Uniform Long-Time Sobolev Bounds for Nonlinear Beam Equations

Definitions and mathematical statement

Let $\mathbb T={\mathbb{R}}/(2\pi{\mathbb{Z}})$ be the spatial circle, and let $m\in[1,2]$. The supplied record displays the nonlinear beam equation \[ \partial_t^2\psi(x,t)+\partial_x^4\psi(x,t)+m\psi(x,t)+f(\psi(x,t))=0, \qquad x\in\mathbb T, \] where $\psi=\psi(x,t)$, the potential $F$ is real analytic in a neighborhood of the origin, $F(0)=0$, and \[ f(\psi):=\partial_\psi F(\psi). \] The torus specifies periodicity in the spatial variable. The record does not specify a time interval, initial data, a solution class, or the scalar field of the unknown. No further vanishing conditions on derivatives of $F$ are supplied.

The recorded question is literally \[ \text{Can one prove } f(t)=c\varepsilon \text{ for some absolute constant }c\,? \] Here the time-dependent quantity denoted by $f(t)$ is undefined and must not be identified without justification with the nonlinearity $f(\psi)$. The parameter $\varepsilon$ is also undefined. The title and supplied curation describe an intended question about uniform long-time Sobolev bounds, but neither the Sobolev norm nor its order, the initial smallness hypothesis, the time scale, the permitted dependence of $c$, or the relevant quantifiers over $m$, $F$, and solutions are recoverable from the statement. In particular, the displayed equality cannot silently be replaced by an inequality.

The record ends with an incomplete fragment mentioning the beam equation after a convenient linear symplectic change of variables. Neither that transformation nor the transformed variables are supplied. Thus the equation and the literal question above preserve the recoverable content; they do not constitute a fully specified Sobolev stability conjecture.

The supplied registry labels the statement unrecoverable and the literature status partially_solved. Its dated assessment attributes to Feola and Massetti a uniform $O(\varepsilon)$ Sobolev bound with constant $8$, on a long sub-exponential time interval and for almost every mass parameter, while saying that the unrestricted absolute-constant bound requested in the record is not established. A separate supplied verification note says that the question is immediately followed by a stability theorem providing a constant multiple of the initial norm on its stated time scale. These are supplied descriptions with missing theorem hypotheses and time-scale formulas, not independently verified mathematical assertions or a complete formulation of the target.

Short English statement

The source asks for a uniform small Sobolev bound for a nonlinear beam equation on the circle, with mass $m\in[1,2]$ and an analytic potential. Its literal formula $f(t)=c\varepsilon$ leaves the measured quantity, smallness parameter, time interval, and quantifiers undefined. The supplied dated curation reports partial progress, but the intended precise statement remains unrecoverable and requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005587$, source problem code catalogue entry, title Uniform Long-Time Sobolev Bounds for Nonlinear Beam Equations. The registry records proposed year $2023$; no individual proposer is supplied. The supplied statement status is unrecoverable, and its literature status is partially_solved.
• Supplied source citation: Dynamische Systeme (2024), Oberwolfach Reports, pp. 1671--1730, DOI 10.4171/owr/2023/30. https://doi.org/10.4171/owr/2023/30. The background describes these reports as documentation of mathematical workshops and their open-problem discussions. The supplied verification note preserves the original record for provenance and promotes no separate clean statement.
• Supplied literature source, labeled Feola--Massetti: https://doi.org/10.1016/j.jde.2023.01.038. The supplied assessment attributes the constant-$8$ Sobolev bound, long sub-exponential time interval, and almost-every-mass qualification to this work; its full theorem statement is not supplied.
• Supplied literature source, labeled original OWR report: https://doi.org/10.4171/owr/2023/30.
• Literature assessment date: 2026-08-22. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment. The record explicitly calls for independent verification of time-sensitive status and cited claims.