On Initial-Boundary Value Problems for Some One Dimensional Quasilinear Wave Equations: Global Existence, Scattering and Rigidity

We consider the initial-boundary value problem on R+×R+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^{+}\times \mathbb {R}^{+}$$\end{document} for some one dimensional systems of quasilinear wave equations with null conditions. We first prove that for homogeneous Dirichlet boundary values and sufficiently small initial data, classical solutions always globally exist. Then we show that the global solution will scatter, i.e., it will converge to some solution of one dimensional linear wave equations as time tends to infinity, in the energy sense. Finally we prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically.