Almost global solutions of 2-D quadratic quasilinear wave equations in exterior domains without the second null condition
Abstract
For 3-D quadratic quasilinear wave equations in exterior domains with homogeneous Dirichlet boundary conditions and small initial data of size ε>0, the authors of two papers [M. Keel, H. Smith, C.D. Sogge, Amer. J. Math. 122 (2000), no. 4, 805-842; J. Amer. Math. Soc. 17 (2004), no. 1, 109-153] have shown that the lifespan T̅ε of smooth solutions satisfies T̅ε=∞ or T̅ε≥ eC/ε, depending on whether the null condition holds or not, respectively. However, for the corresponding 2-D problem in exterior domains, it remains a difficult and unsolved problem to determine the precise lifespan Tε of smooth solutions. In our previous papers, we have established that Tε=∞ holds when both the first and second null conditions are satisfied; and that Tε≥C/ε2 holds when the first null condition is violated. In this paper, we will prove Tε≥ eC/ε2 when the first null condition holds but the second null condition is violated. Combining these results, we have systematically established sharp estimates for the lifespan Tε of 2-D quadratic quasilinear wave equations in exterior domains with homogeneous Dirichlet boundary conditions. Moreover, our series of results for quasilinear wave equations in 2-D exterior domains are consistent with the corresponding results for the classical Cauchy problem.
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BibTeX
@misc{hou2026almost,
title = {{Almost global solutions of 2-D quadratic quasilinear wave equations in exterior domains without the second null condition}},
author = {Fei Hou and Huicheng Yin},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01471},
eprint = {2611.01471},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01471},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Almost global solutions of 2-D quadratic quasilinear wave equations in exterior domains without the second null condition
AU - Fei Hou
AU - Huicheng Yin
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01471
UR - https://arxiv.org/abs/2611.01471
AB - For 3-D quadratic quasilinear wave equations in exterior domains with homogeneous Dirichlet boundary conditions and small initial data of size ε>0, the authors of two papers [M. Keel, H. Smith, C.D. Sogge, Amer. J. Math. 122 (2000), no. 4, 805-842; J. Amer. Math. Soc. 17 (2004), no. 1, 109-153] have shown that the lifespan T̅_ε of smooth solutions satisfies T̅_ε=∞ or T̅_ε≥ e^C/ε, depending on whether the null condition holds or not, respectively. However, for the corresponding 2-D problem in exterior domains, it remains a difficult and unsolved problem to determine the precise lifespan T_ε of smooth solutions. In our previous papers, we have established that T_ε=∞ holds when both the first and second null conditions are satisfied; and that T_ε≥C/ε^2 holds when the first null condition is violated. In this paper, we will prove T_ε≥ e^C/ε^2 when the first null condition holds but the second null condition is violated. Combining these results, we have systematically established sharp estimates for the lifespan T_ε of 2-D quadratic quasilinear wave equations in exterior domains with homogeneous Dirichlet boundary conditions. Moreover, our series of results for quasilinear wave equations in 2-D exterior domains are consistent with the corresponding results for the classical Cauchy problem.
ER -