Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation
Abstract
We study the Cauchy problem vtt−vxx=absvt+vxmabsvtn, v(x,0)=ηφ(x), vt(x,0)=ηψ(x), on R, with compactly supported profiles φ∈ C02(R), ψ∈ C01(R), exponents m,n>1, and amplitude η>0. We show that the sign of P0=ψ+φ' decides the behaviour of small solutions, whatever the sign of ψ−φ'. If P0 is positive at some point, the lifespan T(η) obeys explicit two-sided bounds of order η−(m+n−1) for every η>0, and limη→0+ηm+n−1T(η)=2n/((m+n−1) (maxRP0)m+n−1) . If P0≤0, the solution is global for every amplitude below an explicit threshold. If moreover P0not≡0, the component vt+vx decays at the universal rate (2n/((m+n−1)t))1/(m+n−1), and the gradient of the solution converges uniformly to that of a free wave travelling to the right; if P0≡0, the solution is itself such a travelling wave. The analysis rests on a localization property: the zero set of vt+vx is invariant along its own characteristics, so that the nonlinear source stays in a slab of fixed width moving with speed one, and each characteristic of the other family is forced only during a bounded time. This yields the global existence, the long-time behaviour and the exact value of the limit. The results extend to the endpoint exponents m,n≥1 and to sources f(vt+vx)g(vt), for which T(η) is asymptotic to an Osgood-type integral.
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BibTeX
@misc{kaabi2026characteristic,
title = {{Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation}},
author = {Firas Kaabi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28292},
eprint = {2608.28292},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28292},
}
RIS
TY - UNPB
TI - Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation
AU - Firas Kaabi
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2608.28292
UR - https://arxiv.org/abs/2608.28292
AB - We study the Cauchy problem v_tt−v_xx=absv_t+v_x^mabsv_t^n, v(x,0)=ηφ(x), v_t(x,0)=ηψ(x), on R, with compactly supported profiles φ∈ C_0^2(R), ψ∈ C_0^1(R), exponents m,n>1, and amplitude η>0. We show that the sign of P_0=ψ+φ' decides the behaviour of small solutions, whatever the sign of ψ−φ'. If P_0 is positive at some point, the lifespan T(η) obeys explicit two-sided bounds of order η^−(m+n−1) for every η>0, and lim_η→0^+η^m+n−1T(η)=2^n/((m+n−1) (max_RP_0)^m+n−1) . If P_0≤0, the solution is global for every amplitude below an explicit threshold. If moreover P_0not≡0, the component v_t+v_x decays at the universal rate (2^n/((m+n−1)t))^1/(m+n−1), and the gradient of the solution converges uniformly to that of a free wave travelling to the right; if P_0≡0, the solution is itself such a travelling wave. The analysis rests on a localization property: the zero set of v_t+v_x is invariant along its own characteristics, so that the nonlinear source stays in a slab of fixed width moving with speed one, and each characteristic of the other family is forced only during a bounded time. This yields the global existence, the long-time behaviour and the exact value of the limit. The results extend to the endpoint exponents m,n≥1 and to sources f(v_t+v_x)g(v_t), for which T(η) is asymptotic to an Osgood-type integral.
ER -