The Klein–Gordon equation in the Anti-de Sitter cosmology

Abstract

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than − 1 4 , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as | t | − 3 2 . We investigate the case μ = ν 2 − 1 2 , ν ∈ N ⁎ , which encompasses the gravitational fluctuations, ν = 4 , and the electromagnetic waves, ν = 2 . The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as | t | − 2 − μ + 1 4 , and we get global L p estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L 2 − L ∞ estimates in suitable weighted Sobolev spaces.