Asymptotic Behavior of Global Classical Solutions to the Cauchy Problem on a Semi-Bounded Initial Axis for Quasilinear Hyperbolic Systems

Abstract:

      In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that, when $t$ tends to the infinity, the solution approaches a combination of $C^1$ travelling wave solutions with the algebraic rate $(1 t)^{-\mu}$, provided that the initial data decay with the rate $(1 x)^{-(1 \mu)}$ (resp. $(1-x)^{-(1 \mu)}$) as $x$ tends to $ \infty$ (resp. $-\infty$), where $\mu$ is a positive constant.