Global existence for initial-boundary value problems of one-dimension quasilinear wave equations with null conditions

We consider the initial-boundary value problems on ${\mathbb{R}}^{+}\times {\mathbb{R}}^{+}$ for one-dimension systems of quasilinear wave equations with null conditions. We show that for homogeneous Dirichlet boundary values and sufficiently small initial data, classical solutions always globally exist. The key innovation in the proof is a new framework of bootstrap argument via coupled high-low order energy estimates.