On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations

Archive for Rational Mechanics and Analysis

Volumn 134, Issue 3, 1996, Pages 227-248

  Komech, A ^a,b   Vainberg, B ^a,b  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030523215     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00379535     Document Type: Article

References (37)

1. H. BERESTYCKI & P.-L. LIONS, Nonlinear scalar field equations I, II, Arch. Rational Mech. Anal. 82 (1983), 313-345, 347-375.
2. H. BREZIS & E. LIEB, Minimum action solutions of some vector field equations, Comm. Math. Physics 96 (1984), 91-113.
3. t), I, II. Ann. Scuola Norm. Sup. Pisa Fis. Mat. 26 (1972), 33-65, 67-95.
4. J. GINIBRE & G. VELO, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrodinger equations, Ann. Inst. Henri Poincare 43 (1985), 399-442.
5. J. GINIBRE & G. VELO, Scattering theory in the energy space for a class of non-linear wave equations, Comm. Math. Phys. 123 (1989), 535-573.
6. R. T. GLASSEY & W. A. STRAUSS, Decay of classical Yang-Mills fields, Comm. Math. Phys. 65 (1979), 1-13.
7. R. T. GLASSEY & W. A. STRAUSS, Decay of Yang-Mills field coupled to a scalar field, Comm. Math. Phys. 67 (1979), 51-67.
8. M. GRILLAKIS, J. SHATAH, & W. A. STRAUSS, Stability theory of solitary waves in the presence of symmetry, I, II, J. Func. Anal. 74 (1987), 160-197; 94 (1990), 308-348.
9. M. GRILLAKIS, J. SHATAH, & W. A. STRAUSS, Stability theory of solitary waves in the presence of symmetry, I, II, J. Func. Anal. 74 (1987), 160-197; 94 (1990), 308-348.
10. L. HÖRMANDER, On the fully nonlinear Cauchy problem with small data, in: Microlocal analysis and nonlinear waves, IMA Vols. Math. Appl., 30, Springer, New York, 1991.
11. S. KLAINERMAN, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982), 73-98.
12. A. I. KOMECH, On the stabilization of the interaction of a string with an oscillator, Russian Math. Surv. 46 (1991), 179-180.
13. A. I. KOMECH, Stabilization of the interaction of a strong with a nonlinear oscillator, Moscow Univ. Math. Bull. 46 (1991), 34-39.
14. A. I. KOMECH, On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appls. 196 (1995), 384-409.
15. A. I. KOMECH, On stabilization of string-oscillators interaction, Russian J. Math. Phys. 3 (1995), 227-248
16. P. D. LAX, C. S. MORAWETZ & R. S. PHILLIPS, Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle, Comm. Pure Appl. Math. 16 (1963), 477-486.
17. P. D. LAX & R. S. PHILLIPS, Scattering Theory, Academic Press, New York, London, 1967.
18. P. D. LAX & R. S. PHILLIPS, Scattering theory, Rocky Mountain J. Math. 1 (1967), 173-223.
19. J. L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris, Dunod, 1969.
20. C. S. MORAWETZ, The decay of solutions to exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961), 561-568.
21. C. S. MORAWETZ, Exponential decay of solutions of the wave equation, Comm. Pure Appl. Math. 19 (1966), 439-444.
22. C. S. MORAWETZ & W. A. STRAUSS, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure and Appl. Math. 25 (1972), 1-31.
23. L. E. PAYNE & D. H. SATTINGER, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 272-303.
24. R. RACKE, Zur existenz globaler Lösungen nichtlinearer Wellengleichungen, Jahresber. Deutsch. Math.-Verein. 94 (1992), 63-97.
25. M. REED, Abstract Non-Linear Wave Equations, Springer Lecture Notes in Mathematics 507 (1976), 1-128.
26. I. SEGAL, Dispersion for nonlinear relativistic wave equations, Ann. Sci. Ecole Norm. Sup. 1 (1968), 459-497.
27. J. SHATAH, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), 313-327.
28. J. SHATAH, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985), 701-710.
29. J. SHATAH & W. A. STRAUSS, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173-190.
30. W. A. STRAUSS, Decay and asymptotics for □u = F(u), J. Funct. Anal. 2 (1968), 409-457.
31. W. A. STRAUSS, Nonlinear invariant wave equations, Springer Lecture Notes in Phys. 73 (1978), 197-249.
32. B. R. VAINBERG, On the analytical properties of the resolvent for a certain class of operator-pencils, Math. USSR Sbornik 6 (1968), 241-273.
33. B. R. VAINBERG, Behavior of the solution of the Cauchy problem for hyperbolic equations as t → ∞, Math. USSR Sbornik 7 (1969), 533-567.
34. B. R. VAINBERG, Behavior for large time of solutions of the Klein-Gordon equation, Trans. Moscow Math. Soc. 30 (1974), 139-158.
35. B. R. VAINBERG, On the short-wave asymptotics behavior of solutions of stationary problems and the asymptotic behavior as t → ∞ of solutions of nonstationary problems, Russian Math. Surveys 30 (1975), 1-58.
36. B. R. VAINBERG, Asymptotic behavior as t → ∞ of solutions of exterior mixed problems for hyperbolic equations and quasiclassics, Encyclopedia of Mathematical Sciences, VINITI-Springer Publishers, 34 (1989), 57-92 (in Russian).
37. B. R. VAINBERG, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, London, Paris, 1989.