Asymptotic behavior of global solutions to one-dimension quasilinear wave equations

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.