In this article, we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t -3 local uniform decay rate (Price's law, Price (1972)) for linear waves. As a corollary, we also prove Price's law for certain small perturbations of the Kerr metric. This result was previously established by the second author in (Tataru ) on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.
|Number of pages||34|
|Journal||Advances in Mathematics|
|State||Published - Jun 20 2012|
Bibliographical noteFunding Information:
The first author was partially supported by NSF grant DMS0800678 . The second author was partially supported by NSF grant DMS0354539 and by the Miller Foundation .
- Pointwise decay
- Wave equation
ASJC Scopus subject areas
- Mathematics (all)