The lifespan for 3-dimensional quasilinear wave equations in exterior domains

TL;DR

This paper establishes long-time existence results for 3D quasilinear wave equations in exterior domains with small initial data, allowing for complex geometries including trapped rays, by developing new energy estimates.

Contribution

It extends lifespan results to more general exterior domains with weaker geometric conditions, using innovative energy estimates involving the scaling vector field.

Findings

Lifespan bounds similar to boundaryless case are achieved in exterior domains.

Long-time existence is proved even with trapped rays in the domain.

Energy estimates involving the scaling vector field are effectively utilized.

Abstract

This article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.