Strongly nonmonotone behavior in bounded domains

Determine whether positive solutions of the semilinear heat equation on bounded domains can exhibit strongly nonmonotone behavior of the sup norm analogous to the oscillatory finite-time blow-up constructed on the whole space or the global behavior with liminf equal to zero and limsup equal to infinity.

Background

The paper constructs nonradial positive solutions on the whole space whose L-infinity norm oscillates during finite-time blow-up, and recalls previously known global whole-space solutions whose L-infinity norm has liminf zero and limsup infinity. The authors explicitly leave unresolved whether comparably strong nonmonotone behavior can occur when the spatial domain is bounded.

References

We do not know presently if strongly nonmonotone behaviors similar to Theorem~\ref{thmNonMon} or to supinf can occur in bounded domains.

— Oscillatory blow-up and gradient estimates for semilinear heat equations  (2608.14257 - Quittner et al., 14 Aug 2026) in Section 1, subsection “Discussion and remarks,” immediately before equation (P)

The latter as well as (Pw) are open questions when $\Omega$ is not convex.

— Oscillatory blow-up and gradient estimates for semilinear heat equations  (2608.14257 - Quittner et al., 14 Aug 2026) in Remark 2(i), Section 1, subsection “Discussion and remarks”