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

Ralston, James V.

doi:10.1090/S0002-9947-1974-0352714-9

Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space
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by James V. Ralston PDF

Trans. Amer. Math. Soc. 194 (1974), 27-51 Request permission

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 $|x| > R$. 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 $|x| \leq R$.

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