On the interaction of metric trapping and a boundary

Datchev, Kiril; Metcalfe, Jason; Shapiro, Jacob; Tohaneanu, Mihai

doi:10.1090/proc/15460

On the interaction of metric trapping and a boundary
HTML articles powered by AMS MathViewer

by Kiril Datchev, Jason Metcalfe, Jacob Shapiro and Mihai Tohaneanu

Proc. Amer. Math. Soc. 149 (2021), 3801-3812

DOI: https://doi.org/10.1090/proc/15460

Published electronically: June 4, 2021

PDF | Request permission

Abstract:

By considering a two ended warped product manifold, we demonstrate a bifurcation that can occur when metric trapping interacts with a boundary. In this highly symmetric example, as the boundary passes through the trapped set, one goes from a nontrapping scenario where lossless local energy estimates are available for the wave equation to the case of stably trapped rays where all but a logarithmic amount of decay is lost.

References

Robert Booth, Hans Christianson, Jason Metcalfe, and Jacob Perry, Localized energy for wave equations with degenerate trapping, Math. Res. Lett. 26 (2019), no. 4, 991–1025. MR 4028109, DOI 10.4310/MRL.2019.v26.n4.a3
Nicolas Burq, Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel, Acta Math. 180 (1998), no. 1, 1–29 (French). MR 1618254, DOI 10.1007/BF02392877
Hans Christianson and Jared Wunsch, Local smoothing for the Schrödinger equation with a prescribed loss, Amer. J. Math. 135 (2013), no. 6, 1601–1632. MR 3145005, DOI 10.1353/ajm.2013.0047
Semyon Dyatlov and Maciej Zworski, Mathematical theory of scattering resonances, Graduate Studies in Mathematics, vol. 200, American Mathematical Society, Providence, RI, 2019. MR 3969938, DOI 10.1090/gsm/200
Gustav Holzegel and Jacques Smulevici, Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes, Anal. PDE 7 (2014), no. 5, 1057–1090. MR 3265959, DOI 10.2140/apde.2014.7.1057
Mitsuru Ikawa, Decay of solutions of the wave equation in the exterior of two convex obstacles, Osaka Math. J. 19 (1982), no. 3, 459–509. MR 676233
Mitsuru Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 2, 113–146 (English, with French summary). MR 949013, DOI 10.5802/aif.1137
Joe Keir, Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars, Classical Quantum Gravity 33 (2016), no. 13, 135009, 42. MR 3509155, DOI 10.1088/0264-9381/33/13/135009
Jeremy Marzuola, Jason Metcalfe, Daniel Tataru, and Mihai Tohaneanu, Strichartz estimates on Schwarzschild black hole backgrounds, Comm. Math. Phys. 293 (2010), no. 1, 37–83. MR 2563798, DOI 10.1007/s00220-009-0940-z
Jason Metcalfe and Christopher D. Sogge, Long-time existence of quasilinear wave equations exterior to star-shaped obstacles via energy methods, SIAM J. Math. Anal. 38 (2006), no. 1, 188–209. MR 2217314, DOI 10.1137/050627149
Jason Metcalfe, Jacob Sterbenz, and Daniel Tataru, Local energy decay for scalar fields on time dependent non-trapping backgrounds, Amer. J. Math. 142 (2020), no. 3, 821–883. MR 4101333, DOI 10.1353/ajm.2020.0019
Cathleen S. Morawetz, Exponential decay of solutions of the wave equation, Comm. Pure Appl. Math. 19 (1966), 439–444. MR 204828, DOI 10.1002/cpa.3160190407
Cathleen S. Morawetz, Time decay for the nonlinear Klein-Gordon equations, Proc. Roy. Soc. London Ser. A 306 (1968), 291–296. MR 234136, DOI 10.1098/rspa.1968.0151
James V. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), 807–823. MR 254433, DOI 10.1002/cpa.3160220605
Jan Sbierski, Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes, Anal. PDE 8 (2015), no. 6, 1379–1420. MR 3397001, DOI 10.2140/apde.2015.8.1379
Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218, DOI 10.1090/gsm/138