Does subcritical heat flow always decay faster?
Let $(M, g)$ be a connected, complete, noncompact smooth Riemannian manifold without boundary. For $j \in \{0, +\}$ let $V_j \in C^\infty(M; \mathbb{R})$ be bounded below, let $H_j = -\Delta_g + V_j$ be its Friedrichs realization in $L^2(M, d\operatorname{vol}_g)$, assume $H_j \ge 0$ and $\inf \sigma(H_j) = 0$, and let $k_j(t, x, y) > 0$ be the heat kernel of $e^{-tH_j}$. Assume that for some distinct $x_*, y_* \in M$ $$\int_0^\infty k_+(t, x_*, y_*) \, dt < \infty, \qquad \int_0^\infty k_0(t, x_*, y_*) \, dt = \infty,$$ so that $H_+$ is subcritical and $H_0$ critical, and that $\ker_{L^2} H_0 = \{0\}$, so that $H_0$ is null-critical. Fraas, Krejčiřík and Pinchover noted that the limit below holds when $H_0$ is positive-critical, by Pinchover's large-time asymptotics, or $\inf \sigma(H_+) > 0$ (a remark added to the arXiv version), and proved it, for symmetric operators, when $k_+(t, x, y_1) \le C k_0(t, x, y_1)$ for some $y_1$ and $C$, every $x$ and all large $t$; in particular when $V_+ - V_0$ is nonnegative and nonzero.
Problem. Must $$\lim_{t \to \infty} \, \sup_{(x, y) \in K} \frac{k_+(t, x, y)}{k_0(t, x, y)} = 0$$ for every compact $K \subset M \times M$? No pointwise ordering between $V_+$ and $V_0$ is assumed.
Source: Fraas, Krejčiřík and Pinchover (2010), Conjecture 1, for second-order elliptic operators on noncompact manifolds; restated by Krejčiřík for self-adjoint operators (AIM workshop, 2015). Stated by AIM — Open Applied Problems for Schrödinger operators on complete manifolds, with both spectral bottoms zero and a null-critical reference operator.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG, SP
- Keywords
- AIM Open Applied Problems, criticality theory, heat kernels, Schrödinger operators
References
@article{fraas2010,
author = {Fraas, Martin and Krej{\v{c}}i{\v{r}}{\'\i}k, David and Pinchover, Yehuda},
title = {On some strong ratio limit theorems for heat kernels},
journal = {Discrete Contin. Dyn. Syst.},
volume = {28},
number = {2},
year = {2010},
pages = {495--509},
doi = {10.3934/dcds.2010.28.495},
eprint = {0912.4337},
archivePrefix = {arXiv},
}
@misc{krejcirik2015,
author = {Krej{\v{c}}i{\v{r}}{\'\i}k, David},
title = {Large-time behavior of the heat equation: subcriticality versus criticality},
howpublished = {Open problem, AIM workshop, San Jose, in the series ``Mathematical aspects of the physics with non-self-adjoint operators''},
year = {2015},
url = {https://nsa.fjfi.cvut.cz/problems/04_AIM_2015/2015-18/2015-18_AIM_Krejcirik_2.pdf},
note = {Conjecture 2},
}
@misc{aim-critical-subcritical-heat-ratio,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does subcritical heat flow always decay faster?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/217-critical-subcritical-heat-ratio.md},
note = {Entry 217 at commit a153f46, 2026-10-13},
}
Resolution claims
We construct two nonnegative Schrödinger operators on the line with smooth bounded potentials and spectral bottom zero. One is subcritical and the other null-critical, while the quotient of their diagonal heat kernels has infinite limsup. This gives a negative answer to the heat-kernel comparison question without an ordering assumption on the potentials. The construction preserves small heat-kernel values of successive …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Open → Solution claimed | The registry's curators | A smooth counterexample to critical--subcritical heat-kernel comparison v1 recorded from AIM @ 24edaad, paper/main.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00421, "Does subcritical heat flow always decay faster?", clopen.solutions.