  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.
