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

DOI：10.3770/j.issn:1000-341X.2010.01.004

 作者 单位 韩伟伟 东华大学应用数学系 上海 201620 复旦大学数学科学学院, 上海 200433

在本文中,我们考察拟线性双曲组在半有界初始轴上的柯西问题整体经典解的渐近性态.在整体经典解存在性结果的基础上, 我们证明了当时间$t\rightarrow \infty$时, 只要初始数据当\ $x\rightarrow \infty$(相应地,$x\rightarrow -\infty$)时以速率$(1 x)^{-(1 \mu)}$(相应地,$(1-x)^{-(1 \mu)}$)衰减, 柯西问题的经典解就以速率$(1 t)^{-\mu}$逼近于$C^1$行波解的组合, 其中$\mu$是一个正常数.

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.