Lifespan of Classical Solutions to One-Dimensional Quasilinear Wave Equations

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.