Motivation

Nguyen and Tang study the irreversible catalytic network $\mathcal A+\mathcal B\to\mathcal B+\mathcal C$, $\mathcal B+\mathcal C\to\mathcal A+\mathcal B$, a simple reaction–diffusion system with both a positive and a boundary equilibrium (catalyst used up). They settle every mass regime but one and leave it as a conjecture (AIM problem 416).

Timeline. 2021: Fellner–Morgan–Tang give global classical solutions with uniform bounds. 2026: Nguyen–Tang prove convergence for $M_1<M_2$ and $M_1\ge2M_2$ and local results in between, conjecturing global attraction there.

Setting

On a bounded connected smooth domain $\Omega\subset\mathbb R^n$ with $|\Omega|=1$ and $d_1,d_2,d_3>0$:

$$a_t-d_1\Delta a=b(c-a),\quad b_t-d_2\Delta b=b(c-a),\quad c_t-d_3\Delta c=-b(c-a),$$

with homogeneous Neumann conditions and strictly positive $C^2$ compatible initial data. The masses $M_1=\int_\Omega(a_0+c_0)$, $M_2=\int_\Omega(b_0+c_0)$ are conserved. The equilibria are the positive one $(\frac{M_1}2,\,M_2-\frac{M_1}2,\,\frac{M_1}2)$ and the boundary one $(M_1-M_2,\,0,\,M_2)$.

Formalization target

If $M_2\le M_1<2M_2$, every classical solution converges uniformly on $\Omega$ to the positive equilibrium (no rate required).

Significance

This is the last open regime of Nguyen–Tang's stability analysis. Strict positivity is essential: with $b_0\equiv0$ the solution stays away from the positive equilibrium.

Difficulty

For $M_1\ge M_2$ there is no positive lower bound on $\int_\Omega b$, so entropy dissipation can degenerate near the boundary equilibrium ($b=0$), and the local results do not exclude trajectories approaching it.

Formalization scope

$\mathbb R^n$ is EuclideanSpace ℝ (Fin n) and $\Delta$ is Mathlib's Laplacian. The domain is $\Omega=\{\varphi<0\}$ for a $C^\infty$ function $\varphi$ with $\nabla\varphi\ne0$ on $\{\varphi=0\}$; the Neumann condition is the derivative within $\overline\Omega$ in the direction $\nabla\varphi$. Solutions are continuous on $[0,\infty)\times\overline\Omega$ and $C^{1,2}$ for $t>0$, and $L^\infty$ convergence is TendstoUniformlyOn. The goal quantifies over classical solutions; their existence is proved in the literature (Fellner–Morgan–Tang; Nguyen–Tang, Theorem 2.1) but is not part of this statement.

Selected references

• T. L. Nguyen, B. Q. Tang, Stability analysis of irreversible chemical reaction-diffusion systems with boundary equilibria, Z. Angew. Math. Phys. 77, 2026, 199. DOI
• K. Fellner, J. Morgan, B. Q. Tang, Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions, DCDS-S 14, 2021, 635–651. DOI
• AIM open problems list, problem 416. github.com/MColbrook/AIM