Thresholds and Traces for Supercritical Lane–Emden Blowup

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{R}}^n$ be a smooth bounded domain. The source considers the supercritical Lane--Emden heat equation \[ \partial_t u-\Delta u=|u|^{p-2}u \quad\text{in }\Omega\times(0,T), \qquad p>2^*:=\frac{2n}{n-2}, \] and sets \[ \alpha:=\frac{2}{p-2},\qquad M_{\mathrm{crit}}:=\alpha(n-2-\alpha). \] The displayed Sobolev critical exponent presupposes the dimensional range $n\geq3$; this restriction is implicit rather than separately stated in the supplied record. No initial or boundary conditions, sign restrictions, or solution class are supplied.

1. Can the threshold $M$ in the complete-blow-up result referred to by the source be chosen as \[ M=M_{\mathrm{crit}}=\alpha(n-2-\alpha) \] for supercritical exponents $p$ sufficiently close to $2^*$ from above? This interprets the source's phrase "for sufficiently small $p>2^*$" as a near-critical condition. The record does not specify the underlying complete-blow-up theorem, how $M$ enters its hypotheses, or the allowed dependence of the near-critical range on $n$, $\Omega$, or other data. Consequently this question cannot be given a complete quantified formulation from the supplied record alone.
2. Does every finite-time blow-up solution with bounded energy admit a terminal trace satisfying \[ u(T)\in L^{p,\lambda}_0(\Omega)? \] Here $T<\infty$ denotes the blow-up time referred to in the source. The notation $L^{p,\lambda}_0(\Omega)$ is retained exactly as supplied. Neither $\lambda$, the function-space norm, the meaning of the subscript $0$, the energy functional and its boundedness condition, nor the meaning or topology of the terminal trace is specified. These omissions prevent this question from being a fully defined mathematical assertion.

The supplied literature assessment calls the target a Lorentz space, but provides no definition establishing that interpretation. It must not be silently replaced by a Lorentz-space or Morrey-space convention.

Short English statement

The source asks whether a complete-blow-up threshold equals $\alpha(n-2-\alpha)$ for exponents just above the Sobolev critical exponent, and whether every finite-time blow-up solution with bounded energy has a terminal trace in the stated space $L^{p,\lambda}_0(\Omega)$. The threshold theorem, boundary conditions, solution class, energy condition, and trace space require clarification before either question has a precise complete formulation.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30003175$, title "Thresholds and Traces for Supercritical Lane--Emden Blowup," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Calculus of Variations (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/34, pp. 1943--2008. Supplied source URL and literature-source link, labelled "Oberwolfach Report on calculus of variations": https://doi.org/10.4171/owr/2016/34. The background describes these reports as documenting mathematical workshops and their open-problem discussions. No individual proposer is identified; the supplied proposed year is 2016.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that a complete-blow-up theorem supplies the surrounding construction, but that no primary result located through that date proves the requested critical choice of $M$ or the asserted terminal trace in what the assessment calls the stated Lorentz space. This is supplied dated curation, not an independently verified literature assessment. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and describes its changes as repairs to notation, typography, or omitted definitions while preserving mathematical content. This supplied editorial label does not resolve the missing definitions listed here or constitute independent mathematical verification.