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Asymptotic Formula of the Lifespan of Classical Solutions to 1D Quasilinear Wave Equations
Abstract
We study the lifespan T*(ε) of classical solutions to the Cauchy problem for the one-dimensional quasilinear wave equation utt−(c(u)2ux)x=0, with initial data of size ε>0. We assume that c'(θ)∼ a|θ|p−2 as θ→ 0, where a>0 and p>3 or p=2. We obtain explicit asymptotic formulas for T*(ε) as ε↓0. We treat both the case ψ∈ L1(ℝ) and the case ψ=ψ1+Braketx−α, where ψ1∈ L1(ℝ) and 0<α≤1. Our results identify the dependence of the lifespan scale on the decay exponent, including the borderline case α=1. The proof combines Riccati equations along characteristics with a scaling argument inspired by Lindblad.
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BibTeX
@misc{sugiyama2026asymptotic,
title = {{Asymptotic Formula of the Lifespan of Classical Solutions to 1D Quasilinear Wave Equations}},
author = {Yuusuke Sugiyama and Taro Yamanoi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38353},
eprint = {2610.38353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38353},
}
RIS
TY - UNPB
TI - Asymptotic Formula of the Lifespan of Classical Solutions to 1D Quasilinear Wave Equations
AU - Yuusuke Sugiyama
AU - Taro Yamanoi
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38353
UR - https://arxiv.org/abs/2610.38353
AB - We study the lifespan T^*(ε) 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 ε>0. We assume that c'(θ)∼ a|θ|^p−2 as θ→ 0, where a>0 and p>3 or p=2. We obtain explicit asymptotic formulas for T^*(ε) as ε↓0. We treat both the case ψ∈ L^1(ℝ) and the case ψ=ψ_1+Braketx^−α, where ψ_1∈ L^1(ℝ) and 0<α≤1. Our results identify the dependence of the lifespan scale on the decay exponent, including the borderline case α=1. The proof combines Riccati equations along characteristics with a scaling argument inspired by Lindblad.
ER -