Small-time global control of a slightly superlinear heat equation
Let $\Omega \subset \mathbb{R}^d$ be a bounded connected domain of class $C^2$ and $\omega \subset \Omega$ a nonempty open set. For $\alpha \in [3/2, 2)$ consider $$\partial_t y - \Delta y + |y| \log^\alpha(2 + |y|) = \mathbf{1}_\omega v \ \text{ in } (0, T) \times \Omega, \qquad y = 0 \ \text{ on } (0, T) \times \partial\Omega, \qquad y(0) = y_0,$$ with a control $v \in L^\infty((0, T) \times \omega)$. Without control, solutions can blow up in finite time. Fernández-Cara and Zuazua proved null-controllability in every time $T > 0$ for nonlinearities growing more slowly than $|s| \log^{3/2}(1 + |s|)$, and gave nonlinearities growing like $|s| \log^p(1 + |s|)$, $p > 2$, for which, when $\omega \ne \Omega$, it fails in every time. For the equation above, Le Balc'h proved that there is a time $T$, independent of $y_0$, in which every $y_0 \in L^\infty(\Omega)$ can be steered to $0$, and that in every time $T > 0$ every such $y_0$ can be steered to a nonnegative state.
Problem. For every $\alpha \in [3/2, 2)$, every $T > 0$ and every $y_0 \in L^\infty(\Omega)$, is there $v \in L^\infty((0, T) \times \omega)$ such that this problem has a bounded weak solution on $[0, T]$ with $y(T) = 0$? The time $T$ is prescribed independently of the initial state, and negative initial data are allowed.
Source: Fernández-Cara and Zuazua (2000), whose results leave open growth between |s| log^{3/2}(1 + |s|) and |s| log^2(1 + |s|); for this nonlinearity, Le Balc'h (2020), Remark 2.8. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, blow-up, null controllability, semilinear heat equations
References
@article{fernandezcara2000,
author = {Fern{\'a}ndez-Cara, Enrique and Zuazua, Enrique},
title = {Null and approximate controllability for weakly blowing up semilinear heat equations},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {17},
number = {5},
year = {2000},
pages = {583--616},
doi = {10.1016/S0294-1449(00)00117-7},
}
@article{lebalch2020,
author = {Le Balc'h, K{\'e}vin},
title = {Global null-controllability and nonnegative-controllability of slightly superlinear heat equations},
journal = {J. Math. Pures Appl. (9)},
volume = {135},
year = {2020},
pages = {103--139},
doi = {10.1016/j.matpur.2019.10.009},
eprint = {1810.12232},
archivePrefix = {arXiv},
}
@misc{aim-small-time-superlinear-heat-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Small-time global control of a slightly superlinear heat equation}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/066-small-time-superlinear-heat-control.md},
note = {Entry 066 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00560, "Small-time global control of a slightly superlinear heat equation", clopen.solutions.