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
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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 -