Decay estimates for variable coefficient wave equations in exterior domains

TL;DR

The paper extends Strichartz and localized energy theory to variable-coefficient wave equations in exterior domains, covering long-range perturbations of Minkowski space and obstacles that are star-shaped or strictly convex. It constructs a localized-energy framework with frequency-localized spaces and Hardy-type inequalities, and proves global-in-time localized energy estimates for small perturbations. A conditional global Strichartz theory is then developed by combining local Strichartz estimates with the LE bounds via a parametrix argument, yielding explicit Strichartz bounds once local estimates are assumed. A notable corollary shows global Strichartz estimates for strictly convex obstacles when coefficients are time-independent nearby the obstacle, tying geometry to dispersive control. These results enable long-time dispersive analysis for nonlinear wave equations in exterior domains with variable coefficients.

Abstract

In this article we consider variable coefficient, time dependent wave equations in exterior domains. We prove localized energy estimates if the domain is star-shaped and global in time Strichartz estimates if the domain is strictly convex.

Theorems & Definitions (16)

• Proposition 1
• Definition 2
• Theorem 3
• Definition 4
• Theorem 5
• Corollary 6
• Proposition 7
• Proposition 8
• proof
• Proposition 9