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$ L\sp{2}$ asymptotes for the Klein-Gordon equation


Author: Stuart Nelson
Journal: Proc. Amer. Math. Soc. 27 (1971), 110-116
MSC: Primary 35.79
DOI: https://doi.org/10.1090/S0002-9939-1971-0271561-2
MathSciNet review: 0271561
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Abstract: An approximation $ a(x,t)$ is obtained for solutions $ u(x,t)$ of the Klein-Gordon equation. $ a(x,t)$ can be expressed in terms of the Fourier transforms of the Cauchy data and it is shown that $ \vert\vert a( \cdot ,t) - u( \cdot ,t)\vert{\vert _2} \to 0$ as $ t \to \infty $. This result is applied to show how energy distributes among various conical regions.


References [Enhancements On Off] (What's this?)

  • [1] A. R. Brodsky, On the asymptotic behavior of solutions of the wave equations, Proc. Amer. Math. Soc. 18 (1967), 207-208. MR 35 #3289. MR 0212417 (35:3289)
  • [2] -, Asymptotic decay of solutions to the relativistic wave equation, Thesis, M.I.T., Cambridge, Mass., 1964.
  • [3] W. Littman, Maximal rates of decay of solutions of partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 1273-1275. MR 0245964 (39:7270)
  • [4] -, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766-770. MR 27 #5086. MR 0155146 (27:5086)
  • [5] S. Nelson, On some solutions to the Klein-Gordon equation related to an integral of Sonine, Trans. Amer. Math. Soc. (to appear). MR 0415049 (54:3140)
  • [6] -, $ {L^2}$ asymptotes for Fourier transforms of surface-carried measures, Proc. Amer. Math. Soc. (to appear). MR 0283491 (44:722)
  • [7] I. E. Segal, Quantization and dispersion for non-linear relativistic equations, Proc. Conference Mathematical Theory of Elementary Particles (Dedham, Mass., 1965), M.I.T. Press, Cambridge, Mass., 1966, pp. 79-108. MR 36 #542.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271561-2
Keywords: Klein-Gordon equation, Cauchy problem, asymptotic behavior, $ {L^\infty }$ decay, $ {L^2}$ approximation, energy in conical region, Virial theorem, Riemann-Lebesgue theorem
Article copyright: © Copyright 1971 American Mathematical Society

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