Generalized Price’s law on fractional-order asymptotically flat stationary spacetimes

We obtain estimates on the rate of decay of a solution to the wave equation on a stationary spacetime that tends to Minkowski space at a rate O(|x|), κ ∈ (1,∞)\N. Given suitably smooth and decaying initial data, we show a wave locally enjoys the decay rate O(t).

[1]  M. Tohaneanu,et al.  Price's Law on Nonstationary Space-Times , 2011, 1104.5437.

[2]  M. Tohaneanu,et al.  Strichartz Estimates on Schwarzschild Black Hole Backgrounds , 2008, 0802.3942.

[3]  Arne Jensen,et al.  Spectral properties of Schrödinger operators and time-decay of the wave functions , 1979 .

[4]  R. Price,et al.  Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations , 1972 .

[5]  M. Zworski,et al.  Resolvent Estimates for Normally Hyperbolic Trapped Sets , 2010, 1003.4640.

[6]  D. Tataru,et al.  Decay estimates for variable coefficient wave equations in exterior domains , 2008, 0806.3409.

[7]  I. Rodnianski,et al.  A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds , 2008, 0805.4309.

[8]  Richard B. Melrose,et al.  The Atiyah-Patodi-Singer Index Theorem , 1993 .

[9]  Dejan Gajic,et al.  Price's law and precise late-time asymptotics for subextremal Reissner-Nordstr\"om black holes , 2021 .

[10]  K. Morgan.  The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting , 2020, American Journal of Mathematics.

[11]  D. Tataru,et al.  Local energy decay for scalar fields on time dependent non-trapping backgrounds , 2017, American Journal of Mathematics.

[12]  Wilhelm Schlag,et al.  A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta , 2009, 0908.4292.

[13]  P. Hintz.  A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes , 2020, Communications in Mathematical Physics.

[14]  Shi-Zhuo Looi.  Pointwise decay for the wave equation on nonstationary spacetimes , 2021 .

[15]  S. Aretakis,et al.  Late-time asymptotics for the wave equation on spherically symmetric, stationary spacetimes , 2016, 1612.01566.

[16]  Mihalis Dafermos,et al.  The Red-shift effect and radiation decay on black hole spacetimes , 2005 .

[17]  Dietrich Hafner,et al.  Local Energy Decay for Several Evolution Equations on Asymptotically Euclidean Manifolds , 2010, 1008.2357.

[18]  R. Melrose,et al.  Elliptic Operators of Totally Characteristic Type , 1983 .

[19]  S. Alinhac.  On the Morawetz-Keel-Smith-Sogge Inequality for the Wave Equation on a Curved Background , 2006 .

[20]  I. Rodnianski,et al.  Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case |a| < M , 2014, 1402.7034.

[21]  Jason Metcalfe,et al.  Long-Time Existence of Quasilinear Wave Equations Exterior to Star-Shaped Obstacles via Energy Methods , 2006, SIAM J. Math. Anal..

[22]  J. Bony,et al.  Improved local energy decay for the wave equation on asymptotically Euclidean odd dimensional manifolds in the short range case , 2011, Journal of the Institute of Mathematics of Jussieu.

[23]  C. Sogge,et al.  Concerning the wave equation on asymptotically Euclidean manifolds , 2008, 0901.0022.

[24]  I. Rodnianski,et al.  Lectures on black holes and linear waves , 2008, 0811.0354.

[25]  L. Andersson,et al.  Hidden symmetries and decay for the wave equation on the Kerr spacetime , 2009, 0908.2265.

[26]  D. Tataru,et al.  Global parametrices and dispersive estimates for variable coefficient wave equations , 2007, 0707.1191.

[27]  R. Melrose.  Transformation of boundary problems , 1981 .

[28]  Jan Sbierski.  Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes , 2013, 1311.2477.

[29]  Andrew Hassell,et al.  Resolvent at low energy III: The spectral measure , 2010, 1009.3084.

[30]  J. Wunsch.  Resolvent estimates with mild trapping , 2012, 1209.0843.

[31]  S. Yau,et al.  Decay of Solutions of the Wave Equation in the Kerr Geometry , 2005, gr-qc/0504047.

[32]  Dejan Gajic,et al.  Late-time tails and mode coupling of linear waves on Kerr spacetimes , 2021, Advances in Mathematics.

[33]  Cathleen S. Morawetz,et al.  The decay of solutions of the exterior initial-boundary value problem for the wave equation , 1961 .

[34]  I. Rodnianski,et al.  A new physical-space approach to decay for the wave equation with applications to black hole spacetimes , 2009, 0910.4957.

[35]  R. Melrose.  Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidian Spaces , 2020, Spectral and scattering theory.

[36]  J. Ralston.  Solutions of the wave equation with localized energy , 1969 .

[37]  D. Tataru.  Local decay of waves on asymptotically flat stationary space-times , 2009, 0910.5290.

[38]  R. Booth,et al.  Localized energy for wave equations with degenerate trapping , 2017, Mathematical Research Letters.

[39]  N. Burq.  Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel , 1998 .

[40]  J. Luk.  A Vector Field Method Approach to Improved Decay for Solutions to the Wave Equation on a Slowly Rotating Kerr Black Hole , 2010, 1009.0671.

[41]  A. Vasy.  Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces, a Lagrangian approach , 2018, Pure and Applied Analysis.