The Glassey conjecture on asymptotically flat manifolds

TL;DR

This paper proves the Glassey conjecture for 3D asymptotically flat manifolds with small perturbations of flat metrics and extends results to radial cases in higher dimensions, using energy and Sobolev estimates.

Contribution

It verifies the Glassey conjecture on asymptotically flat manifolds and extends the results to radial higher-dimensional cases with advanced analytical techniques.

Findings

Verification of the Glassey conjecture in 3D asymptotically flat manifolds.

Extension of the conjecture verification to radial higher-dimensional cases.

Development of estimates using local energy, KSS, and weighted Sobolev techniques.

Abstract

We verify the 3-dimensional Glassey conjecture on asymptotically flat manifolds $(R^{1+3}, g)$, where the metric $g$ is certain small space-time perturbation of the flat metric, as well as the nontrapping asymptotically Euclidean manifolds. Moreover, for radial asymptotically flat manifolds $(R^{1+n}, g)$ with $n\ge 3$, we verify the Glassey conjecture in the radial case. High dimensional wave equations with higher regularity are also discussed. The main idea is to exploit local energy and KSS estimates with variable coefficients, together with the weighted Sobolev estimates including trace estimates.