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

TL;DR

This paper proves that classical solutions to certain one-dimensional quasilinear wave equations with null conditions exist globally when starting from small initial data and homogeneous Dirichlet boundary conditions, using a novel bootstrap energy estimate approach.

Contribution

It introduces a new bootstrap framework with coupled high-low order energy estimates for proving global existence of solutions.

Findings

Global existence of solutions under small initial data

Effective bootstrap energy estimate method

Applicable to one-dimensional quasilinear wave equations with null conditions

Abstract

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.