𝐿² asymptotes for the Klein-Gordon equation

Nelson, Stuart

doi:10.1090/S0002-9939-1971-0271561-2

$L^{2}$ asymptotes for the Klein-Gordon equation
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by Stuart Nelson

Proc. Amer. Math. Soc. 27 (1971), 110-116

DOI: https://doi.org/10.1090/S0002-9939-1971-0271561-2

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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 $||a( \cdot ,t) - u( \cdot ,t)|{|_2} \to 0$ as $t \to \infty$. This result is applied to show how energy distributes among various conical regions.

References

A. R. Brodsky, On the asymptotic behavior of solutions of the wave equations, Proc. Amer. Math. Soc. 18 (1967), 207–208. MR 212417, DOI 10.1090/S0002-9939-1967-0212417-X

Asymptotic decay of solutions to the relativistic wave equation

Walter Littman, Maximal rates of decay of solutions of partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 1273–1275. MR 245964, DOI 10.1090/S0002-9904-1969-12391-8
Walter Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963), 766–770. MR 155146, DOI 10.1090/S0002-9904-1963-11025-3
Stuart Nelson, On some solutions to the Klein-Gordon equation related to an integral of Sonine, Trans. Amer. Math. Soc. 154 (1971), 227–237. MR 415049, DOI 10.1090/S0002-9947-1971-0415049-9
Stuart Nelson, $L^{2}$ asymptotes for Fourier transforms of surface- carried measures, Proc. Amer. Math. Soc. 28 (1971), 134–136. MR 283491, DOI 10.1090/S0002-9939-1971-0283491-0

Quantization and dispersion for non-linear relativistic equations

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