Global classical solvability for high-growth deterministic reaction–diffusion systems

Establish global-in-time classical solutions for realistic deterministic reaction–diffusion systems arising from chemical reaction networks with reaction nonlinearities of arbitrarily high polynomial growth, thereby resolving the finite-time blow-up question in the deterministic setting.

Background

The paper studies reaction–diffusion systems generated by chemical reaction networks, whose mass-action nonlinearities can have arbitrarily high polynomial growth. Although global classical solvability is known in several low-growth or specially structured regimes, the deterministic theory does not generally prevent finite-time blow-up for realistic networks with super-quadratic reaction terms.

The authors address this unresolved deterministic situation by introducing suitably chosen transport noise and proving global classical solutions with high probability for complex balanced networks without boundary equilibria. Their stochastic result therefore provides a regularization-by-noise theorem rather than a resolution of the corresponding deterministic problem.

References

The existence of global classical solutions for reaction--diffusion systems arising from chemical reaction networks remains a major open problem in the deterministic setting, especially for reactions with high polynomial growth.

— Global classical solutions by transport noise for reaction-diffusion systems with entropy dissipation  (2608.13332 - Agresti et al., 13 Aug 2026) in Abstract; Section 1, paragraph beginning “Since then, the quest to determine the optimal growth rate”

To the best of our knowledge, global classical well-posedness for the corresponding deterministic reaction--diffusion system with arbitrary positive diffusion coefficients remains open.

— Global classical solutions by transport noise for reaction-diffusion systems with entropy dissipation  (2608.13332 - Agresti et al., 13 Aug 2026) in Section 1, Subsection “Concrete reaction networks with large stoichiometric coefficients,” paragraph beginning “As a second example”