Global classical continuation in the chemotaxis-consumption system
Let $n \ge 3$, let $\Omega \subset \mathbb{R}^n$ be a bounded connected domain with smooth boundary, and let $u_0, v_0 \in W^{1,\infty}(\Omega)$ with $u_0 \ge 0$, $u_0 \not\equiv 0$ and $v_0 > 0$ on $\overline\Omega$. Consider $$u_t = \Delta u - \nabla \cdot (u \nabla v), \qquad v_t = \Delta v - uv \qquad \text{in } \Omega \times (0, \infty),$$ with $\partial_\nu u = \partial_\nu v = 0$ on $\partial\Omega$ and $(u, v)(\cdot, 0) = (u_0, v_0)$. A global classical solution is a pair $$u \in C^0(\overline\Omega \times [0, \infty)) \cap C^{2,1}(\overline\Omega \times (0, \infty)), \qquad v \in C^{2,1}(\overline\Omega \times (0, \infty)) \cap \bigcap_{n < q < \infty} C^0([0, \infty); W^{1,q}(\Omega)),$$ with $u \ge 0$ and $v > 0$ for $t > 0$, satisfying the equations and the boundary conditions pointwise and attaining the initial data in these spaces; no compatibility condition at $t = 0$ is imposed beyond the assumptions on the data. For $n = 2$ there is a unique global classical solution for all such data (Tao and Winkler for convex domains; Jiang, Wu and Zheng in general). In every dimension $n \ge 2$ there is one, bounded, when $\|v_0\|_{L^\infty(\Omega)} \le \pi/\sqrt{2(n+1)}$ (Tao; Baghaei and Khelghati). For data of any size and $n \ge 3$, Lankeit and Winkler constructed global weak solutions that are classical on $\overline\Omega \times [T, \infty)$ for some $T > 0$, after Tao and Winkler for convex three-dimensional domains; Jiang, Wu and Zheng obtained blow-up criteria and lower blow-up estimates for local classical solutions in three dimensions.
Problem. For every $n \ge 3$, every such $\Omega$ and all such $u_0, v_0$, without any smallness assumption, does the maximal local classical solution extend to all positive times, that is, is there a global classical solution in the above sense? No bound uniform in time is required.
Source: Lankeit and Winkler (2023), §2, the question after Theorem 2.2 of their survey; stated in this form, for every smooth bounded domain and with the solution class of their (2.8), by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, chemotaxis-consumption systems, global classical solutions
References
@article{lankeit2023,
author = {Lankeit, Johannes and Winkler, Michael},
title = {Depleting the signal: Analysis of chemotaxis-consumption models---{A} survey},
journal = {Stud. Appl. Math.},
volume = {151},
number = {4},
year = {2023},
pages = {1197--1229},
doi = {10.1111/sapm.12625},
eprint = {2304.02449},
archivePrefix = {arXiv},
}
@article{taowinkler2012,
author = {Tao, Youshan and Winkler, Michael},
title = {Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant},
journal = {J. Differential Equations},
volume = {252},
number = {3},
year = {2012},
pages = {2520--2543},
doi = {10.1016/j.jde.2011.07.010},
}
@article{jiang2015,
author = {Jiang, Jie and Wu, Hao and Zheng, Songmu},
title = {Global existence and asymptotic behavior of solutions to a chemotaxis-fluid system on general bounded domains},
journal = {Asymptot. Anal.},
volume = {92},
number = {3-4},
year = {2015},
pages = {249--258},
doi = {10.3233/ASY-141276},
}
@article{tao2011,
author = {Tao, Youshan},
title = {Boundedness in a chemotaxis model with oxygen consumption by bacteria},
journal = {J. Math. Anal. Appl.},
volume = {381},
number = {2},
year = {2011},
pages = {521--529},
doi = {10.1016/j.jmaa.2011.02.041},
}
@article{baghaei2017,
author = {Baghaei, Khadijeh and Khelghati, Ali},
title = {Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant},
journal = {C. R. Math. Acad. Sci. Paris},
volume = {355},
number = {6},
year = {2017},
pages = {633--639},
doi = {10.1016/j.crma.2017.04.009},
}
@article{lankeit2025,
author = {Lankeit, Johannes and Winkler, Michael},
title = {Chemotaxis-consumption interaction: Solvability and asymptotics in general high-dimensional domains},
journal = {Nonlinear Anal.},
volume = {260},
year = {2025},
pages = {113853},
doi = {10.1016/j.na.2025.113853},
eprint = {2502.17338},
archivePrefix = {arXiv},
}
@article{jiang2018,
author = {Jiang, Jie and Wu, Hao and Zheng, Songmu},
title = {Blow-up for a three dimensional {K}eller--{S}egel model with consumption of chemoattractant},
journal = {J. Differential Equations},
volume = {264},
number = {8},
year = {2018},
pages = {5432--5464},
doi = {10.1016/j.jde.2018.01.004},
eprint = {1610.06684},
archivePrefix = {arXiv},
}
@misc{aim-chemotaxis-consumption-classical-continuation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global classical continuation in the chemotaxis-consumption system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/324-chemotaxis-consumption-classical-continuation.md},
note = {Entry 324 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00579, "Global classical continuation in the chemotaxis-consumption system", clopen.solutions.