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Transactions of the American Mathematical Society

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Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space


Author: James V. Ralston
Journal: Trans. Amer. Math. Soc. 194 (1974), 27-51
MSC: Primary 35L45
DOI: https://doi.org/10.1090/S0002-9947-1974-0352714-9
MathSciNet review: 0352714
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Abstract: This paper deals with symmetric hyperbolic systems, $ \partial u/\partial t = Lu$, where L is equal to the homogeneous, constant coefficient operator $ {L_0}$ for $ \vert x\vert > R$. Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of $ \partial u/\partial t = Lu$ and solutions of $ \partial u/\partial t = {L_0}u$ are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the $ {L^2}$-solutions of the equation $ Lu + (i\lambda + \varepsilon )u = g$ for smooth g with support in $ \vert x\vert \leq R$.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0352714-9
Article copyright: © Copyright 1974 American Mathematical Society

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