Asymptotic Formula of the Lifespan of Classical Solutions to 1D Quasilinear Wave Equations

Abstract

We study the lifespan \(T^*(\varepsilon)\) of classical solutions to the Cauchy problem for the one-dimensional quasilinear wave equation $u_{tt}-(c(u)^2u_x)_x=0$, with initial data of size $\varepsilon>0$. We assume that $c'(\theta)\sim a|\theta|^{p-2}$ as $\theta \to 0$, where \(a>0\) and \(p>3\) or $p=2$. We obtain explicit asymptotic formulas for $T^*(\varepsilon)$ as $\varepsilon\downarrow0$. We treat both the case $\psi \in L^1(\mathbb R)$ and the case $\psi=\psi_1+\Braket{x}^{-\alpha}$, where \(\psi_1\in L^1(\mathbb R)\) and \(0<\alpha\le1\). Our results identify the dependence of the lifespan scale on the decay exponent, including the borderline case $\alpha=1$. The proof combines Riccati equations along characteristics with a scaling argument inspired by Lindblad.