Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
Abstract
This paper investigates the 2D quasilinear wave equation ∑α,β=02gαβ(∂ϕ)∂αβ2ϕ=0 with the short pulse initial data (ϕ, ∂tϕ)|t=1=(δ2−ε0ϕ0((r−1)/δ,ω), δ1−ε0ϕ1((r−1)/δ,ω)), where 0<ε0<1 is a fixed constant and δ>0 is a small parameter. In our previous work [Ding Bingbing, Xin Zhouping, Yin Huicheng, Sci. China Math., Vol. 69 (2026), No. 3, 667-736], we have established the global existence of a smooth solution ϕ under the k-th order null condition (k≥ 2) when the (k+1)-th order null condition fails, provided that 0<ε0<εk*=k/(k+1). As a natural continuation, the present paper demonstrates that for ε0≥ εk*, the smooth solution ϕ blows up in finite time. Consequently, the threshold ε0=εk* is shown to be the critical exponent that precisely characterizes the global existence and finite-time blowup for smooth short pulse solutions to the aforementioned 2D quasilinear wave equation, under the assumption that the k-th order null condition holds while the (k+1)-th order null condition does not. A key novelty of our approach lies in deriving sharp orders of δ for almost all quantities in both energy and pointwise estimates. This is achieved by utilizing the higher-order null structure in conjunction with a new good unknown recently introduced by us and the transport equation for the difference between this good unknown and its corrector.
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BibTeX
@misc{ding2026blowup,
title = {{Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data}},
author = {Bingbing Ding and Zhouping Xin and Huicheng Yin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18358},
eprint = {2610.18358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18358},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Blowup of smooth solutions to 2D quasilinear wave equations with higher-order null conditions and short pulse initial data
AU - Bingbing Ding
AU - Zhouping Xin
AU - Huicheng Yin
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18358
UR - https://arxiv.org/abs/2610.18358
AB - This paper investigates the 2D quasilinear wave equation ∑_α,β=0^2g^αβ(∂ϕ)∂_αβ^2ϕ=0 with the short pulse initial data (ϕ, ∂_tϕ)|_t=1=(δ^2−ε_0ϕ_0((r−1)/δ,ω), δ^1−ε_0ϕ_1((r−1)/δ,ω)), where 0<ε_0<1 is a fixed constant and δ>0 is a small parameter. In our previous work [Ding Bingbing, Xin Zhouping, Yin Huicheng, Sci. China Math., Vol. 69 (2026), No. 3, 667-736], we have established the global existence of a smooth solution ϕ under the k-th order null condition (k≥ 2) when the (k+1)-th order null condition fails, provided that 0<ε_0<ε_k^*=k/(k+1). As a natural continuation, the present paper demonstrates that for ε_0≥ ε_k^*, the smooth solution ϕ blows up in finite time. Consequently, the threshold ε_0=ε_k^* is shown to be the critical exponent that precisely characterizes the global existence and finite-time blowup for smooth short pulse solutions to the aforementioned 2D quasilinear wave equation, under the assumption that the k-th order null condition holds while the (k+1)-th order null condition does not. A key novelty of our approach lies in deriving sharp orders of δ for almost all quantities in both energy and pointwise estimates. This is achieved by utilizing the higher-order null structure in conjunction with a new good unknown recently introduced by us and the transport equation for the difference between this good unknown and its corrector.
ER -