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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

$ 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.


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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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