Some Problems On Partial Differential Equations In Elastodynamics

The study on partial differential equations in elastodynamics is an important subject in both theory and applications. This thesis deals with some problems of partial differential equations related elastodynamics. The main contribution of the thesis is to prove the global existence of solution to second order quasilinear hyperbolic systems with the linear elastodynamic system as main part. Another contribution of this thesis is to obtain the 2-dimensional model for the linearly elastodynamic shallow shells with variable thickness by means of the asymptotic method.At present, there have been some important works on the existence of solutions to the nonlinear elastodynamic system. In 1988. F.John [12] 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. By applying energy estimates and Klainerman-Sobolev inequality, S.Klainerman and T.Sideris [32] showed the same result in 1996. In 2000, R.Agemi [2] and T.Sideris [48] introduced the null conditions for the nonlinear elastodynamic system in different ways respectively and proved independently the global existence of classical solutions to the initial value problem with small initial data. J.Xin and T.Qin [60] proved the almost global existence of solutions to the initial-boundary value problem of nonlinear elastic waves in the exterior of a star-shaped domain. Recently, J.Metcalfe and B.Thamoses [44] proved the global existence of solutions to the initial-boundary value problem of nonlinear elastic waves satisfying null condition in exterior domain.As regards elastic shallow shells, there have been some works on the static shallow shells. Ciarlet and Miara [9] justified the two-dimensional model of shallow shells with constant thickness in Cartesian coordinates. Afterwards Busse, Ciarlet and Miara [4] discussed the same problem in curvilinear coordinates. Then Sabu [47] identified the two-dimensional model of a shallow shell with variable thickness. There are a few works in the dynamic case. Xiao [58, 59] presented the two-dimensional dynamic models of both linear membranes with restrictions on the boundary and flexural shells. And then Ye [61] improved Xiao's results on membrane shells and extended it to generalized dynamic membrane shells.Now. we state our results:(1) We prove the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as main part and the nonlinear terms depending on the first power of u.(2) We prove the global existence of classical solutions to a kind of quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as main part in divergent form and the nonlinear terms depending on the second power of u.(3) We discuss the Cauchy problem of a kind of quasilinear hyperbolic systems, with the linear elastodynamic system as main part and the nonlinear terms conclude terms of the first order derivatives of u. A system of new null conditions are given, by which we show the global existence of the problem.(4) By applying the asymptotic method, we identify the two-dimensional model of a elastodynamic shallow shell with variable thickness.