The Glassey conjecture for nontrapping obstacles

TL;DR

This paper proves the three-dimensional Glassey conjecture for exterior domains with asymptotically Euclidean metrics under local energy assumptions, covering various obstacle configurations and manifold types.

Contribution

It verifies the Glassey conjecture in exterior domains with nontrapping obstacles and Euclidean-like metrics, extending previous results to broader geometric settings.

Findings

Verification of the Glassey conjecture for exterior domains with nontrapping obstacles.

Extension of results to higher dimensions and various obstacle geometries.

Validation of local energy assumptions in multiple important cases.

Abstract

We verify the three-dimensional Glassey conjecture for exterior domain (M, g), where the metric g is asymptotically Euclidean, provided that certain local energy assumption is satisfied. The radial Glassey conjecture exterior to a ball is also verified for dimension three or higher. The local energy assumption is satisfied for many important cases, including exterior domain with nontrapping obstacles and flat metric, exterior domain with star-shaped obstacles and small asymptotically Euclidean metric, as well as the nontrapping asymptotically Euclidean manifolds (R^n, g).