Year: 2026
Type: preprint
Abstract: In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation \[ \partial_{tt}u-\partial_{xx} u= Θ_1(\partial_t u)^2 + Θ_2(\partial_x u)^2. \] For any $Θ_1 Θ_2 < 0$, $β<1$, we construct infinitely many weak solutions $u \in C_{t,x}^β$ 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.
Source: arXiv (Cornell University)
Authors: Qu, Peng, Zhang, Mingxin
Institution: Fudan University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4305158243, G4478161941
OpenAlex ID: W7213244561