The Well-posedness Of Boundary Problem For Nonlinear Wave Equations And Elastodynamic System In 3D

This paper mainly studies the well-posedness of boundary problem for nonlinear wave equations and elastodynamic system in 3D.Firstly,we prove the existence and regularity of solutions for Dirichlet problem to hyperbolic system of second order outside a domain.Secondly,we consider the local existence of solutions for third boundary problem of nonlinear elastodynamic system outside a domain.Thirdly,we study the almost global existence of Neumann problem for nonlinear elastodynamic system outside a star-shaped domain.Lastly,we derive the almost global existence for Neumann problem of quasilinear wave equations outside star-shaped domains in 3D.This paper is composed of five sections.In section 1,we introduce the physical background and the research status of nonlinear wave equations and elastodynamic system,and outline the main content of this paper.In section 2,we study the Dirichlet problem to a kind of hyperbolic system of second order outside a domain,which can be applied to elastokinetics.Firstly,the existence of solutions to this problem is proved via the semigroup theory.Secondly,the regularity of solutions is given by iteration.In section 3,we consider the local existence of solutions for exterior problem of nonlinear elastodynamic system with third boundary condition.In order to prove this problem,the existence of exterior problem for linear variable coefficient hyperbolic system of second order with the third boundary condition in Sobolev space is proved via the method of linear evolution operators and integro-differential equations.In section 4,we prove the almost global existence of nonlinear elastodynamic system outside a star-shaped domain with Neumann condition and give a lower bound for the lifespan of the solutions.The key steps of the proof are point-wise estimates and weightedL~2 estimates.In section 5,we discuss the Neumann problem for quasilinear wave equations outside star-shaped domains in 3D.The almost global existence of solutions to this problem is proved and a lower bound for the lifespan of the solutions is given.