Global null control of a heat equation with cubic growth
Let $\Omega \subset \mathbb{R}^d$ be a bounded connected smooth domain and $\omega \Subset \Omega$ a nonempty open set, and consider $$\partial_t y - \Delta y = y^3 + \mathbf{1}_\omega u \ \text{ in } (0,T) \times \Omega, \qquad y = 0 \ \text{ on } (0,T) \times \partial\Omega, \qquad y(0,\cdot) = y_0,$$ with a control $u \in L^\infty((0,T) \times \omega)$. Without control, solutions of this equation can blow up in finite time. For every $T > 0$, initial data $y_0$ that are small in $L^\infty(\Omega)$ can be steered to zero at time $T$. Fernández-Cara and Zuazua proved that blow-up can be prevented and the state steered to zero when $y^3$ is replaced by a nonlinearity $f(y)$ with $|f(y)| = o\bigl(|y| \log^{3/2}(1 + |y|)\bigr)$, and that for every $p > 2$ some $f$ with $|f(y)| \sim |y| \log^p(1 + |y|)$ has, for every $T > 0$, initial data whose solution blows up before time $T$ whatever the control. Coron observed that their argument does not seem to give a negative answer for $y^3$.
Problem. For every $T > 0$ and every $y_0 \in L^\infty(\Omega)$, is there a control $u \in L^\infty((0,T) \times \omega)$ for which the system has a bounded weak solution on all of $[0,T]$ with $y(T,\cdot) = 0$? The control may depend on $y_0$ and $T$, and no bound on its size is imposed.
Source: Coron (2007), Open Problems 7.14 and 7.15, for boundary control in one space dimension; Le Balc'h and Takahashi (2024), §1.1, remarks after Theorem 1.1, for interior control. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- OC — Optimization and Control
- Secondary
- AP
- Keywords
- AIM Open Applied Problems, blow-up, null controllability, semilinear heat equations
References
@book{coron2007,
author = {Coron, Jean-Michel},
title = {Control and Nonlinearity},
series = {Mathematical Surveys and Monographs},
volume = {136},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2007},
doi = {10.1090/surv/136},
}
@article{lebalch2024,
author = {Le Balc'h, K{\'e}vin and Takahashi, Tak{\'e}o},
title = {Null-controllability of cascade reaction-diffusion systems with odd coupling terms},
journal = {Ann. Math. Blaise Pascal},
volume = {31},
number = {2},
year = {2024},
pages = {239--271},
doi = {10.5802/ambp.430},
}
@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} 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-global-cubic-heat-null-control,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global null control of a heat equation with cubic growth}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/140-global-cubic-heat-null-control.md},
note = {Entry 140 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-00526, "Global null control of a heat equation with cubic growth", clopen.solutions.