Nonlinear Stability of Periodic Reaction–Diffusion Waves

Definitions and mathematical statement

Let $u^*$ denote the periodic solution of a reaction--diffusion PDE appearing in the supplied record, written there as $u^*(t)$. The equation, spatial domain, boundary conditions, dependent-variable space, and period are not specified. In particular, the record does not determine whether periodicity is temporal, spatial, or associated with a moving wave.

For a perturbation $v$ on a spatial domain $\Omega$, boundedness means that its pointwise norm has finite supremum, or finite essential supremum if the intended function space identifies functions almost everywhere. Neither $\Omega$ nor the choice between these conventions is supplied. The phrase "spatially localized" is also not assigned a specific decay or integrability condition.

Spectral stability concerns the spectrum of an appropriate linearization about $u^*$. The record does not specify the linearized operator or evolution operator, its domain and ambient function space, the treatment of neutral modes, or any required spectral bounds. Nonlinear stability likewise requires a perturbation topology and a choice between stability relative to a fixed solution and stability allowing phase or other symmetry adjustments. These choices are unresolved.

Source-level question. Given a spectrally stable periodic solution $u^*(t)$ of a reaction--diffusion PDE, does spectral stability imply nonlinear stability for a larger class of perturbations than spatially localized perturbations, in particular for bounded perturbations?

The intended smallness requirement, stability norm, time interval, admissible perturbation class, and quantification over equations and periodic solutions are not supplied. Thus this question preserves the source formulation, but a fully quantified implication cannot be specified without additional hypotheses and definitions.

Short English statement

The question asks whether spectral stability of a periodic reaction--diffusion solution guarantees nonlinear stability when perturbations may be bounded without being spatially localized. The supplied record leaves the equation, domain, periodicity, spectral criterion, perturbation smallness, and meaning of nonlinear stability unspecified, so the precise implication requires clarification.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30001128$, source problem code catalogue entry, title "Nonlinear Stability of Periodic Reaction--Diffusion Waves," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2008.
• Supplied report citation: Dynamics of Patterns (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/57, pp. 3201--3271. Original OWR report: https://doi.org/10.4171/owr/2008/57. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "nonlinear modulational stability": https://arxiv.org/abs/1203.3795. No authors or fuller bibliographic details are supplied for this reference.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem "partially solved" and reports that spectral stability implies nonlinear stability in several localized-perturbation and viscous-wave settings, while a general theorem covering the report's larger class of bounded perturbations is unavailable. This is supplied dated curation, described as literature triage derived from the OWR curation; neither the status nor the cited claims have been independently verified here.
• The registry labels the statement corrected_verified and says its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This is a supplied verification assertion, not an independent verification in this draft.