Lifespan of Classical Solutions to One-Dimensional Quasilinear Wave Equations

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 $c(θ)$ 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 $c(θ)$ is flat at the origin ($c'(θ)$ and any higher order derivatives vanish 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.