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Sharp non-uniqueness of weak solutions to the semilinear wave equation
Abstract
In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation ∂ttu−∂xx u= Θ1(∂t u)2 + Θ2(∂x u)2. For any Θ1 Θ2 < 0, β<1, we construct infinitely many weak solutions u ∈ Ct,xβ with zero initial data. To our knowledge, this is the first non-uniqueness result for a semilinear wave equation. At this low level of regularity Ct,xβ(β<1), the quadratic term is defined by Littlewood--Paley decomposition in Ct Hx−s. Our construction is based on a frequency-localized convex integration scheme. Together with the classical local uniqueness theory in Ct,x1, this non-uniqueness result is sharp.
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BibTeX
@misc{qu2026sharp,
title = {{Sharp non-uniqueness of weak solutions to the semilinear wave equation}},
author = {Peng Qu and Mingxin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15442},
eprint = {2609.15442},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15442},
}
RIS
TY - UNPB
TI - Sharp non-uniqueness of weak solutions to the semilinear wave equation
AU - Peng Qu
AU - Mingxin Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15442
UR - https://arxiv.org/abs/2609.15442
AB - In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation ∂_ttu−∂_xx u= Θ_1(∂_t u)^2 + Θ_2(∂_x u)^2. For any Θ_1 Θ_2 < 0, β<1, we construct infinitely many weak solutions u ∈ C_t,x^β with zero initial data. To our knowledge, this is the first non-uniqueness result for a semilinear wave equation. At this low level of regularity C_t,x^β(β<1), the quadratic term is defined by Littlewood--Paley decomposition in C_t H_x^−s. Our construction is based on a frequency-localized convex integration scheme. Together with the classical local uniqueness theory in C_t,x^1, this non-uniqueness result is sharp.
ER -