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The divergence set for the wave equation in higher dimensions
Abstract
It is shown that if u solves the wave equation in ℝ4+1 with initial data u(·,0) = u0(·) ∈ Hs and ∂tu(·,0) = u1(· ) ∈ Hs−1, where 0.5 < s ≤ 0.55, then u(x,t) → u0(x) and ∂tu(x,t) → u1(x) pointwise as t → 0, for all x outside an exceptional set of Hausdorff dimension at most 6−4s. In a very small range of s, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in ℝn+1 is obtained for n ≥ 4 and 1/2 < s < n/4.
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- A rigidity result for the weighted p-Laplace equation in higher dimensions
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- Isolated singularities of solutions to the Yamabe equation in higher dimensions II
BibTeX
@misc{du2026divergence,
title = {{The divergence set for the wave equation in higher dimensions}},
author = {Xiumin Du and Terence L. J. Harris and Jianhui Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19691},
eprint = {2609.19691},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19691},
}
RIS
TY - UNPB
TI - The divergence set for the wave equation in higher dimensions
AU - Xiumin Du
AU - Terence L. J. Harris
AU - Jianhui Li
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19691
UR - https://arxiv.org/abs/2609.19691
AB - It is shown that if u solves the wave equation in ℝ^4+1 with initial data u(·,0) = u_0(·) ∈ H^s and ∂_tu(·,0) = u_1(· ) ∈ H^s−1, where 0.5 < s ≤ 0.55, then u(x,t) → u_0(x) and ∂_tu(x,t) → u_1(x) pointwise as t → 0, for all x outside an exceptional set of Hausdorff dimension at most 6−4s. In a very small range of s, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in ℝ^n+1 is obtained for n ≥ 4 and 1/2 < s < n/4.
ER -