The Exterior Problem For The Nonlinear Elastodynamic System

In this paper, we deal with the exterior problem for the nonlinear elastodynamic system. The nonlinear elastodynamic system is a very important model from both theoretical and practical points of view. Many famous mathematicians have worked in this area, but most results are about the Cauchy problem. In 1988, F. John [10] proved the almost global existence of solutions to the initial value problem for the nonlinear elastodynamic system by applying the estimates on the fundamental solution of the linear elastic operator. In 1996, S. Klainerman and T. C. Sideris [23] showed the same result by applying energy estimates and Klainerman-Sobolev inequalities. Recently, R. Agemi [1] and T. C. Sideris [33] introduced the null conditions for the nonlinear elastodynamic system in different ways respectively and proved the global existence of classical solutions to the initial value problem with small initial data.Now we state our results.(1) We prove the local existence of classical solutions to the exterior problem for the nonlinear elastodynamic system. 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.(2) We prove the almost global existence of classical solutions to the Cauchy problem for the nonlinear elastodynamic system and give the lower bound of the lifespan of solutions. Although this result is not new, our method and derived estimates are different. These estimates will be used in the study of the exterior problem.(3) We deal with the Dirichlet initial-boundary problem for the nonlinear elastodynamic system outside of a star-shaped domain. We prove the almost global existence of solutions to this problem with small initial data and give the lower bound of the lifespan of solutions. This is the main result in our paper.(4) We derive the same null condition as in [1] for the nonlinear elastodynamic system in a simpler way and prove the equivalence of the null conditions introduced in both [1] and [33].