Decay of linear waves on higher-dimensional Schwarzschild black holes

Abstract

We consider solutions to the linear wave equation on higher dimensional Schwarzschild black hole spacetimes and prove robust nondegenerate energy decay estimates that are in principle required in a nonlinear stability problem. More precisely, it is shown that for solutions to the wave equation ${\square }_g\varphi =0$ on the domain of outer communications of the Schwarzschild spacetime manifold $\left({\mathcal{ℳ}}_m^n,g\right)$ (where $n\ge 3$ is the spatial dimension, and $m>0$ is the mass of the black hole) the associated energy flux $E\left[\varphi \right]\left({\Sigma }_{\tau }\right)$ through a foliation of hypersurfaces ${\Sigma }_{\tau }$ (terminating at future null infinity and to the future of the bifurcation sphere) decays, $E\left[\varphi \right]\left({\Sigma }_{\tau }\right)\le CD∕{\tau }^2$, where $C$ is a constant depending on $n$ and $m$, and $D<\infty$ is a suitable higher-order initial energy on ${\Sigma }_0$; moreover we improve the decay rate for the first-order energy to $E\left[{\partial }_t\varphi \right]\left({\Sigma }_{\tau }^R\right)\le C{D}_{\delta }∕{\tau }^{4-2\delta }$ for any $\delta >0$, where ${\Sigma }_{\tau }^R$ denotes the hypersurface ${\Sigma }_{\tau }$ truncated at an arbitrarily large fixed radius $R<\infty$ provided the higher-order energy $D_{\delta }$ on ${\Sigma }_0$ is finite. We conclude our paper by interpolating between these two results to obtain the pointwise estimate $|\varphi {|}_{{\Sigma }_{\tau }^R}\le C{D}_{\delta }^{\prime }∕{\tau }^{\frac{3}{2}-\delta }$. In this work we follow the new physical-space approach to decay for the wave equation of Dafermos and Rodnianski (2010).