The asymptotic behavior of solutions is a significant subject in the theory of wave equations. In this paper we are concerned with the asymptotic behavior of the unique global solution to the Cauchy problem for one-dimension quasilinear wave equations with null conditions. By applying the small-data-global-existence result and exploiting the strength of weights, we not only provide sharper convergence from the quasilinear case to the linear case but also study the rigidity aspect of the scattering problem for quasilinear waves.

中文翻译：

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一维拟线性波动方程整体解的渐近行为

解的渐近行为是波动方程理论中的重要课题。在本文中，我们关注具有零条件的一维拟线性波动方程的柯西问题的唯一整体解的渐近行为。通过应用小数据全局存在的结果并利用权重的强度，我们不仅提供了从拟线性情况到线性情况的更清晰的收敛，而且还研究了拟线性波散射问题的刚度方面。