The Strauss conjecture on asymptotically flat space-times

TL;DR

This paper proves the Strauss conjecture on asymptotically flat space-times, including black hole backgrounds, using weighted Strichartz estimates and localized energy assumptions, extending results to various perturbations of Minkowski space.

Contribution

It establishes the Strauss conjecture on asymptotically flat manifolds under localized energy estimates, covering black hole backgrounds and perturbations of Minkowski space.

Findings

Validates Strauss conjecture in 3 and 4 dimensions for black hole backgrounds.

Introduces weighted Strichartz estimates near infinity.

Demonstrates applicability to small perturbations of Minkowski space.

Abstract

By assuming a certain localized energy estimate, we prove the existence portion of the Strauss conjecture on asymptotically flat manifolds, possibly exterior to a compact domain, when the spatial dimension is 3 or 4. In particular, this result applies to the 3 and 4-dimensional Schwarzschild and Kerr (with small angular momentum) black hole backgrounds, long range asymptotically Euclidean spaces, and small time-dependent asymptotically flat perturbations of Minkowski space-time. We also permit lower order perturbations of the wave operator. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and the assumed localized energy estimate, which is used in the remaining compact set.