Instability of nonlinear bound states. (English) Zbl 0603.35007

Authors’ summary: We establish a sharp instability theorem for the bound states of lowest energy of the nonlinear Klein-Gordon equation, \(u_{tt}-\Delta u+f(u)=0\), and the nonlinear Schrödinger equation, \(- iu_ t-\Delta u+f(u)=0\).
Reviewer: U.F.Wodarzik


35B35 Stability in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35G20 Nonlinear higher-order PDEs
35L70 Second-order nonlinear hyperbolic equations
35K55 Nonlinear parabolic equations
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[1] Anderson, D.: Stability of time-dependent particlelike solutions in nonlinear field theories II. J. Math. Phys.12, 945-952 (1971)
[2] Berestycki, H., Cazenave, T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires. C.R. Acad. Sci.293, 489-492 (1981) · Zbl 0492.35010
[3] Berestycki, H., Lions, P. L.: Nonlinear scalar field equations. Arch. Rat. Mech. Anal.82, 313-375 (1983) · Zbl 0556.35046
[4] Cazenave, T., Lions, P. L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys.85, 549-561 (1982) · Zbl 0513.35007
[5] Glassey, R.: On the blowing-up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys.18, 1794-7 (1977) · Zbl 0372.35009
[6] Keller, C.: Stable and unstable manifolds for the nonlinear wave equation with dissipation. J. Diff. Eqs.50, 330-347 (1983) · Zbl 0515.35061
[7] Lee, T. D.: Particles physics and introduction to field theory. New York: Harwood Academic 1981
[8] Makhankov, V. G.: Dynamics of classical solutions (in non-integrable systems). Phys. Rep.35, 1-128 (1978)
[9] Payne, L., Sattinger, D.: Saddle points and instability of nonlinear hyperbolic equations. Israel J. Math.22, 273-303 (1975) · Zbl 0317.35059
[10] Shatah, J.: Stable standing waves of nonlinear Klein-Gordon equations. Commun. Math. Phys.91, 313-327 (1983) · Zbl 0539.35067
[11] Shatah, J.: Unstable ground states of nonlinear Klein-Gordon equations. Trans. A.M.S. (1985) · Zbl 0617.35072
[12] Strauss, W.: On weak solutions of semi-linear hyperbolic equations. Anais Acad. Brasil. Cienc.42, 645-651 (1970) · Zbl 0217.13104
[13] Strauss, W.: Existence of solitary waves in higher dimensions. Commun. Math. Phys.55, 149-162 (1977) · Zbl 0356.35028
[14] Strauss, W.: Stable and unstable states of nonlinear wave equations. Contemp. Math.17, 429-441 (1983) · Zbl 0588.35027
[15] Weinstein, M.: Stability analysis of ground states of nonlinear Schrödinger equations. preprint · Zbl 0583.35028
[16] Berestycki, H., Gallouet, T., Kavian, O.: Equations des champs scalaires euclidiens non linéaires dans le plan. C.R. Dokl. Acad. Sci.297, 307-310 (1983) · Zbl 0544.35042
[17] Brezis, H., Lieb, E.: Minimum action solutions of some vector field equations. Commun. Math. Phys.96, 97-113 (1984) · Zbl 0579.35025
[18] Pecher, H.: Low-energy scattering for nonlinear Klein-Gordon equations. preprint · Zbl 0588.35061
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