Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremalcase $|a| < M$

Abstract

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: the cases $|a| \ll M$ or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal $|a|< M$ case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al. (ed.), World Scientific, Singapore, 2011, pp. 132189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincaré], together with a streamlined continuity argument in the parameter $a$, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notation so that it can be read independently of previous work.

Note: To view the article, click on the URL link for the DOI number.

  • [alexakis] Go to document S. Alexakis, A. D. Ionescu, and S. Klainerman, "Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces," Comm. Math. Phys., vol. 299, iss. 1, pp. 89-127, 2010.
    @article{alexakis, mrkey = {2672799},
      author = {Alexakis, S. and Ionescu, A. D. and Klainerman, S.},
      title = {Uniqueness of smooth stationary black holes in vacuum: small perturbations of the {K}err spaces},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {299},
      year = {2010},
      number = {1},
      pages = {89--127},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {53C80 (83C57)},
      mrnumber = {2672799},
      doi = {10.1007/s00220-010-1072-1},
      zblnumber = {1200.83059},
      }
  • [alinhac] Go to document S. Alinhac, "Energy multipliers for perturbations of the Schwarzschild metric," Comm. Math. Phys., vol. 288, iss. 1, pp. 199-224, 2009.
    @article{alinhac, mrkey = {2491622},
      author = {Alinhac, Serge},
      title = {Energy multipliers for perturbations of the {S}chwarzschild metric},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {288},
      year = {2009},
      number = {1},
      pages = {199--224},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J47 (83C47)},
      mrnumber = {2491622},
      mrreviewer = {Atanas G. Stefanov},
      doi = {10.1007/s00220-009-0770-z},
      zblnumber = {1196.53053},
      }
  • [anblue] Go to document L. Andersson and P. Blue, "Hidden symmetries and decay for the wave equation on the Kerr spacetime," Ann. of Math., vol. 182, iss. 3, pp. 787-853, 2015.
    @article{anblue, mrkey = {3418531},
      author = {Andersson, Lars and Blue, Pieter},
      title = {Hidden symmetries and decay for the wave equation on the {K}err spacetime},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {182},
      year = {2015},
      number = {3},
      pages = {787--853},
      issn = {0003-486X},
      mrclass = {58J45 (35B40 35B45 35L10 35R01)},
      mrnumber = {3418531},
      doi = {10.4007/annals.2015.182.3.1},
      zblnumber = {06514748},
      }
  • [anblue2] L. Andersson and P. Blue, Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior, 2013.
    @misc{anblue2,
      author = {Andersson, Lars and Blue, Pieter},
      title = {Uniform energy bound and asymptotics for the {M}axwell field on a slowly rotating {K}err black hole exterior},
      arxiv = {1310.2664},
      year = {2013},
      }
  • [AndGlam] Go to document N. Andersson and K. Glampedakis, "Superradiance resonance cavity outside rapidly rotating black holes," Phys. Rev. Lett., vol. 84, iss. 20, pp. 4537-4540, 2000.
    @article{AndGlam, mrkey = {1757382},
      author = {Andersson, Nils and Glampedakis, Kostas},
      title = {Superradiance resonance cavity outside rapidly rotating black holes},
      journal = {Phys. Rev. Lett.},
      fjournal = {Physical Review Letters},
      volume = {84},
      year = {2000},
      number = {20},
      pages = {4537--4540},
      issn = {0031-9007},
      coden = {PRLTAO},
      mrclass = {83C57},
      mrnumber = {1757382},
      doi = {10.1103/PhysRevLett.84.4537},
      }
  • [aretakisKerr] Go to document S. Aretakis, "Decay of axisymmetric solutions of the wave equation on extreme Kerr backgrounds," J. Funct. Anal., vol. 263, iss. 9, pp. 2770-2831, 2012.
    @article{aretakisKerr, mrkey = {2967306},
      author = {Aretakis, Stefanos},
      title = {Decay of axisymmetric solutions of the wave equation on extreme {K}err backgrounds},
      journal = {J. Funct. Anal.},
      fjournal = {Journal of Functional Analysis},
      volume = {263},
      year = {2012},
      number = {9},
      pages = {2770--2831},
      issn = {0022-1236},
      coden = {JFUAAW},
      mrclass = {58J45 (35L05 83C57)},
      mrnumber = {2967306},
      mrreviewer = {Semyon Dyatlov},
      doi = {10.1016/j.jfa.2012.08.015},
      zblnumber = {1254.83021},
      }
  • [aretakisHor] Go to document S. Aretakis, "Horizon instability of extremal black holes," Adv. Theor. Math. Phys., vol. 19, pp. 507-530, 2015.
    @article{aretakisHor, MRKEY={3418509},
      author = {Aretakis, Stefanos},
      title = {Horizon instability of extremal black holes},
      journal = {Adv. Theor. Math. Phys.},
      volume = {19},
      year = {2015},
      pages = {507--530},
      mrnumber ={3418509},
      doi = {10.4310/ATMP.2015.v19.n3.a1},
     }
  • [aretakis] Go to document S. Aretakis, "Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations I," Comm. Math. Phys., vol. 307, iss. 1, pp. 17-63, 2011.
    @article{aretakis, mrkey = {2835872},
      author = {Aretakis, Stefanos},
      title = {Stability and instability of extreme {R}eissner-{N}ordström black hole spacetimes for linear scalar perturbations {I}},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {307},
      year = {2011},
      number = {1},
      pages = {17--63},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {53C80 (58J45 83C57)},
      mrnumber = {2835872},
      doi = {10.1007/s00220-011-1254-5},
      zblnumber = {1229.85002},
      }
  • [aretakis2] Go to document S. Aretakis, "Stability and instability of extreme Reissner-Nordström black hole spacetimes for linear scalar perturbations II," Ann. Henri Poincaré, vol. 12, iss. 8, pp. 1491-1538, 2011.
