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Dynamics of the reversible Gray-Scott model and convergence to its irreversible limit

Philippe Laurençot, Christoph Walker

TL;DR

This work analyzes a reversible Gray-Scott reaction-diffusion system with four species, establishing global well-posedness in $L_p$ and revealing a Lyapunov-driven dichotomy: trajectories converge to one of two spatially homogeneous equilibria. It provides a detailed stability picture, proving exponential stability of the interior equilibrium and employing center-manifold theory to understand the boundary equilibrium's dynamics. A rigorous singular-limit result connects the reversible model to the classical irreversible Gray-Scott model, showing convergence under a coordinated vanishing of certain rates and diffusion coefficients via duality-based $L_{2+\delta}$ estimates. Overall, the paper clarifies how reversibility shapes long-time behavior, relocating pattern formation to transient dynamics and linking reversible dynamics to the classical model in a precise limit.

Abstract

Well-posedness of a reversible variant of the Gray-Scott model is shown, along with the convergence of each trajectory to one of the two spatially homogeneous steady states. The principle of linearized stability provides the local attractivity at an exponential rate of the stable steady state, while the long-term limit is identified with the help of center manifold theory. Finally, convergence to the classical Gray-Scott model is proved for an appropriate choice of parameters.

Dynamics of the reversible Gray-Scott model and convergence to its irreversible limit

TL;DR

This work analyzes a reversible Gray-Scott reaction-diffusion system with four species, establishing global well-posedness in and revealing a Lyapunov-driven dichotomy: trajectories converge to one of two spatially homogeneous equilibria. It provides a detailed stability picture, proving exponential stability of the interior equilibrium and employing center-manifold theory to understand the boundary equilibrium's dynamics. A rigorous singular-limit result connects the reversible model to the classical irreversible Gray-Scott model, showing convergence under a coordinated vanishing of certain rates and diffusion coefficients via duality-based estimates. Overall, the paper clarifies how reversibility shapes long-time behavior, relocating pattern formation to transient dynamics and linking reversible dynamics to the classical model in a precise limit.

Abstract

Well-posedness of a reversible variant of the Gray-Scott model is shown, along with the convergence of each trajectory to one of the two spatially homogeneous steady states. The principle of linearized stability provides the local attractivity at an exponential rate of the stable steady state, while the long-term limit is identified with the help of center manifold theory. Finally, convergence to the classical Gray-Scott model is proved for an appropriate choice of parameters.

Paper Structure

This paper contains 8 sections, 17 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Well-Posedness and Liapunov Functionals
  4. Long-Term Behavior
  5. Long-Term Convergence
  6. Stability of Equilibria
  7. Boundary Steady State
  8. Convergence to the Classical Gray-Scott Model

Key Result

Theorem 2.1

Let $p\in (n,\infty)$ and $\varrho>0$. Given $\mathbf{u}^0\in \mathcal{Z}_{p,\varrho}$ there exists a unique global strong solution $\mathbf{u}=\mathbf{u}(\cdot;\mathbf{u}^0)$ to U in $\mathcal{Z}_{p,\varrho}$; that is, Moreover, for any $t_0>0$, and, if $\mathbf{u}^0\in \mathcal{Z}_{p,\varrho}^+$, then $\mathbf{u}(t)\in \mathcal{Z}_{p,\varrho}^+$ for all $t\ge 0$. In fact, the mapping $(t,\math

Theorems & Definitions (33)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • proof : Proof of \ref{['T1']}
  • Lemma 4.1
  • ...and 23 more