Lifespan of Classical Solutions to One-Dimensional Quasilinear Wave Equations

TL;DR

This paper investigates the lifespan bounds of classical solutions to a one-dimensional quasilinear wave equation, revealing algebraic and exponential lifespan extensions depending on initial data smallness and the flatness of the wave speed function.

Contribution

It establishes new bounds on solution lifespan for the equation, especially when the wave speed function is flat at the origin, using characteristic and Riemann invariant methods.

Findings

Lifespan extends algebraically with small initial data when c(θ) tends to zero.

Lifespan extends exponentially when c(θ) is flat at the origin.

Method based on Lax's characteristics and Riemann invariants.

Abstract

In this paper, we consider the upper and lower bounds of the lifespan of classical solutions of the Cauchy problem for the one-dimensional quasilinear wave equation $u_{tt}-c(u_x)^2u_{xx}=0$ where the derivative of $c(\theta)$ tends to $0$ near the origin. In particular, our result shows that the lifespan of the solution extends algebraically depending on the smallness of the initial data. Furthermore, we also show that when $c(\theta)$ is flat at the origin, the lifespan extends exponentially depending on the smallness of the initial data. Our proof is based on the method of Lax's characteristics and Riemann invariants.