Global Existence And Asymptotic Behavior Of Classical Solutions Of Quasilinear Hyperbolic Systems

This thesis mainly investigates the global existence and asymptotic behavior of clas-sical solutions to quasilinear hyperbolic systems. Moreover, we apply our conclusions to study some equations arising from physics and geometry. Especially, we investigate the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n) and the system of motion of an elastic string. Finally, we study the Einstein's filed equation with the energy-momentum tensors for perfect fluid. We use the Ori's model to construct a new exact solution——time machine solution. This thesis is organized as follows:In chapter 1, we recall the history and some important results on the studies of first order quasilinear hyperbolic systems and state the main results obtained in this thesis.In chapter 2, we study the Goursat problem of reducible reducible quasilinear hyper-bolic systems. Based on the global existence of classical solutions to the systems, under some suitable assumptions, we prove that the solution approaches a combination of Lips-chitz continuous and piecewise C1 traveling wave solutions. As an application, we apply the result to the equation for time-like extremal surfaces in the Minkowski space-time R1+(1+n)In chapter 3, we investigate global existence and asymptotic behavior for the classical solution to the generalized Goursat problem. Firstly, we simply recall the present situation of the study on generalized Goursat problem. Under some degenerate assumptions, we use F.John's formula to gain the global existence of classical solutions. Then we prove that the solution tends to a combination of traveling wave solution which are Lipschitz continuous and piecewise C1. Finally, we give an example about the system of motion of an elastic string.In chapter 4, by the matric of Ori's model, we construct the time machine solution to the Einstein's field equations with the energy-momentum tensors for perfect fluid. In fact, this kind of time machine is more simple than Ori's.