    @article{aretakis2, mrkey = {2855176},
      author = {Aretakis, Stefanos},
      title = {Stability and instability of extreme {R}eissner-{N}ordström black hole spacetimes for linear scalar perturbations {II}},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {12},
      year = {2011},
      number = {8},
      pages = {1491--1538},
      issn = {1424-0637},
      mrclass = {83C57 (83C10)},
      mrnumber = {2855176},
      doi = {10.1007/s00023-011-0110-7},
      zblnumber = {1242.83049},
      }
  • [2bachelots] Go to document A. Bachelot and A. Motet-Bachelot, "Les résonances d’un trou noir de Schwarzschild," Ann. Inst. H. Poincaré Phys. Théor., vol. 59, iss. 1, pp. 3-68, 1993.
    @article{2bachelots, mrkey = {1244181},
      author = {Bachelot, Alain and Motet-Bachelot, Agn{è}s},
      title = {Les résonances d'un trou noir de {S}chwarzschild},
      journal = {Ann. Inst. H. Poincaré Phys. Théor.},
      fjournal = {Annales de l'Institut Henri Poincaré. Physique Théorique},
      volume = {59},
      year = {1993},
      number = {1},
      pages = {3--68},
      issn = {0246-0211},
      coden = {AHPAAO},
      mrclass = {83C57 (35Q75 47N20)},
      mrnumber = {1244181},
      mrreviewer = {Philippe-A. Dionne},
      url = {http://www.numdam.org/item?id=AIHPA_1993__59_1_3_0},
      zblnumber = {0793.53094},
      }
  • [BlueSof0] Go to document P. Blue and A. Soffer, "Semilinear wave equations on the Schwarzschild manifold. I. Local decay estimates," Adv. Differential Equations, vol. 8, iss. 5, pp. 595-614, 2003.
    @article{BlueSof0, mrkey = {1972492},
      author = {Blue, P. and Soffer, A.},
      title = {Semilinear wave equations on the {S}chwarzschild manifold. {I}. {L}ocal decay estimates},
      journal = {Adv. Differential Equations},
      fjournal = {Advances in Differential Equations},
      volume = {8},
      year = {2003},
      number = {5},
      pages = {595--614},
      issn = {1079-9389},
      mrclass = {58J45 (35Q75 83C57)},
      mrnumber = {1972492},
      zblnumber = {1044.58033},
      URL = {http://projecteuclid.org/euclid.ade/1355926842},
      }
  • [BlueSof] P. Blue and A. Soffer, Errata for “Global existence and scattering for the nonlinear Sch"rodinger equation on Schwarzschild manifolds”, “Semilinear wave equations on the Schwarzschild manifold I: Local Decay Estimates”, and “The wave equation on the Schwarzschild metric II: Local Decay for the spin 2 Regge Wheeler equation”, 6 pages.
    @misc{BlueSof,
      author = {Blue, P. and Soffer, A.},
      title = {Errata for ``{G}lobal existence and scattering for the nonlinear {S}ch\"rodinger equation on {S}chwarzschild manifolds'', ``{S}emilinear wave equations on the {S}chwarzschild manifold {I: L}ocal {D}ecay {E}stimates'', and ``{T}he wave equation on the {S}chwarzschild metric {II: L}ocal {D}ecay for the spin 2 {R}egge {W}heeler equation'', 6 pages},
      arxiv = {gr-qc/0608073},
      sortyear = {2017},
      }
  • [BlueSter] Go to document P. Blue and J. Sterbenz, "Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space," Comm. Math. Phys., vol. 268, iss. 2, pp. 481-504, 2006.
    @article{BlueSter, mrkey = {2259204},
      author = {Blue, Pieter and Sterbenz, Jacob},
      title = {Uniform decay of local energy and the semi-linear wave equation on {S}chwarzschild space},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {268},
      year = {2006},
      number = {2},
      pages = {481--504},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (83C47)},
      mrnumber = {2259204},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00220-006-0101-6},
      zblnumber= {1123.58018},
      }
  • [bh] Go to document J. Bony and D. Häfner, "Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric," Comm. Math. Phys., vol. 282, iss. 3, pp. 697-719, 2008.
    @article{bh, mrkey = {2426141},
      author = {Bony, Jean-Fran{ç}ois and H{ä}fner, Dietrich},
      title = {Decay and non-decay of the local energy for the wave equation on the de {S}itter-{S}chwarzschild metric},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {282},
      year = {2008},
      number = {3},
      pages = {697--719},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (35L70 83C57)},
      mrnumber = {2426141},
      mrreviewer = {Tanya J. Christiansen},
      doi = {10.1007/s00220-008-0553-y},
      zblnumber = {1159.35007},
      }
  • [carter] B. Carter, "Black hole equilibrium states," in Black Holes/Les Astres Occlus, New York: Gordon and Breach, 1973, pp. 57-214.
    @incollection{carter, mrkey = {0465047},
      author = {Carter, Brandon},
      title = {Black hole equilibrium states},
      booktitle = {Black Holes/{L}es Astres Occlus},
      venue = {\'{E}cole d'\'{E}té {P}hys. {T}héor., {L}es {H}ouches, 1972},
      pages = {57--214},
      publisher = {Gordon and Breach},
      address = {New York},
      year = {1973},
      mrclass = {83.53},
      mrnumber = {0465047},
      mrreviewer = {Andrew King},
      }
  • [cartersep2] Go to document B. Carter, "Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations," Comm. Math. Phys., vol. 10, pp. 280-310, 1968.
    @article{cartersep2, mrkey = {0239841},
      author = {Carter, Brandon},
      title = {Hamilton-{J}acobi and {S}chrödinger separable solutions of {E}instein's equations},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {10},
      year = {1968},
      pages = {280--310},
      issn = {0010-3616},
      mrclass = {83.53},
      mrnumber = {0239841},
      mrreviewer = {W. Israel},
      url = {http://projecteuclid.org/euclid.cmp/1103841118},
      zblnumber = {0162.59302},
      }
  • [chandrasekhar] S. Chandrasekhar, The Mathematical Theory of Black Holes, New York: The Clarendon Press, Oxford University Press, 1983, vol. 69.
