Large data global well-posedness for a one-dimensional quasilinear wave equation

TL;DR

This paper establishes the global well-posedness of large initial data solutions for a one-dimensional quasilinear wave equation, addressing an open problem in the mathematical analysis of nonlinear wave phenomena.

Contribution

It proves global existence of smooth solutions for large data in a specific quasilinear wave equation, using a novel comparison principle for Riemann variables.

Findings

Global well-posedness for large initial data is achieved.

A new comparison principle for Riemann variables is developed.

Partial resolution of an open problem by Glassey, Hunter, and Zheng.

Abstract

In this paper, we prove global well-posedness with large initial data for the one-dimensional quasilinear wave equation $$ u_{tt}=c(u)^2u_{xx}, \qquad (t,x)\in (0,T)\times\R, $$ where \(c\) is a positive, bounded, monotonically increasing function with bounded derivative. This result gives a partial resolution of an open problem posed by Glassey, Hunter and Zheng on the global existence of smooth solutions to this equation for large initial data. Our proof is based on upper and lower estimates for the Riemann variables via a new comparison principle.