Nonlinear Klein-Gordon equation on Schwarzschild-like metrics
Using a method similar to [\textit{F. Cagnac} and \textit{Y. Choquet-Bruhat}, J. Math. Pures Appl., IX. Sér. 63, 377-390 (1984; Zbl 0554.58049)] we solve the global Cauchy problem for \[ \square u+ m^2 u+ \lambda|u|^2 u= 0,\quad \lambda\geq 0, \] on a general Schwarzschild-type metric which covers all the usual cases of spherical black-holes. Outside the black-hole, the curves \(x= cst.\) are not uniformly time-like and we need to prove a Sobolev embedding result before getting the local existence of solutions for arbitrarily large initial data with minimum regularity. Global existence is then given by the conservation of energy. We study the asymptotic behavior of the fields at the horizon using a Kruskal-Szekeres coordinate transformation. In the neighbourhood of infinity, we only consider the behavior of massless fields and this is done by means of a Penrose conformal compactification.
scientific article; zbMATH DE number 891034