Abstract
We consider nonlinear wave and Klein-Gordon equations with general nonlinear terms, localized in space. Conditions are found which provide asymptotic stability of stationary solutions in local energy norms. These conditions are formulated in terms of spectral properties of the Schrödinger operator corresponding to the linearized problem. They are natural extensions to partial differential equations of the known Lyapunov condition. For the nonlinear wave equation in three-dimensional space we find asymptotic expansions, as t→∞, of the solutions which are close enough to a stationary asymptotically stable solution.
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References
H. Berestycki & P.-L. Lions, Nonlinear scalar field equations I, II, Arch. Rational Mech. Anal. 82 (1983), 313–345, 347–375.
H. Brezis & E. Lieb, Minimum action solutions of some vector field equations, Comm. Math. Physics 96 (1984), 91–113.
J. M. Chadam, Asymptotics for \(\square u\) = m 2 u + G(x, t, u, u x , u t ), I, II. Ann. Scuola Norm. Sup. Pisa Fis. Mat. 26 (1972), 33–65, 67–95.
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.
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.
R. T. Glassey & W. A. Strauss, Decay of classical Yang-Mills fields, Comm. Math. Phys. 65 (1979), 1–13.
R. T. Glassey & W. A. Strauss, Decay of Yang-Mills field coupled to a scalar field, Comm. Math. Phys. 67 (1979), 51–67.
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.
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.
S. Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982), 73–98.
A. I. Komech, On the stabilization of the interaction of a string with an oscillator, Russian Math. Surv. 46 (1991), 179–180.
A. I. Komech, Stabilization of the interaction of a string with a nonlinear oscillator, Moscow Univ. Math. Bull. 46 (1991), 34–39.
A. I. Komech, On stabilization of string-nonlinear oscillator interaction, J. Math. Anal. Appls. 196 (1995), 384–409.
A. I. Komech, On stabilization of string-oscillators interaction, Russian J. Math. Phys. 3 (1995), 227–248
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.
P. D. Lax & R. S. Phillips, Scattering Theory, Academic Press, New York, London, 1967.
P. D. Lax & R. S. Phillips, Scattering theory, Rocky Mountain J. Math. 1 (1967), 173–223.
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris, Dunod, 1969.
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.
C. S. Morawetz, Exponential decay of solutions of the wave equation, Comm. Pure Appl. Math. 19 (1966), 439–444.
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.
L. E. Payne & D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 272–303.
R. Racke, Zur existenz globaler Lösungen nichtlinearer Wellengleichungen, Jahresber. Deutsch. Math.-Verein. 94 (1992), 63–97.
M. Reed, Abstract Non-Linear Wave Equations, Springer Lecture Notes in Mathematics 507 (1976), 1–128.
I. Segal, Dispersion for nonlinear relativistic wave equations, Ann. Sci. Ecole Norm. Sup. 1 (1968), 459–497.
J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Comm. Math. Phys. 91 (1983), 313–327.
J. Shatah, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 290 (1985), 701–710.
J. Shatah & W. A. Strauss, Instability of nonlinear bound states, Comm. Math. Phys. 100 (1985), 173–190.
W. A. Strauss, Decay and asymptotics for \(\square u\) = F(u), J. Funct. Anal. 2 (1968), 409–457.
W. A. Strauss, Nonlinear invariant wave equations, Springer Lecture Notes in Phys. 73 (1978), 197–249.
B. R. Vainberg, On the analytical properties of the resolvent for a certain class of operator-pencils, Math. USSR Sbornik 6 (1968), 241–273.
B. R. Vainberg, Behavior of the solution of the Cauchy problem for hyperbolic equations as t→∞, Math. USSR Sbornik 7 (1969), 533–567.
B. R. Vainberg, Behavior for large time of solutions of the Klein-Gordon equation, Trans. Moscow Math. Soc. 30 (1974), 139–158.
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.
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).
B. R. Vainberg, Asymptotic Methods in Equations of Mathematical Physics, Gordon and Breach, New York, London, Paris, 1989.
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Komech, A., Vainberg, B. On asymptotic stability of stationary solutions to nonlinear wave and Klein-Gordon equations. Arch. Rational Mech. Anal. 134, 227–248 (1996). https://doi.org/10.1007/BF00379535
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DOI: https://doi.org/10.1007/BF00379535