Global classical continuation for a reversible mass-action reaction
Let $n, I, J \ge 1$ and let $\Omega \subset \mathbb{R}^n$ be a bounded connected domain with smooth boundary. For distinct species $A_1, \dots, A_I, B_1, \dots, B_J$ consider the single reversible reaction $$\sum_{i=1}^{I} \alpha_i A_i \rightleftharpoons \sum_{j=1}^{J} \beta_j B_j, \qquad \alpha_i, \beta_j \in \{1, 2, \dots\},$$ with mass-action rate $R(a, b) = k_+ \prod_i a_i^{\alpha_i} - k_- \prod_j b_j^{\beta_j}$, $k_\pm > 0$, and the system $$\partial_t a_i - d_i \Delta a_i = -\alpha_i R(a, b), \qquad \partial_t b_j - e_j \Delta b_j = \beta_j R(a, b) \qquad \text{in } (0, \infty) \times \Omega,$$ with constant diffusivities $d_i, e_j > 0$, homogeneous Neumann boundary conditions, and nonnegative smooth initial data on $\overline{\Omega}$ satisfying them. A classical solution is continuous up to $t = 0$, $C^1$ in time and $C^2$ in space for $t > 0$, and satisfies the equations and boundary conditions pointwise. Fischer proved that such systems have global renormalized solutions (2015), and that a renormalized solution coincides with the classical one as long as the latter exists (2017). Fellner, Morgan and Tang proved that classical solutions are global in every dimension for quasi-positive systems with mass control and quadratic, or slightly superquadratic, growth (2020); this covers the reactions with $\sum_i \alpha_i \le 2$ and $\sum_j \beta_j \le 2$.
Problem. For every such reaction, domain, choice of coefficients and initial data, must the unique local nonnegative classical solution extend to every finite time, with $$\sup_{0 \le t \le T} \Bigl( \sum_{i=1}^{I} \|a_i(t)\|_{L^\infty(\Omega)} + \sum_{j=1}^{J} \|b_j(t)\|_{L^\infty(\Omega)} \Bigr) < \infty \quad \text{for each } T < \infty?$$ The bound may depend on $T$, the data and all coefficients. No bound on the reaction orders, smallness of the data or closeness of the diffusivities is assumed.
Source: Fischer (2017), §1, who writes that global existence of smooth solutions is conjectured for a single reversible reaction with mass-action kinetics. Stated by AIM — Open Applied Problems for a reaction between disjoint sets of species.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Keywords
- AIM Open Applied Problems, mass-action kinetics, reaction–diffusion systems
References
@article{fischer2017,
author = {Fischer, Julian},
title = {Weak--strong uniqueness of solutions to entropy-dissipating reaction--diffusion equations},
journal = {Nonlinear Anal.},
volume = {159},
year = {2017},
pages = {181--207},
doi = {10.1016/j.na.2017.03.001},
eprint = {1703.00730},
archivePrefix = {arXiv},
}
@article{fischer2015,
author = {Fischer, Julian},
title = {Global existence of renormalized solutions to entropy-dissipating reaction--diffusion systems},
journal = {Arch. Ration. Mech. Anal.},
volume = {218},
number = {1},
year = {2015},
pages = {553--587},
doi = {10.1007/s00205-015-0866-x},
}
@article{fellner2020,
author = {Fellner, Klemens and Morgan, Jeff and Tang, Bao Quoc},
title = {Global classical solutions to quadratic systems with mass control in arbitrary dimensions},
journal = {Ann. Inst. H. Poincar{\'e} C Anal. Non Lin{\'e}aire},
volume = {37},
number = {2},
year = {2020},
pages = {281--307},
doi = {10.1016/j.anihpc.2019.09.003},
eprint = {1808.01315},
archivePrefix = {arXiv},
}
@misc{aim-reversible-reaction-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 for a reversible mass-action reaction}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/304-reversible-reaction-classical-continuation.md},
note = {Entry 304 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-00637, "Global classical continuation for a reversible mass-action reaction", clopen.solutions.