The Klein-Gordon Equation in Anti-de Sitter Cosmology

Abstract

This paper deals with the Klein-Gordon equation on the Poincar\'e chart of the 5-dimensional Anti-de Sitter universe. When the mass $\mu$ 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 $\mid t\mid^{-{3}{2}}$. We investigate the case $\mu={\nu^2-1}{2}$, $\nu\in\NN^*$, which encompasses the gravitational fluctuations, $\nu=4$, and the electromagnetic waves, $\nu=2$. The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as $\mid t\mid^{-2-\sqrt{\mu+{1}{4}}}$, and we get global $L^p$ estimates of Strichartz type. When $\nu$ 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 adress 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 $L^2-L^{\infty}$ estimates in suitable weighted Sobolev spaces.