Notes on the wave equation on asymptotically Euclidean manifolds. (English) Zbl 0997.58013

Summary: The author discusses the asymptotics of solutions of the wave equation on an asymptotically Euclidean manifold, when the initial data have compact support. By going over to an appropriate conformal metric, it is shown that (just as for the ordinary wave equation) such a solution has a forward (“future”) and a backward (“past”) radiation field. The same method is then used to define “end points” of bicharacteristics that begin and end above the boundary (the analogue of the sphere at infinity in the Euclidean case), and to derive a relation between the wave front sets of the two radiation fields. The extension of these results to solutions with finite energy is also briefly discussed.


58J45 Hyperbolic equations on manifolds
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[1] Friedlander, F. G., Radiation fields and hyperbolic scattering theory, Math. Proc. Cambridge Philos. Soc., 88, 483-515 (1980) · Zbl 0465.35068
[4] Leray, J., Hyperbolic Differential Equations (1953), The Institute for Advanced Study: The Institute for Advanced Study Princeton
[5] Lax, P. D.; Phillips, R. S., Scattering Theory (1967), Academic Press: Academic Press New York · Zbl 0214.12002
[6] Melrose, R. B., Geometric Scattering Theory (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0849.58071
[7] Melrose, R. B.; Zworski, M., Scattering metrics and geodesic flow at infinity, Invent. Math., 124, 389-436 (1996) · Zbl 0855.58058
[8] Wunsch, J., Propagation of singularities and growth for Schrödinger operators, Duke Math. J., 98, 137-186 (1999) · Zbl 0953.35121
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