Table of Contents
International Journal of Partial Differential Equations
Volume 2013, Article ID 748761, 10 pages
http://dx.doi.org/10.1155/2013/748761
Research Article

An Initial Boundary Value Problem for the Zakharov Equation

1Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210008, China
2Department of Mathematics, Nanjing Xiaozhuang College, Nanjing 210008, China
3Department of Mathematics, Wellesley College, Wellesley, MA 02481, USA

Received 15 February 2013; Accepted 20 June 2013

Academic Editor: Nikolai A. Kudryashov

Copyright © 2013 Quankang Yang and Charles Bu. 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. G. Boling and S. Longjun, “The global existence and uniqueness of classical solutions of periodic initial boundary value problems of Zakharov equations,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 2, pp. 310–324, 1982 (Chinese). View at Google Scholar
  2. G. Boling, “The initial boundary value problem for generalized Zakharov system,” Applied Mathematics-A Journal of Chinese Universities, vol. 9, no. 1, pp. 1–12, 1994 (Chinese). View at Google Scholar
  3. J. Colliander, “Wellposedness for Zakharov systems with generalized nonlinearity,” Journal of Differential Equations, vol. 148, no. 2, pp. 351–363, 1998. View at Google Scholar · View at Scopus
  4. T. Ozawa, K. Tsutaya, and Y. Tsutsumi, “Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions,” Mathematische Annalen, vol. 313, no. 1, pp. 127–140, 1999. View at Google Scholar · View at Scopus
  5. L. Yongsheng, “On the initial boundary value problem for two dimensional systems of Zakharov equations and of complex-Schrödinger-real-Boussinesq equations,” Journal of Partial Differential Equations, vol. 5, no. 2, pp. 81–93, 1992. View at Google Scholar
  6. C. E. Kenig, G. Ponce, and L. Vega, “On the zakharov and zakharov-schulman systems,” Journal of Functional Analysis, vol. 127, no. 1, pp. 204–234, 1995. View at Publisher · View at Google Scholar · View at Scopus
  7. G. C. Papanicolaou, C. Sulem, P. L. Sulem, and X. P. Wang, “Singular solutions of the Zakharov equations for Langmuir turbulence,” Physics of Fluids B, vol. 3, no. 4, pp. 969–980, 1991. View at Google Scholar · View at Scopus
  8. L. Glangetas and F. Merle, “Existence of self-similar blow-up solutions for Zakhrov equation in dimension two. Part I,” Communications in Mathematical Physics, vol. 160, no. 1, pp. 173–215, 1994. View at Publisher · View at Google Scholar · View at Scopus
  9. G. Boling, “Initial boundary value problem for one class of system of multidimensional nonlinear Schrödinger-Boussinesq type equation,” Journal of Mathematical Research and Exposition, vol. 8, no. 1, pp. 61–71, 1988. View at Google Scholar
  10. G. Boling and S. Longjun, “The global solution of initial value problem for nonlinear Schrödinger-Boussinesq equation in 3-Dimensions,” Acta Mathematicae Applicatae Sinica, vol. 6, no. 1, pp. 11–21, 1990. View at Google Scholar
  11. Z. Zhou, “Spectral method for Zakharov equation with periodic boundary conditions,” Acta Mathematicae Applicatae Sinica, vol. 5, no. 3, pp. 279–288, 1989. View at Publisher · View at Google Scholar · View at Scopus
  12. G. Boling, “Global smooth solutions for the system of Zakharov equations in nonhonogeneous medium,” Northeastern Mathematical Journal, vol. 6, no. 4, pp. 379–390, 1990. View at Google Scholar
  13. G. Boling and Y. Guangwei, “Global smooth solution for the Klein-Gordon-Zakharov equations,” Journal of Mathematical Physics, vol. 36, no. 8, pp. 4119–4124, 1995. View at Google Scholar · View at Scopus
  14. G. Boling, “On the initial boundary value problem for some more extensive Zakharov equations,” Journal of Mathematics, vol. 7, no. 3, pp. 269–275, 1987 (Chinese). View at Google Scholar
  15. L. Yongsheng and G. Boling, “Attractor of dissipative radially symmetric Zakharov equations outside a ball,” Acta Mathematicae Applicatae Sinica, vol. 27, pp. 803–818, 2004. View at Google Scholar
  16. G. Boling, “On global solution for a class of systems of multi-dimensional generalized Zakharov type equation,” Acta Mathematicae Applicatae Sinica, vol. 10, no. 4, pp. 419–433, 1994. View at Publisher · View at Google Scholar · View at Scopus
  17. V. Masselin, “A result on the blow-up rate for the Zakharov system in dimension 3,” SIAM Journal on Mathematical Analysis, vol. 33, no. 2, pp. 440–447, 2001. View at Google Scholar · View at Scopus
  18. P. A. Robinson, D. L. Newman, and M. V. Goldman, “Three-dimensional strong langmuir turbulence and wave collapse,” Physical Review Letters, vol. 61, no. 6, pp. 702–705, 1988. View at Publisher · View at Google Scholar · View at Scopus
  19. R. O. Dendy, Plasma Dynamics, Oxford University Press, Oxford, UK, 1990.
  20. P. M. Bellan, Fundamentals of Plasmas Physics, Cambridge University Press, Cambridge, UK, 2006.
  21. V. E. Zakharov, “The collapse of langmuir waves,” Journal of Experimental and Theoretical Physics, vol. 35, pp. 908–914, 1972. View at Google Scholar
  22. L. Nirenberg, “On elliptic partial differential equations,” Annali della Scuola Normale Superiore di Pisa, vol. 13, pp. 115–162, 1959. View at Google Scholar
  23. J. Simon, “Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure,” SIAM Journal on Mathematical Analysis, vol. 21, no. 5, pp. 1093–1117, 1990. View at Google Scholar