Global attraction for complex-balanced reaction–diffusion with boundary equilibria
Let $\Omega \subset \mathbb{R}^d$ be a bounded, connected, smooth domain with $|\Omega| = 1$. A finite mass-action network on $N$ species consists of reactions $y_r \to y'_r$ between complexes $y_r, y'_r \in \mathbb{N}_0^N$ with rates $k_r > 0$; with $z^y = \prod_i z_i^{y_i}$ put $$f(z) = \sum_r k_r z^{y_r} (y'_r - y_r), \qquad S = \operatorname{span}\{y'_r - y_r\}_r.$$ The network is assumed complex balanced: some $c_* \in (0, \infty)^N$ satisfies, for every complex $y$, $$\sum_{r : y_r = y} k_r c_*^{y_r} = \sum_{r : y'_r = y} k_r c_*^{y_r}.$$ Then every stoichiometric class $(c + S) \cap [0, \infty)^N$ with $c \in (0, \infty)^N$ contains exactly one positive equilibrium, but it may also contain boundary equilibria, which have some zero components. For diffusion constants $d_i > 0$ consider $$\partial_t u_i = d_i \Delta u_i + f_i(u) \quad \text{in } \Omega, \qquad \partial_\nu u_i = 0 \quad \text{on } \partial\Omega.$$ Let $u_0$ be smooth, strictly positive and compatible with the boundary condition, suppose that its classical solution exists for all $t \ge 0$ with $\sup_{t \ge 0} \|u(t)\|_{L^\infty} < \infty$, and let $c_\infty$ be the positive equilibrium with $\int_\Omega u_0 - c_\infty \in S$. Desvillettes, Fellner and Tang proved exponential convergence to $c_\infty$ for networks without boundary equilibria and, for suitable initial data, for two particular networks with boundary equilibria (2017). Craciun, Jin, Pantea and Tudorascu proved convergence to the positive equilibrium for a three-species network and a class of two-species reversible networks with boundary equilibria (2021).
Problem. Must $\|u(t) - c_\infty\|_{L^1(\Omega)} \to 0$ as $t \to \infty$, even when the stoichiometric class of $c_\infty$ contains boundary equilibria? No uniform positive lower bound on $u$ is assumed, and the diffusion constants are arbitrary; they need not be large or nearly equal.
Source: Desvillettes, Fellner and Tang (2017), Remark 3.6, for general complex balanced reaction–diffusion systems with boundary equilibria. Stated by AIM — Open Applied Problems for globally bounded classical solutions with smooth, strictly positive initial data.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, chemical reaction networks, convergence to equilibrium, reaction–diffusion systems
References
@article{desvillettes2017,
author = {Desvillettes, Laurent and Fellner, Klemens and Tang, Bao Quoc},
title = {Trend to equilibrium for reaction-diffusion systems arising from complex balanced chemical reaction networks},
journal = {SIAM J. Math. Anal.},
volume = {49},
number = {4},
year = {2017},
pages = {2666--2709},
doi = {10.1137/16M1073935},
eprint = {1604.04536},
archivePrefix = {arXiv},
}
@article{craciun2021,
author = {Craciun, Gheorghe and Jin, Jiaxin and Pantea, Casian and Tudorascu, Adrian},
title = {Convergence to the complex balanced equilibrium for some chemical reaction-diffusion systems with boundary equilibria},
journal = {Discrete Contin. Dyn. Syst. Ser. B},
volume = {26},
number = {3},
year = {2021},
pages = {1305--1335},
doi = {10.3934/dcdsb.2020164},
eprint = {1812.07707},
archivePrefix = {arXiv},
}
@misc{aim-complex-balanced-pde-global-attractor,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Global attraction for complex-balanced reaction–diffusion with boundary equilibria}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/484-complex-balanced-pde-global-attractor.md},
note = {Entry 484 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-00581, "Global attraction for complex-balanced reaction–diffusion with boundary equilibria", clopen.solutions.