Inhomogeneous Quasilinear Hyperbolic System With Small The Cauchy Problem Of The Initial Value Of Total Variation

In this paper, we study the global existence, uniqueness and the asymptotic behav-ior of classical solution to the Cauchy problem for quasilinear hyperbolic system with inhomogeneous term. The system will have strictly hyperbolic and linearly degenerate characteristic fields. This paper is divided into two parts. In the first part, we will intro-duce several basic concepts and prerequisites, like strictly hyperbolic, linearly degenerate, normalized coordinates and match condition for inhomogeneous term. By the existence and uniqueness of local C1solution to the Cauchy problem, to show the global existence and uniqueness of the solution it is sufficient to establish a priori estimate. We diagonal-ize the hyperbolic system firstly by John’s formula on the decomposition of waves and then establish priori estimate of the solution using L1norm estimate on cross-wave. By this way we can prove that if the initial data has sufficiently small BV norm, the Cauchy problem for quasilinear hyperbolic system with inhomogeneous term admits a unique global C1solution. Based on the existence result on the global classical solution, in the second part we will show the asymptotic behavior of the solution, which is, when the time t tends to infinity, the solution approaches to a function which is the combination of C1traveling waves. Meanwhile we will also show some continuous properties of the limit function.