On one-dimension quasilinear wave equations with null conditions

TL;DR

This paper proves that small initial data lead to global classical solutions for one-dimensional quasilinear wave equations with null conditions, extending previous semilinear results and showing solutions are asymptotically free.

Contribution

It extends the global existence results from semilinear to quasilinear wave equations with null conditions in one dimension, using advanced energy estimates.

Findings

Global classical solutions exist for small initial data.

Solutions are asymptotically free in the energy sense.

Employs weighted energy estimates and bootstrap methods.

Abstract

In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu's seminal work [G. Luli, S. Yang, P. Yu, On one-dimension semi-linear wave equations with null conditions, Adv. Math.329 (2018) 174-188] from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense. In order to achieve these goals, we will employ Luli, Yang and Yu's weighted energy estimates with positive weights, introduce some space-time weighted energy estimates and pay some special attentions to the highest order energies, then use some suitable bootstrap process to close the argument.