## Abstract

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)[54]) 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 [65]) 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.

Original language | English |
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Pages (from-to) | 995-1028 |

Number of pages | 34 |

Journal | Advances in Mathematics |

Volume | 230 |

Issue number | 3 |

DOIs | |

State | Published - Jun 20 2012 |

### Bibliographical note

Funding 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 .

## Keywords

- Kerr
- Nonstationary
- Pointwise decay
- Wave equation

## ASJC Scopus subject areas

- Mathematics (all)