The Klein-Gordon equation in anti-de Sitter spacetime - Seminario ...

272 K. Yagdjian and A. Galstianto a constant faster than any given polynomial rate, where the decay is measured withrespect to natural future-directed advanced and retarded time coordinates.Catania and Georgiev [4] studied the Cauchy problem for the semilinear waveequation □ g φ=|φ| p in the Schwarzschild metric (3+1)-dimensional spacetime, thatis the case of Λ = 0 in 0 < M bh < 1/3 √ Λ. They established that the problem in theRegge–Wheeler coordinates is locally well-posed in H σ for any σ ∈ [1, p+1). Thenfor the special choice of the initial data they proved the blow-up of the solution in twocases: (a) p∈(1,1+ √ 2) and small initial data supported far away from the black hole;(b) p ∈(2,1+ √ 2) and large data supported near the black hole. In both cases, theyalso gave an estimate from above for the lifespan of the solution.In the present paper we focus on another limit case as M bh → 0 in the interval0 < M bh < 1/3 √ Λ, namely, we set M bh = 0 to ignore completely influence of theblack hole. Thus, the line element in the de Sitter spacetime has the form) ) −1ds 2 =−(1− r2R 2 c 2 dt 2 +(1− r2R 2 dr 2 + r 2( dθ 2 + sin 2 θdφ 2) .The Lamaître–Robertson transformation [9]r ′ =r√1−r 2 /R 2 e−ct/R , t ′ = t+ R ( )2c ln 1− r2R 2 , θ ′ = θ, φ ′ = φ,leads to the following form for the line element:ds 2 =−c 2 dt ′ 2 + e2ct ′ /R ( dr ′ 2 + r ′2 dθ ′ 2 + r ′2 sin 2 θ ′ dφ ′ 2 ) .By defining coordinates x ′ , y ′ , z ′ connected with r ′ , θ ′ , φ ′ by the usual equations connectingCartesian coordinates and polar coordinates in a Euclidean space, the line elementmay be written [9, Sec.134]ds 2 =−c 2 dt ′ 2 + e2ct ′ /R ( dx ′ 2 + dy′2 + dz′2 ) .The new coordinates r ′ , θ ′ , φ ′ , t ′ can take all values from −∞ to ∞. Here R is the“radius” of the universe. In the Robertson–Walker spacetime [3, 7] one can choosecoordinates so that the metric has the formds 2 =−dt 2 + S 2 (t)dσ 2 .In particular, the metric in the de Sitter and anti-de Sitter spacetime in the Lamaître–Robertson coordinates [9] has this form with S(t)=e t and S(t)=e −t , respectively.In the paper [16], we study the Cauchy problem for the Klein–Gordon equationin Robertson–Walker spacetime by applying the Lamaître-Robertson transformationand by employing the fundamental solutions constructed there for the Klein–Gordonoperator in Robertson–Walker spacetime, that is forS := ∂ 2 t − e−2t △+M 2 . The fundamentalsolutionE =E(x,t;x 0 ,t 0 ), that is solution ofSE = δ(x−x 0 ,t− t 0 ), with