Elastic waves in exterior domains, Part II: Global existence with a null structure

In this article, we prove that solutions to a problem in nonlinear elasticity corresponding to small initial displacements exist globally in the exterior of a nontrapping obstacle. The medium is assumed to be homogeneous, isotropic, and hyperelastic, and the nonlinearity is assumed to satisfy a null condition. The techniques contained herein would allow for more complicated geometries provided that there is a sufficient decay of local energy for the linearized problem.