    @book{chandrasekhar, mrkey = {0700826},
      author = {Chandrasekhar, Subrahmanyan},
      title = {The Mathematical Theory of Black Holes},
      series = {Intern. Ser. Monogr. Physics},
      volume = {69},
      note = {Oxford Science Publications},
      publisher = {The Clarendon Press, Oxford University Press},
      address = {New York},
      year = {1983},
      pages = {xxi+646},
      isbn = {0-19-851291-0},
      mrclass = {83-02 (83C15)},
      mrnumber = {0700826},
      mrreviewer = {Hans Stephani},
      zblnumber = {0511.53076},
      }
  • [civin] Go to document D. Civin, Stability at charged rotating black holes for linear scalar perturbations, 2014.
    @misc{civin,
      author = {Civin, D.},
      title = {Stability at charged rotating black holes for linear scalar perturbations},
      note = {Ph.D. thesis, University of Cambridge},
      url={http://www.repository.cam.ac.uk/handle/1810/247397},
      year = {2014},
      }
  • [book2] Go to document D. Christodoulou, "Symplectic geometry and partial differential equations," in Differential Equations: La Pietra 1996, Providence, RI: Amer. Math. Soc., 1999, vol. 65, pp. 27-37.
    @incollection{book2, mrkey = {1662748},
      author = {Christodoulou, Demetrios},
      title = {Symplectic geometry and partial differential equations},
      booktitle = {Differential Equations: {L}a {P}ietra 1996},
      venue = {{F}lorence},
      series = {Proc. Sympos. Pure Math.},
      volume = {65},
      pages = {27--37},
      publisher = {Amer. Math. Soc.},
      address = {Providence, RI},
      year = {1999},
      mrclass = {58E30 (35A15 37K05)},
      mrnumber = {1662748},
      mrreviewer = {Charles-Michel Marle},
      doi = {10.1090/pspum/065/1662748},
      zblnumber = {0930.35013},
      }
  • [ck] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton, NJ: Princeton Univ. Press, 1993, vol. 41.
    @book{ck, mrkey = {1316662},
      author = {Christodoulou, Demetrios and Klainerman, Sergiu},
      title = {The Global Nonlinear Stability of the {M}inkowski Space},
      series = {Princeton Math. Ser.},
      volume = {41},
      publisher = {Princeton Univ. Press},
      address = {Princeton, NJ},
      year = {1993},
      pages = {x+514},
      isbn = {0-691-08777-6},
      mrclass = {83C05 (35Q75 58G16 83C35)},
      mrnumber = {1316662},
      mrreviewer = {Alan D. Rendall},
      zblnumber = {0827.53055},
      }
  • [cbh] Go to document M. Dafermos, "The interior of charged black holes and the problem of uniqueness in general relativity," Comm. Pure Appl. Math., vol. 58, iss. 4, pp. 445-504, 2005.
    @article{cbh, mrkey = {2119866},
      author = {Dafermos, Mihalis},
      title = {The interior of charged black holes and the problem of uniqueness in general relativity},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {58},
      year = {2005},
      number = {4},
      pages = {445--504},
      issn = {0010-3640},
      coden = {CPAMA},
      mrclass = {83C57 (83C05)},
      mrnumber = {2119866},
      mrreviewer = {Lars {\AA}ke Andersson},
      doi = {10.1002/cpa.20071},
      zblnumber = {1071.83037},
      }
  • [dr1] Go to document M. Dafermos and I. Rodnianski, "A proof of Price’s law for the collapse of a self-gravitating scalar field," Invent. Math., vol. 162, iss. 2, pp. 381-457, 2005.
    @article{dr1, mrkey = {2199010},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A proof of {P}rice's law for the collapse of a self-gravitating scalar field},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {162},
      year = {2005},
      number = {2},
      pages = {381--457},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {83C05 (35Q75 83C22 83C75)},
      mrnumber = {2199010},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00222-005-0450-3},
      zblnumber = {1088.83008},
      }
  • [dr3] Go to document M. Dafermos and I. Rodnianski, "The red-shift effect and radiation decay on black hole spacetimes," Comm. Pure Appl. Math., vol. 62, iss. 7, pp. 859-919, 2009.
    @article{dr3, mrkey = {2527808},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {The red-shift effect and radiation decay on black hole spacetimes},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {62},
      year = {2009},
      number = {7},
      pages = {859--919},
      issn = {0010-3640},
      coden = {CPAMA},
      mrclass = {83C57 (35B40 35L05)},
      mrnumber = {2527808},
      mrreviewer = {Guilherme De Berredo-Peixoto},
      doi = {10.1002/cpa.20281},
      zblnumber = {1169.83008},
      }
  • [dr4] M. Dafermos and I. Rodnianski, The wave equation on Schwarzschild-de Sitter spacetimes, 2007.
    @misc{dr4,
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {The wave equation on {S}chwarzschild-de {S}itter spacetimes},
      arxiv = {0709.2766v1},
      year = {2007},
      }
  • [dr5] M. Dafermos and I. Rodnianski, A note on energy currents and decay for the wave equation on a Schwarzschild background, 2007.
    @misc{dr5,
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A note on energy currents and decay for the wave equation on a {S}chwarzschild background},
      arxiv = {0710.0171v1},
      year = {2007},
      }
  • [dr6] Go to document M. Dafermos and I. Rodnianski, "A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds," Invent. Math., vol. 185, iss. 3, pp. 467-559, 2011.
    @article{dr6, mrkey = {2827094},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A proof of the uniform boundedness of solutions to the wave equation on slowly rotating {K}err backgrounds},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {185},
      year = {2011},
      number = {3},
      pages = {467--559},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {58J45 (35Q76 35R01 83C05 83C57)},
      mrnumber = {2827094},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1007/s00222-010-0309-0},
      zblnumber = {1226.83029},
      }
  • [jnotes] Go to document M. Dafermos and I. Rodnianski, "Lectures on black holes and linear waves," in Evolution Equations, Amer. Math. Soc., 2013, vol. 17, pp. 97-205.
