×

Asymptotic properties of the massive scalar field in the external Schwarzschild spacetime. (English) Zbl 1136.35326

Summary: The asymptotic properties of the solution to the Klein-Gordon equation will be studied in the Schwarzschild spacetime background. The results are based on the global Sobolev-type inequalities and the generalized energy estimates.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
83C57 Black holes
35Q75 PDEs in connection with relativity and gravitational theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bachelot, A., Asymptotic completeness for the Klein-Gordon equation on the Schwarzschild metric, Ann. Inst. H. Poincaré Phys. Théor., 61, 411-441 (1994) · Zbl 0809.35141
[2] Bizoń, P.; Wasserman, A., On existence of mini-boson stars, Comm. Math. Phys., 215, 357-373 (2000) · Zbl 0982.83026
[3] Blue, P.; Soffer, A., Semilinear wave equations on the Schwarzschild manifold: I. Local decay estimates, Adv. Differential Equations, 8, 595-614 (2003) · Zbl 1044.58033
[4] Burko, L. M.; Ori, A., Late-time evolution of nonlinear gravitational collapse, Phys. Rev. D, 56, 7820-7832 (1997)
[5] Christodoulou, D.; Klainerman, S., Asymptotic properties of linear field equations in Minkowski space, Comm. Pure Appl. Math., 43, 137-199 (1990) · Zbl 0715.35076
[6] Christodoulou, D.; Klainerman, S., The global nonlinear stability of the Minkowski space, (Princeton Mathematical Series, vol. 41 (1993)) · Zbl 0733.35105
[7] Dafermos, M.; Rodnianski, I., The red-shift effect and radiation decay on black hole spacetimes · Zbl 1169.83008
[8] Dimock, J., Scattering for the wave equation on the Schwarzschild metric, Gen. Relativity Gravitation, 17, 353-369 (1985) · Zbl 0618.35088
[9] Finster, F.; Smoller, J.; Yau, S.-T., Non-existence of time-periodic solutions of the Dirac equation in a Reissner-Nordstrom black hole background, J. Math. Phys., 41, 4, 2173-2194 (2000) · Zbl 0983.83021
[10] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., Decay rates and probability estimates for massive Dirac particles in the Kerr-Newman black hole geometry, Comm. Math. Phys., 230, 201-244 (2002) · Zbl 1026.83029
[11] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry, Adv. Theoret. Math. Phys., 7, 1, 25-52 (2003)
[12] Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T., An integral spectral representation of the propagator for the wave equation in the Kerr geometry, Comm. Math. Phys., 260, 2, 257-298 (2005) · Zbl 1089.83017
[13] Gundlach, C.; Price, R. H.; Pullin, J., Large-time behavior of stellar collapse and expansion. I. Linearized perturbations, Phys. Rev. D, 49, 883-889 (1994)
[14] Hörmander, L., Lectures on Nonlinear Hyperbolic Differential Equations (1997), Springer-Verlag: Springer-Verlag Berlin · Zbl 0881.35001
[15] Inglese, W.; Nicolò, F., Asymptotic properties of the electromagnetic field in the external Schwarzschild spacetime, Ann. H. Poincaré, 1, 895-944 (2000) · Zbl 0979.83029
[16] Kay, B. S.; Wald, R. M., Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation 2-sphere, Classical Quantum Gravity, 4, 893-898 (1987) · Zbl 0647.53065
[17] Klainerman, S., Uniform decay estimates and the Lorentz invariance of the classical wave equation, Comm. Pure Appl. Math., 38, 321-332 (1985) · Zbl 0635.35059
[18] Klainerman, S., Remark on the asymptotic behavior of the Klein Gordon equation in \(R^{n + 1}\), Comm. Pure Appl. Math., 46, 137-144 (1993) · Zbl 0805.35104
[19] Klainerman, S.; Nicolò, F., On local and global aspects of the Cauchy problem in general relativity, Classical Quantum Gravity, 16, R73-R157 (1999) · Zbl 0944.83001
[20] Koyama, H.; Tomimatsu, A., Asymptotic power-law tails of massive scalar fields in a Reissner-Nordström background, Phys. Rev. D, 63, 064032 (2001)
[21] Koyama, H.; Tomimatsu, A., Asymptotic tails of massive scalar fields in a Schwarzschild background, Phys. Rev. D, 64, 044014 (2001)
[22] Morawetz, C., The limiting amplitude principle, Comm. Pure Appl. Math., 15, 349-362 (1962) · Zbl 0196.41202
[23] Nicolas, J.-P., Non linear Klein-Gordon equation on Schwarzschild-like metrics, J. Math. Pures Appl., 74, 35-58 (1995) · Zbl 0853.35123
[24] Price, R. H., Nonspherical perturbations of relativistic gravitational collapse. I. Scalar and gravitational perturbations, Phys. Rev. D, 5, 2419-2438 (1972)
[25] Randall, L.; Sundrum, R., An alternative to compactification, Phys. Rev. Lett., 83, 4690-4693 (1999) · Zbl 0946.81074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.