Sharp Free-Energy Decay for Fast Diffusion

Definitions and mathematical statement

Let $u(t,\cdot)$ denote the fast-diffusion solution referred to in the supplied record, and let $\mathcal F_\sigma$ denote its free-energy functional. Write, wherever this functional is defined, \[ \mathcal E(t):=\mathcal F_\sigma[u(t,\cdot)],\qquad t\in[0,\infty). \] The source question is to obtain sharp estimates for the decay of $\mathcal E(t)$ valid for every $t\geq0$, including $t=0$.

The supplied record does not specify the fast-diffusion equation or its normalization, the spatial domain or dimension, the diffusion exponent, initial or boundary conditions, the admissible solution class, the definition of $\mathcal F_\sigma$, or the meaning and possible time dependence of $\sigma$. It also does not specify what constitutes a sharp estimate, or which parameters and initial-data quantities may enter it. Consequently, this is a source-level question with unresolved hypotheses and definitions; it does not determine a precise quantitative inequality or a universally quantified conjecture over a specified class of solutions.

Short English statement

Determine sharp bounds for the decay of the free energy $\mathcal F_\sigma[u(t,\cdot)]$ of the intended fast-diffusion solution, valid at every time $t\geq0$. The equation, admissible data, functional, parameter $\sigma$, and criterion of sharpness require clarification from the source before a precise mathematical formulation is possible.

Sources

• Local UnsolvedMath registry, permanent ID $30001941$, title: Sharp Free-Energy Decay for Fast Diffusion; source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The supplied record lists proposed year 2011 and no named proposer.
• Supplied report citation: Variational Methods for Evolution (2012), Oberwolfach Reports, DOI 10.4171/owr/2011/55, pp. 3145--3216. https://doi.org/10.4171/owr/2011/55. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. No problem-specific page or passage is supplied.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2011/55.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the source explicitly calls sharp all-time decay estimates an open question, while later results provide bounds without settling the stated sharp problem. This is supplied dated curation, not independently verified here; no references for those later results are supplied. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry assigns the statement status corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This verification assertion is supplied provenance and has not been independently checked here.