Instability of standing wave, global existence and blowup for the Klein-Gordon-Zakharov system with different-degree nonlinearities

Journal of Differential Equations

Volumn 246, Issue 10, 2009, Pages 4097-4128

  Gan, Zaihui ^a,b   Guo, Boling ^b   Zhang, Jian ^a  

^a SICHUAN NORMAL UNIVERSITY   (China)

^b INSTITUTE OF APPLIED PHYSICS AND COMPUTATIONAL MATHEMATICS   (China)

Author keywords

Blowup; Global existence; Instability; Klein Gordon Zakharov system; Standing wave

Indexed keywords

EID: 63449110898     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.03.003     Document Type: Article

References (22)

1. Bellan P.M. Fundamentals of Plasmas Physics (2006), Cambridge Univ. Press, Cambridge
2. Bergé L., Bidégaray B., and Colin T. A perturbative analysis of the time-envelope approximation in strong Langmuir turbulence. Phys. D 95 (1996) 351-379
3. Dendy R.O. Plasma Dynamics (1990), Oxford Univ. Press, Oxford
4. Gan Z., and Zhang J. Instability of standing waves for Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. J. Math. Anal. Appl. 307 (2005) 219-231
5. Gan Z. Orbital instability of standing waves for the Klein-Gordon-Zakharov system. Adv. Nonlinear Stud. 8 (2008) 413-428
6. Guo B. Global smooth solution for the system of Zakharov equations in nonhomogeneous medium. Northeast. Math. J. 6 (1990) 379-390
7. Guo B., and Yuan G. Global smooth solution for the Klein-Gordon-Zakharov equations. J. Math. Phys. 36 8 (1995) 4119-4124
8. t t = - A u + F (u). Trans. Amer. Math. Soc. 192 (1974) 1-21
9. Li J. Exact explicit travelling wave solutions for (n + 1)-dimensional Klein-Gordon-Zakharov equations. Chaos Solitons Fractals 34 (2007) 867-871
10. Masmoudi N., and Nakanishi K. From the Klein-Gordon-Zakharov system to the nonlinear Schrödinger equation. J. Hyperbolic Differ. Equ. 2 4 (2005) 975-1008
11. Masmoudi N., and Nakanishi K. Energy convergence for singular limits of Zakharov type system. Invent. Math. 172 (2008) 535-583
12. Ohta M., and Todorova G. Strong instability of standing waves for the nonlinear Klein-Gordon equation and the Klein-Gordon-Zakharov system. SIAM J. Math. Anal. 38 6 (2007) 1912-1931
13. Ozawa T., Tsutaya K., and Tsutsumi Y. Well-posedness in energy space for the Cauchy problem of the Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions. Math. Ann. 313 (1999) 127-140
14. Ozawa T., Tsutaya K., and Tsutsumi Y. Normal form and global solutions for the Klein-Gordon-Zakharov equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 4 (1995) 459-503
15. Payne L.E., and Sattinger D.H. Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math. 22 3-4 (1975) 273-303
16. Shatah J. Normal forms and quadratic nonlinear Klein-Gordon equations. Comm. Pure Appl. Math. 38 (1985) 685-696
17. Thornhill S.G., and ter Haar D. Langmuir turbulence and modulational instability. Phys. Rep. 43 (1978) 43-99
18. Tsutaya K. Global existence of small amplitude solutions for the Klein-Gordon-Zakharov equations. Nonlinear Anal. 27 12 (1996) 1373-1380
19. Strauss W. Existence of solitary waves in high-dimensions. Comm. Math. Phys. 55 (1977) 149-164
20. Zakharov V.E. Collapse of Langmuir waves. Sov. Phys. JETP 35 (1972) 908-914
21. Zhang J., and Gan Z. Sharp conditions of global existence for Klein-Gordon-Zakharov equations in three space dimensions. Adv. Math. 34 2 (2005) 241-244
22. Zhang J. On the standing wave in coupled nonlinear Klein-Gordon equations. Math. Methods Appl. Sci. 26 (2003) 11-25