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Decay of solutions of the wave equation in the Kerr geometry. (English) Zbl 1194.83015
Commun. Math. Phys. 264, No. 2, 465-503 (2006); erratum ibid. 280, No. 2, 563-573 (2008).
Summary: We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in time in \(L ^{\infty } _{loc}\). The proof is based on a representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable \(\omega \) on the real line. This integral representation involves solutions of the radial and angular ODEs which arise in the separation of variables.

MSC:
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
58J45 Hyperbolic equations on manifolds
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