    @incollection{jnotes, mrkey = {3098640},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {Lectures on black holes and linear waves},
      booktitle = {Evolution Equations},
      series = {Clay Math. Proc.},
      volume = {17},
      pages = {97--205},
      publisher = {Amer. Math. Soc.},
      year = {2013},
      mrclass = {83C57 (35Q75 35R01 58J99 83C05)},
      mrnumber = {3098640},
      mrreviewer = {Hans-Peter K{ü}nzle},
      url = {http://www.claymath.org/library/proceedings/cmip017c.pdf},
     }
  • [icmp] Go to document M. Dafermos and I. Rodnianski, "A new physical-space approach to decay for the wave equation with applications to black hole spacetimes," in XVIth International Congress on Mathematical Physics, Hackensack, NJ: World Sci. Publ., 2010, pp. 421-432.
    @incollection{icmp, mrkey = {2730803},
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {A new physical-space approach to decay for the wave equation with applications to black hole spacetimes},
      booktitle = {X{VI}th {I}nternational {C}ongress on {M}athematical {P}hysics},
      pages = {421--432},
      publisher = {World Sci. Publ.},
      address = {Hackensack, NJ},
      year = {2010},
      mrclass = {58J45 (35L10 83C05 83C57)},
      mrnumber = {2730803},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1142/9789814304634_0032},
      zblnumber = {1211.83019},
      }
  • [stabi] Go to document M. Dafermos and I. Rodnianski, "The black hole stability problem for linear scalar perturbations," in Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, relax Damour et al., T., Ed., Singapore: World Scientific, 2012, pp. 132-189.
    @incollection{stabi,
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {The black hole stability problem for linear scalar perturbations},
      booktitle = {Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity},
      editor = {{\relax Damour et al.},
      T.},
      publisher = {World Scientific},
      address = {Singapore},
      year = {2012},
      pages = {132--189},
      doi = {10.1142/9789814374552_0008},
      }
  • [dr7] M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: The cases $|a|\ll M$ or axisymmetry, 2010.
    @misc{dr7,
      author = {Dafermos, Mihalis and Rodnianski, Igor},
      title = {Decay for solutions of the wave equation on {K}err exterior spacetimes {I--II}: The cases {$|a|\ll M$} or axisymmetry},
      arxiv = {1010.5132},
      year = {2010},
      }
  • [scattering] M. Dafermos, G. Holzegel, and I. Rodnianski, A scattering theory construction of dynamical black hole spacetimes, 2013.
    @misc{scattering,
      author = {Dafermos, Mihalis and Holzegel, G. and Rodnianski, I.},
      title = {A scattering theory construction of dynamical black hole spacetimes},
      year = {2013},
      note = {to appear in \emph{JDG}},
      ARXIV={1306.5364},
     }
  • [dyatlov1] Go to document S. Dyatlov, "Quasi-normal modes and exponential energy decay for the Kerr-de Sitter black hole," Comm. Math. Phys., vol. 306, iss. 1, pp. 119-163, 2011.
    @article{dyatlov1, mrkey = {2819421},
      author = {Dyatlov, Semyon},
      title = {Quasi-normal modes and exponential energy decay for the {K}err-de {S}itter black hole},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {306},
      year = {2011},
      number = {1},
      pages = {119--163},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {58J45 (83C57)},
      mrnumber = {2819421},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1007/s00220-011-1286-x},
      zblnumber = {1223.83029},
      }
  • [dyatlov2] Go to document S. Dyatlov, "Exponential energy decay for Kerr–de Sitter black holes beyond event horizons," Math. Res. Lett., vol. 18, iss. 5, pp. 1023-1035, 2011.
    @article{dyatlov2, mrkey = {2875874},
      author = {Dyatlov, Semyon},
      title = {Exponential energy decay for {K}err--de {S}itter black holes beyond event horizons},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {18},
      year = {2011},
      number = {5},
      pages = {1023--1035},
      issn = {1073-2780},
      mrclass = {58J45 (35B40 35Q75 83C57)},
      mrnumber = {2875874},
      mrreviewer = {Piotr T. Chru{\'s}ciel},
      doi = {10.4310/MRL.2011.v18.n5.a19},
      zblnumber = {1253.83020},
      }
  • [dyatlov-last] Go to document S. Dyatlov, "Asymptotics of linear waves and resonances with applications to black holes," Comm. Math. Phys., vol. 335, iss. 3, pp. 1445-1485, 2015.
    @article{dyatlov-last, mrkey = {3320319},
      author = {Dyatlov, Semyon},
      title = {Asymptotics of linear waves and resonances with applications to black holes},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {335},
      year = {2015},
      number = {3},
      pages = {1445--1485},
      issn = {0010-3616},
      mrclass = {83C57},
      mrnumber = {3320319},
      mrreviewer = {Giovanni Preti},
      doi = {10.1007/s00220-014-2255-y},
      zblnumber = {1315.83022},
      }
  • [fksy] Go to document F. Finster, N. Kamran, J. Smoller, and S. -T. Yau, "Decay of solutions of the wave equation in the Kerr geometry," Comm. Math. Phys., vol. 264, iss. 2, pp. 465-503, 2006.
    @article{fksy, mrkey = {2215614},
      author = {Finster, F. and Kamran, N. and Smoller, J. and Yau, S.-T.},
      title = {Decay of solutions of the wave equation in the {K}err geometry},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {264},
      year = {2006},
      number = {2},
      pages = {465--503},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {83C05 (35B40 35L15 58J45)},
      mrnumber = {2215614},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00220-006-1525-8},
      zblnumber = {1194.83015},
      }
  • [fksy2] Go to document F. Finster, N. Kamran, J. Smoller, and S. -T. Yau, "Decay of solutions of the wave equation in the Kerr geometry," Comm. Math. Phys., vol. 264, iss. 2, pp. 465-503, 2006.
