Positive equilibrium attraction for an irreversible catalytic reaction
Let $\Omega \subset \mathbb{R}^n$ be a bounded connected smooth domain with $|\Omega| = 1$, and let $d_1, d_2, d_3 > 0$. Consider the unique global bounded classical solution of $$a_t - d_1 \Delta a = b(c - a), \qquad b_t - d_2 \Delta b = b(c - a), \qquad c_t - d_3 \Delta c = -b(c - a)$$ with homogeneous Neumann boundary conditions, where $a_0, b_0, c_0 \in C^2(\overline{\Omega})$ are strictly positive and satisfy the Neumann compatibility conditions. Put $$M_1 = \int_\Omega (a_0 + c_0), \qquad M_2 = \int_\Omega (b_0 + c_0).$$ Nguyen and Tang studied this system for the irreversible network $A + B \to C$, $B + C \to A + 2B$, in which $B$ acts as a catalyst. They proved that for nonnegative bounded data it has a unique global classical solution, bounded uniformly in time; the integrals of $a + c$ and $b + c$ are conserved. For $M_2 \le M_1 < 2M_2$ the system has two nonnegative spatially homogeneous equilibria with these masses, the positive one $(M_1/2,\, M_2 - M_1/2,\, M_1/2)$ and the boundary equilibrium $(M_1 - M_2,\, 0,\, M_2)$, and they proved that the positive one is locally exponentially stable. For $M_1 < M_2$, where the positive equilibrium is the only nonnegative one, they proved that every solution converges to it exponentially.
Problem. Assume $M_2 \le M_1 < 2M_2$. Is it true that $$\lim_{t \to \infty} \Bigl[\|a(t) - M_1/2\|_\infty + \|b(t) - (M_2 - M_1/2)\|_\infty + \|c(t) - M_1/2\|_\infty\Bigr] = 0?$$
Source: Nguyen and Tang (2026), the conjecture after Table 1, for nonnegative data. Stated by AIM — Open Applied Problems for strictly positive C^2 data compatible with the Neumann condition.
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{nguyen2026,
author = {Nguyen, Thi Lien and Tang, Bao Quoc},
title = {Stability analysis of irreversible chemical reaction-diffusion systems with boundary equilibria},
journal = {Z. Angew. Math. Phys.},
volume = {77},
number = {7},
year = {2026},
pages = {Paper No. 199},
doi = {10.1007/s00033-026-02847-0},
eprint = {2410.22928},
archivePrefix = {arXiv},
}
@misc{aim-catalytic-reaction-diffusion-positive-attractor,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Positive equilibrium attraction for an irreversible catalytic reaction}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/416-catalytic-reaction-diffusion-positive-attractor.md},
note = {Entry 416 at commit 24edaad, 2026-10-15},
}
Resolution claims
An irreversible reaction--diffusion system can have both a positive equilibrium and a stationary state at which the catalyst is absent. For the three-species system studied by Nguyen and Tang, we prove that every strictly positive compatible initial triple converges uniformly to the positive equilibrium whenever the conserved masses satisfy \(M_2\le M_1<2M_2\). The result holds on bounded connected smooth domains in …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/manuscript.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00782, "Positive equilibrium attraction for an irreversible catalytic reaction", clopen.solutions.