Local existence of solutions for the Neumann problem of the nonlinear elastodynamic system outside a domain

In this paper, we study the mixed initial–boundary value problem of Neumann type for the nonlinear elastic wave equation outside a domain. The local existence of solutions to this problem is proved by iteration. To get this result, we prove the existence of solutions for the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.