TY - JOUR

T1 - Small-amplitude nonlinear waves on a black hole background

AU - Dafermos, Mihalis

AU - Rodnianski, Igor

N1 - Funding Information:
M.D. is supported in part by NSF grant DMS-0302748. I.R. is supported in part by NSF grant DMS-0406627. Part of this research was conducted when I.R. was a long-term prize fellow of the Clay Mathematics Institute.

PY - 2005/9

Y1 - 2005/9

N2 - Let G(x) be a C0 function such that G(x) ≤ K x p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordström metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max{ν, 1})-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.

AB - Let G(x) be a C0 function such that G(x) ≤ K x p for x ≤ c, for constants K, c > 0. We consider spherically symmetric solutions of □gφ = G(φ) where g is a Schwarzschild or more generally a Reissner-Nordström metric, and such that φ and ∇φ are compactly supported on a complete Cauchy surface. It is proven that for p > 4, such solutions do not blow up in the domain of outer communications, provided the initial data are small. Moreover, φ ≤ C(max{ν, 1})-1, where ν denotes an Eddington-Finkelstein advanced time coordinate.

KW - Black hole

KW - Non-linear wave equation

KW - Reissner-Nordström

KW - Schwarzschild

UR - http://www.scopus.com/inward/record.url?scp=24044503074&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=24044503074&partnerID=8YFLogxK

U2 - 10.1016/j.matpur.2005.04.004

DO - 10.1016/j.matpur.2005.04.004

M3 - Article

AN - SCOPUS:24044503074

SN - 0021-7824

VL - 84

SP - 1147

EP - 1172

JO - Journal des Mathematiques Pures et Appliquees

JF - Journal des Mathematiques Pures et Appliquees

IS - 9

ER -