    @article{fksy2, mrkey = {2215614},
      author = {Finster, F. and Kamran, N. and Smoller, J. and Yau, S.-T.},
      title = {Decay of solutions of the wave equation in the {K}err geometry},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {264},
      year = {2006},
      number = {2},
      pages = {465--503},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {83C05 (35B40 35L15 58J45)},
      mrnumber = {2215614},
      mrreviewer = {Simone Calogero},
      doi = {10.1007/s00220-006-1525-8},
      zblnumber = {1194.83015},
      }
  • [franzen] Go to document A. Franzen, The wave equation on black hole interiors, 2014.
    @misc{franzen,
      author = {Franzen, A.},
      title = {The wave equation on black hole interiors},
      note = {Ph.D.~thesis, Universiteit Utrecht},
      year = {2014},
      url = {http://dspace.library.uu.nl/bitstream/handle/1874/324945/Franzen.pdf},
      }
  • [gannot] Go to document O. Gannot, "Quasinormal modes for Schwarzschild-AdS black holes: exponential convergence to the real axis," Comm. Math. Phys., vol. 330, iss. 2, pp. 771-799, 2014.
    @article{gannot, mrkey = {3223487},
      author = {Gannot, Oran},
      title = {Quasinormal modes for {S}chwarzschild-{A}d{S} black holes: exponential convergence to the real axis},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {330},
      year = {2014},
      number = {2},
      pages = {771--799},
      issn = {0010-3616},
      mrclass = {83C57},
      mrnumber = {3223487},
      mrreviewer = {Joseph Katz},
      doi = {10.1007/s00220-014-2002-4},
      zblnumber = {1295.85001},
      }
  • [he:lssst] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-time, New York: Cambridge Univ. Press, 1973, vol. 1.
    @book{he:lssst, mrkey = {0424186},
      author = {Hawking, S. W. and Ellis, G. F. R.},
      title = {The Large Scale Structure of Space-time},
      note = {Cambridge Monog. Math. Phys.},
      volume = {1},
      publisher = {Cambridge Univ. Press},
      address = {New York},
      year = {1973},
      pages = {xi+391},
      mrclass = {83.58},
      mrnumber = {0424186},
      mrreviewer = {Michael P. Ryan, Jr.},
      zblnumber = {0265.53054},
      }
  • [kostakis2] G. Holzegel, Ultimately Schwarzschildean spacetimes and the black hole stability problem, 2010.
    @misc{kostakis2,
      author = {Holzegel, G.},
      title = {Ultimately {S}chwarzschildean spacetimes and the black hole stability problem},
      arxiv = {1010.3216},
      year = {2010},
      }
  • [holz-smul] Go to document G. Holzegel and J. Smulevici, "Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes," Comm. Pure Appl. Math., vol. 66, iss. 11, pp. 1751-1802, 2013.
    @article{holz-smul, mrkey = {3103910},
      author = {Holzegel, Gustav and Smulevici, Jacques},
      title = {Decay properties of {K}lein-{G}ordon fields on {K}err-{A}d{S} spacetimes},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {66},
      year = {2013},
      number = {11},
      pages = {1751--1802},
      issn = {0010-3640},
      mrclass = {83C57 (35B40 35L10 35Q75 35R01)},
      mrnumber = {3103910},
      doi = {10.1002/cpa.21470},
      zblnumber = {1277.83023},
      }
  • [holz-smul2] Go to document G. Holzegel and J. Smulevici, "Quasimodes and a lower bound on the uniform energy decay rate for Kerr-AdS spacetimes," Anal. PDE, vol. 7, iss. 5, pp. 1057-1090, 2014.
    @article{holz-smul2, mrkey = {3265959},
      author = {Holzegel, Gustav and Smulevici, Jacques},
      title = {Quasimodes and a lower bound on the uniform energy decay rate for {K}err-{A}d{S} spacetimes},
      journal = {Anal. PDE},
      fjournal = {Analysis \& PDE},
      volume = {7},
      year = {2014},
      number = {5},
      pages = {1057--1090},
      issn = {2157-5045},
      mrclass = {58J45 (35B40 35R01 83C57)},
      mrnumber = {3265959},
      mrreviewer = {Piotr T. Chru{\'s}ciel},
      doi = {10.2140/apde.2014.7.1057},
      zblnumber = {1300.83030},
      }
  • [kw:lss] Go to document B. S. Kay and R. M. Wald, "Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation $2$-sphere," Classical Quantum Gravity, vol. 4, iss. 4, pp. 893-898, 1987.
    @article{kw:lss, mrkey = {0895907},
      author = {Kay, Bernard S. and Wald, Robert M.},
      title = {Linear stability of {S}chwarzschild under perturbations which are nonvanishing on the bifurcation {$2$}-sphere},
      journal = {Classical Quantum Gravity},
      fjournal = {Classical and Quantum Gravity},
      volume = {4},
      year = {1987},
      number = {4},
      pages = {893--898},
      issn = {0264-9381},
      coden = {CQGRDG},
      mrclass = {83C99 (81E20)},
      mrnumber = {0895907},
      mrreviewer = {Derek Raine},
      url = {http://stacks.iop.org/0264-9381/4/893},
      zblnumber = {0647.53065},
      }
  • [muchT] Go to document S. Klainerman, "Uniform decay estimates and the Lorentz invariance of the classical wave equation," Comm. Pure Appl. Math., vol. 38, iss. 3, pp. 321-332, 1985.
    @article{muchT, mrkey = {0784477},
      author = {Klainerman, Sergiu},
      title = {Uniform decay estimates and the {L}orentz invariance of the classical wave equation},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {38},
      year = {1985},
      number = {3},
      pages = {321--332},
      issn = {0010-3640},
      coden = {CPAMA},
      mrclass = {35L70 (35B40)},
      mrnumber = {0784477},
      mrreviewer = {C. Bardos},
      doi = {10.1002/cpa.3160380305},
      zblnumber = {0635.35059},
      }
  • [labasoffer] I. Łaba and A. Soffer, "Global existence and scattering for the nonlinear Schrödinger equation on Schwarzschild manifolds," Helv. Phys. Acta, vol. 72, iss. 4, pp. 274-294, 1999.
    @article{labasoffer, mrkey = {1732864},
      author = {{\L}aba, I. and Soffer, A.},
      title = {Global existence and scattering for the nonlinear {S}chrödinger equation on {S}chwarzschild manifolds},
      journal = {Helv. Phys. Acta},
      fjournal = {Helvetica Physica Acta. Physica Theoretica. Societatis Physicae Helveticae Commentaria Publica},
      volume = {72},
      year = {1999},
      number = {4},
      pages = {274--294},
      issn = {0018-0238},
      coden = {HPACAK},
      mrclass = {58J50 (35Q75 81Q05)},
      mrnumber = {1732864},
      mrreviewer = {Ricardo Weder},
      zblnumber = {0976.58019},
      }
  • [lauletal] P. Laul, J. Metcalfe, S. Tikare, and M. Tohaneanu, Localized energy estimates on Myers–Perry space-times, 2014.
    @misc{lauletal,
      author = {Laul, P and Metcalfe, J. and Tikare, S. and Tohaneanu, M.},
      title = {Localized energy estimates on {M}yers--{P}erry space-times},
      arxiv = {1401.0465},
      year = {2014},
      }
  • [luciet] Go to document J. Lucietti and H. S. Reall, "Gravitational instability of an extreme Kerr black hole," Phys. Rev. D, vol. 86, p. 104030, 2012.
    @article{luciet,
      author = {Lucietti, J. and Reall, H. S.},
      title = {Gravitational instability of an extreme {K}err black hole},
      journal = {Phys. Rev. D},
      volume = {86},
      issue = {10},
      pages = {104030, 7 pp.},
      year = {2012},
      publisher = {American Physical Society},
      doi = {10.1103/PhysRevD.86.104030},
      }
  • [luk] Go to document J. Luk, "Improved decay for solutions to the linear wave equation on a Schwarzschild black hole," Ann. Henri Poincaré, vol. 11, iss. 5, pp. 805-880, 2010.
    @article{luk, mrkey = {2736525},
      author = {Luk, Jonathan},
      title = {Improved decay for solutions to the linear wave equation on a {S}chwarzschild black hole},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {11},
      year = {2010},
      number = {5},
      pages = {805--880},
      issn = {1424-0637},
      mrclass = {58J45 (83C57)},
      mrnumber = {2736525},
      mrreviewer = {Ng Ibohal},
      doi = {10.1007/s00023-010-0043-6},
      zblnumber = {1208.83068},
      }
  • [luk2] Go to document J. Luk, "A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole," Anal. PDE, vol. 5, iss. 3, pp. 553-625, 2012.
    @article{luk2, mrkey = {2994507},
      author = {Luk, Jonathan},
      title = {A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating {K}err black hole},
      journal = {Anal. PDE},
      fjournal = {Analysis \& PDE},
      volume = {5},
      year = {2012},
      number = {3},
      pages = {553--625},
      issn = {2157-5045},
      mrclass = {83C57 (35B40 35Q75 35R01 58J99)},
      mrnumber = {2994507},
      mrreviewer = {Micha{\l} Eckstein},
      doi = {10.2140/apde.2012.5.553},
      zblnumber = {1267.83065},
      }
  • [luk3] Go to document J. Luk, "The null condition and global existence for nonlinear wave equations on slowly rotating Kerr spacetimes," J. Eur. Math. Soc. $($JEMS$)$, vol. 15, iss. 5, pp. 1629-1700, 2013.
    @article{luk3, mrkey = {3082240},
      author = {Luk, Jonathan},
      title = {The null condition and global existence for nonlinear wave equations on slowly rotating {K}err spacetimes},
      journal = {J. Eur. Math. Soc. $($JEMS$)$},
      fjournal = {Journal of the European Mathematical Society (JEMS)},
      volume = {15},
      year = {2013},
      number = {5},
      pages = {1629--1700},
      issn = {1435-9855},
      mrclass = {83C10 (35A01 35L72 35R01 53C80)},
      mrnumber = {3082240},
      mrreviewer = {Willie W. Wong},
      doi = {10.4171/JEMS/400},
      zblnumber = {1280.35154},
      }
  • [sbvm] Go to document R. Melrose, A. Sá Barreto, and A. Vasy, "Asymptotics of solutions of the wave equation on de Sitter-Schwarzschild space," Comm. Partial Differential Equations, vol. 39, iss. 3, pp. 512-529, 2014.
    @article{sbvm, mrkey = {3169793},
      author = {Melrose, Richard and S{á} Barreto, Ant{ô}nio and Vasy, Andr{á}s},
      title = {Asymptotics of solutions of the wave equation on de {S}itter-{S}chwarzschild space},
      journal = {Comm. Partial Differential Equations},
      fjournal = {Communications in Partial Differential Equations},
      volume = {39},
      year = {2014},
      number = {3},
      pages = {512--529},
      issn = {0360-5302},
      mrclass = {58J45 (35L05 35P25 35R01 83C30)},
      mrnumber = {3169793},
      mrreviewer = {Olga S. Rozanova},
      doi = {10.1080/03605302.2013.866958},
      zblnumber = {1286.35145},
      }
  • [mora2] Go to document C. S. Morawetz, "Time decay for the nonlinear Klein-Gordon equations," Proc. Roy. Soc. Ser. A, vol. 306, pp. 291-296, 1968.
    @article{mora2, mrkey = {0234136},
      author = {Morawetz, Cathleen S.},
      title = {Time decay for the nonlinear {K}lein-{G}ordon equations},
      journal = {Proc. Roy. Soc. Ser. A},
      volume = {306},
      year = {1968},
      pages = {291--296},
      mrclass = {35.79 (81.00)},
      mrnumber = {0234136},
      mrreviewer = {R. C. Gilbert},
      doi = {10.1098/rspa.1968.0151},
      zblnumber = {0157.41502},
      }
  • [moschidis] G. Moschidis, The $r^{p}$-weighted energy method of Dafermos and Rodnianski in general asymptotically flat spacetimes and applications, 2015.
    @misc{moschidis,
      author = {Moschidis, G.},
      title = {The $r^{p}$-weighted energy method of {D}afermos and {R}odnianski in general asymptotically flat spacetimes and applications},
      arxiv={1509.08489},
      year={2015},
      }
  • [murataetal] Go to document K. Murata, H. S. Reall, and N. Tanahashi, "What happens at the horizon(s) of an extreme black hole?," Classical Quantum Gravity, vol. 30, iss. 23, p. 235007, 2013.
    @article{murataetal, mrkey = {3129282},
      author = {Murata, Keiju and Reall, Harvey S. and Tanahashi, Norihiro},
      title = {What happens at the horizon(s) of an extreme black hole?},
      journal = {Classical Quantum Gravity},
      fjournal = {Classical and Quantum Gravity},
      volume = {30},
      year = {2013},
      number = {23},
      pages = {235007, 36},
      issn = {0264-9381},
      mrclass = {83C57 (83-08)},
      mrnumber = {3129282},
      doi = {10.1088/0264-9381/30/23/235007},
      zblnumber = {1284.83093},
      }
  • [Ralston] Go to document J. V. Ralston, "Solutions of the wave equation with localized energy," Comm. Pure Appl. Math., vol. 22, pp. 807-823, 1969.
    @article{Ralston, mrkey = {0254433},
      author = {Ralston, James V.},
      title = {Solutions of the wave equation with localized energy},
      journal = {Comm. Pure Appl. Math.},
      fjournal = {Communications on Pure and Applied Mathematics},
      volume = {22},
      year = {1969},
      pages = {807--823},
      issn = {0010-3640},
      mrclass = {35.76},
      mrnumber = {0254433},
      mrreviewer = {Teruo Ikebe},
      doi = {10.1002/cpa.3160220605},
      zblnumber = {0209.40402},
      }
  • [Sbierski] Go to document J. Sbierski, "Characterisation of the energy of Gaussian beams on Lorentzian manifolds: with applications to black hole spacetimes," Anal. PDE, vol. 8, iss. 6, pp. 1379-1420, 2015.
    @article{Sbierski, mrkey = {3397001},
      author = {Sbierski, Jan},
      title = {Characterisation of the energy of {G}aussian beams on {L}orentzian manifolds: with applications to black hole spacetimes},
      journal = {Anal. PDE},
      fjournal = {Analysis \& PDE},
      volume = {8},
      year = {2015},
      number = {6},
      pages = {1379--1420},
      issn = {2157-5045},
      mrclass = {58J45 (35R01 83C57)},
      mrnumber = {3397001},
      doi = {10.2140/apde.2015.8.1379},
      zblnumber = {06488452},
      }
  • [schlue] Go to document V. Schlue, "Decay of linear waves on higher-dimensional Schwarzschild black holes," Anal. PDE, vol. 6, iss. 3, pp. 515-600, 2013.
    @article{schlue, mrkey = {3080190},
      author = {Schlue, Volker},
      title = {Decay of linear waves on higher-dimensional {S}chwarzschild black holes},
      journal = {Anal. PDE},
      fjournal = {Analysis \& PDE},
      volume = {6},
      year = {2013},
      number = {3},
      pages = {515--600},
      issn = {2157-5045},
      mrclass = {83C57 (35L10 35Q75 35R01 58J45 83E15)},
      mrnumber = {3080190},
      mrreviewer = {H{é}ctor H. Hern{á}ndez},
      doi = {10.2140/apde.2013.6.515},
      zblnumber = {1326.35382},
      }
  • [schlue2] Go to document V. Schlue, "Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies," Comm. Math. Phys., vol. 334, iss. 2, pp. 977-1023, 2015.
    @article{schlue2, mrkey = {3306609},
      author = {Schlue, Volker},
      title = {Global results for linear waves on expanding {K}err and {S}chwarzschild de {S}itter cosmologies},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {334},
      year = {2015},
      number = {2},
      pages = {977--1023},
      issn = {0010-3616},
      mrclass = {83C35 (83F05)},
      mrnumber = {3306609},
      mrreviewer = {Robyn Arianrhod},
      doi = {10.1007/s00220-014-2154-2},
      zblnumber = {1310.83043},
      }
  • [shlapRot] Go to document Y. Shlapentokh-Rothman, "Quantitative mode stability for the wave equation on the Kerr spacetime," Ann. Henri Poincaré, vol. 16, iss. 1, pp. 289-345, 2015.
    @article{shlapRot, mrkey = {3296646},
      author = {Shlapentokh-Rothman, Yakov},
      title = {Quantitative mode stability for the wave equation on the {K}err spacetime},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {16},
      year = {2015},
      number = {1},
      pages = {289--345},
      issn = {1424-0637},
      mrclass = {58J45 (35L10 35R01 83C05)},
      mrnumber = {3296646},
      mrreviewer = {Marcel Dossa},
      doi = {10.1007/s00023-014-0315-7},
      zblnumber = {1308.83104},
      }
  • [shlapRot2] Go to document Y. Shlapentokh-Rothman, "Exponentially growing finite energy solutions for the Klein-Gordon equation on sub-extremal Kerr spacetimes," Comm. Math. Phys., vol. 329, iss. 3, pp. 859-891, 2014.
    @article{shlapRot2, mrkey = {3212872},
      author = {Shlapentokh-Rothman, Yakov},
      title = {Exponentially growing finite energy solutions for the {K}lein-{G}ordon equation on sub-extremal {K}err spacetimes},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {329},
      year = {2014},
      number = {3},
      pages = {859--891},
      issn = {0010-3616},
      mrclass = {35R01 (35L10 35Q75 35Q76)},
      mrnumber = {3212872},
      mrreviewer = {Marcelo Mendes Disconzi},
      doi = {10.1007/s00220-014-2033-x},
      zblnumber = {1294.83062},
      }
  • [Starobinsky] Go to document A. Starobinsky, "Amplification of waves during reflection from a black hole," Soviet Phys. JETP, vol. 37, pp. 28-32, 1973.
    @article{Starobinsky,
      author = {Starobinsky, A.},
      title = {Amplification of waves during reflection from a black hole},
      journal = {Soviet Phys. JETP},
      volume = {37},
      year = {1973},
      pages = {28--32},
      url = {http://www.jetp.ac.ru/cgi-bin/dn/e_037_01_0028.pdf},
      }
  • [tatar] Go to document D. Tataru, "Local decay of waves on asymptotically flat stationary space-times," Amer. J. Math., vol. 135, iss. 2, pp. 361-401, 2013.
    @article{tatar, mrkey = {3038715},
      author = {Tataru, Daniel},
      title = {Local decay of waves on asymptotically flat stationary space-times},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {135},
      year = {2013},
      number = {2},
      pages = {361--401},
      issn = {0002-9327},
      mrclass = {35L15 (35B40 35R01 83C10)},
      mrnumber = {3038715},
      mrreviewer = {Calvin Tadmon},
      doi = {10.1353/ajm.2013.0012},
      zblnumber = {1266.83033},
      }
  • [tattoh] Go to document D. Tataru and M. Tohaneanu, "A local energy estimate on Kerr black hole backgrounds," Int. Math. Res. Not., vol. 2011, iss. 2, pp. 248-292, 2011.
    @article{tattoh, mrkey = {2764864},
      author = {Tataru, Daniel and Tohaneanu, Mihai},
      title = {A local energy estimate on {K}err black hole backgrounds},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices. IMRN},
      year = {2011},
      number = {2},
      pages = {248--292},
      issn = {1073-7928},
      mrclass = {58J45 (35Q75 83C57)},
      mrnumber = {2764864},
      mrreviewer = {Alan D. Rendall},
      doi = {10.1093/imrn/rnq069},
      volume = {2011},
      zblnumber = {1209.83028},
      }
  • [vasy] Go to document A. Vasy, "Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)," Invent. Math., vol. 194, iss. 2, pp. 381-513, 2013.
    @article{vasy, mrkey = {3117526},
      author = {Vasy, Andr{á}s},
      title = {Microlocal analysis of asymptotically hyperbolic and {K}err-de {S}itter spaces (with an appendix by {S}emyon {D}yatlov)},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {194},
      year = {2013},
      number = {2},
      pages = {381--513},
      issn = {0020-9910},
      mrclass = {58J47 (35R01 83C57)},
      mrnumber = {3117526},
      mrreviewer = {Davide Batic},
      doi = {10.1007/s00222-012-0446-8},
      zblnumber = {1315.35015},
      }
  • [whiting] Go to document B. F. Whiting, "Mode stability of the Kerr black hole," J. Math. Phys., vol. 30, iss. 6, pp. 1301-1305, 1989.
    @article{whiting, mrkey = {0995773},
      author = {Whiting, Bernard F.},
      title = {Mode stability of the {K}err black hole},
      journal = {J. Math. Phys.},
      fjournal = {Journal of Mathematical Physics},
      volume = {30},
      year = {1989},
      number = {6},
      pages = {1301--1305},
      issn = {0022-2488},
      coden = {JMAPAQ},
      mrclass = {83C25 (83C15)},
      mrnumber = {0995773},
      doi = {10.1063/1.528308},
      zblnumber = {0689.53041},
      }
  • [WunschZworski] Go to document J. Wunsch and M. Zworski, "Resolvent estimates for normally hyperbolic trapped sets," Ann. Henri Poincaré, vol. 12, iss. 7, pp. 1349-1385, 2011.
    @article{WunschZworski, mrkey = {2846671},
      author = {Wunsch, Jared and Zworski, Maciej},
      title = {Resolvent estimates for normally hyperbolic trapped sets},
      journal = {Ann. Henri Poincaré},
      fjournal = {Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics},
      volume = {12},
      year = {2011},
      number = {7},
      pages = {1349--1385},
      issn = {1424-0637},
      mrclass = {58J40 (35S05 47A40 83C57)},
      mrnumber = {2846671},
      mrreviewer = {Sandro Coriasco},
      doi = {10.1007/s00023-011-0108-1},
      zblnumber = {1228.81170},
      }
  • [Yang] Go to document S. Yang, "Global solutions of nonlinear wave equations in time dependent inhomogeneous media," Arch. Ration. Mech. Anal., vol. 209, iss. 2, pp. 683-728, 2013.
    @article{Yang, mrkey = {3056620},
      author = {Yang, Shiwu},
      title = {Global solutions of nonlinear wave equations in time dependent inhomogeneous media},
      journal = {Arch. Ration. Mech. Anal.},
      fjournal = {Archive for Rational Mechanics and Analysis},
      volume = {209},
      year = {2013},
      number = {2},
      pages = {683--728},
      issn = {0003-9527},
      mrclass = {35L72 (35L15)},
      mrnumber = {3056620},
      mrreviewer = {Calvin Tadmon},
      doi = {10.1007/s00205-013-0631-y},
      zblnumber = {1284.35290},
      }
  • [shiwu2] Go to document S. Yang, "Global stability of solutions to nonlinear wave equations," Selecta Math., vol. 21, iss. 3, pp. 833-881, 2015.
    @article{shiwu2, mrkey = {3366921},
      author = {Yang, Shiwu},
      title = {Global stability of solutions to nonlinear wave equations},
      journal = {Selecta Math.},
      fjournal = {Selecta Mathematica. New Series},
      volume = {21},
      year = {2015},
      number = {3},
      pages = {833--881},
      issn = {1022-1824},
      mrclass = {35L72 (35B35 35L15)},
      mrnumber = {3366921},
      doi = {10.1007/s00029-014-0165-7},
      zblnumber = {1326.35045},
      }
  • [shiwu] S. Yang, On the quasilinear wave equations in time dependent inhomogeneous media, 2013.
    @misc{shiwu,
      author = {Yang, Shiwu},
      title = {On the quasilinear wave equations in time dependent inhomogeneous media},
      arxiv = {1312.7246},
      year = {2013},
      }

Authors

Mihalis Dafermos

University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Cambridge, United Kingdom and Princeton University, Princeton, NJ

Igor Rodnianski

Princeton University, Princeton, NJ

Yakov Shlapentokh-Rothman

Princeton University, Princeton, NJ