## Abstract and Applied Analysis

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

Masahiro Ikeda

#### Abstract

We study the final state problem for the Dirac-Klein-Gordon equations (DKG) in two space dimensions. We prove that if the nonresonance mass condition is satisfied, then the wave operator for DKG is well defined from a neighborhood at the origin in lower order weighted Sobolev space to some Sobolev space.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 273959, 11 pages.

Dates
First available in Project Euclid: 26 February 2014

https://projecteuclid.org/euclid.aaa/1393449754

Digital Object Identifier
doi:10.1155/2013/273959

Mathematical Reviews number (MathSciNet)
MR3091223

Zentralblatt MATH identifier
1294.35114

#### Citation

Ikeda, Masahiro. Final State Problem for the Dirac-Klein-Gordon Equations in Two Space Dimensions. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 273959, 11 pages. doi:10.1155/2013/273959. https://projecteuclid.org/euclid.aaa/1393449754

#### References

• 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.
• 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.
• 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.
• H. Pecher, “Unconditional well-posedness for the Dirac-Klein-Gordon system in two space dimensions,” http://arxiv.org/abs/ 1001.3065.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• B. Marshall, W. Strauss, and S. Wainger, “${L}^{p}-{L}^{q}$ estimates for the Klein-Gordon equation,” Journal de Mathématiques Pures et Appliquées, vol. 59, no. 4, pp. 417–440, 1980.
• K. Yajima, “Existence of solutions for Schrödinger evolution equations,” Communications in Mathematical Physics, vol. 110, no. 3, pp. 415–426, 1987.
• 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.
• 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.
• 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.
• 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.
• 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.
• Y. Tsutsumi, “Stability of constant equilibrium for the Maxwell-Higgs equations,” Funkcialaj Ekvacioj, vol. 46, no. 1, pp. 41–62, 2003.