Sunagawa, “Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance,” Journal of Differential Equations, vol. 251, no. 9, pp. 2549–2567, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. N. Hayashi, M. Ikeda, and P. I. Naumkin, “Wave operator for the system of the Dirac-Klein-Gordon equations,” Mathematical Methods in the Applied Sciences, vol. 34, no. 8, pp. 896–910, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. Ikeda, A. Shimomura, and H. Sunagawa, “A remark on the algebraic normal form method applied to the Dirac-Klein-Gordon system in two space dimensions,” RIMS Kôkyûroku Bessatsu B, vol. 33, pp. 87–96, 2012.
  12. P. D'Ancona, D. Foschi, and S. Selberg, “Null structure and almost optimal local regularity for the Dirac-Klein-Gordon system,” Journal of the European Mathematical Society, vol. 9, no. 4, pp. 877–899, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. N. Hayashi and P. I. Naumkin, “Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 9, pp. 3826–3833, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. B. Marshall, W. Strauss, and S. Wainger, “LpLq estimates for the Klein-Gordon equation,” Journal de Mathématiques Pures et Appliquées, vol. 59, no. 4, pp. 417–440, 1980. View at Zentralblatt MATH · View at MathSciNet
  15. K. Yajima, “Existence of solutions for Schrödinger evolution equations,” Communications in Mathematical Physics, vol. 110, no. 3, pp. 415–426, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Tsutsumi, “Global solutions for the Dirac-Proca equations with small initial data in 3+1 space time dimensions,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 485–499, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Klainerman, “The null condition and global existence to nonlinear wave equations,” in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, vol. 23 of Lectures in Applied Mathematics, pp. 293–326, American Mathematical Society, Providence, RI, USA, 1986. View at Zentralblatt MATH · View at MathSciNet
  19. S. Katayama, “A note on global existence of solutions to nonlinear Klein-Gordon equations in one space dimension,” Journal of Mathematics of Kyoto University, vol. 39, no. 2, pp. 203–213, 1999. View at Zentralblatt MATH · View at MathSciNet
  20. R. Kosecki, “The unit condition and global existence for a class of nonlinear Klein-Gordon equations,” Journal of Differential Equations, vol. 100, no. 2, pp. 257–268, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. Y. Tsutsumi, “Stability of constant equilibrium for the Maxwell-Higgs equations,” Funkcialaj Ekvacioj, vol. 46, no. 1, pp. 41–62, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet