Nonlinear Klein-Gordon equation on Schwarzschild-like metrics
type: Item
instance of: scholarly article
title: Nonlinear Klein-Gordon equation on Schwarzschild-like metrics
zbMATH Open document ID: 0853.35123
author: Jean-Philippe Nicolas
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1996-06-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 83C57
zbMATH DE Number: 891034
zbMATH Keywords: global existence
zbMATH Keywords: Cauchy problem
zbMATH Keywords: local existence
zbMATH Keywords: asymptotic behavior of the fields
zbMATH Keywords: Penrose conformal compactification
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 891034