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Global Analysis of the Gray-Scott Model with Fractional-Classical Diffusion

Md Shah Alam

TL;DR

The paper addresses the Gray-Scott reaction-diffusion system under mixed diffusion, combining a fractional Laplacian for $u$ with a local Laplacian for $v$ on a bounded domain under Neumann conditions. It employs semigroup theory and duality estimates to prove global existence of component-wise nonnegative solutions and uniform-in-time $L^\infty$ bounds, with a bootstrap from $L^1$ to higher norms. The authors demonstrate that fractional diffusion qualitatively reshapes pattern formation compared to the purely local model, supported by numerical simulations across varying fractional order $s$. The work provides a rigorous analytical framework for nonlocal diffusion in reaction-diffusion systems and informs modeling of pattern formation where different species diffuse at different (local vs nonlocal) rates.

Abstract

We analyze the Gray-Scott reaction--diffusion system on $Ω\subset\mathbb{R}^n$ ($n\ge 1$) with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of component-wise nonnegative solutions and establish uniform-in-time bounds. Numerical simulations illustrate pattern formation and highlight qualitative differences between the purely local and mixed-diffusion models.

Global Analysis of the Gray-Scott Model with Fractional-Classical Diffusion

TL;DR

The paper addresses the Gray-Scott reaction-diffusion system under mixed diffusion, combining a fractional Laplacian for with a local Laplacian for on a bounded domain under Neumann conditions. It employs semigroup theory and duality estimates to prove global existence of component-wise nonnegative solutions and uniform-in-time bounds, with a bootstrap from to higher norms. The authors demonstrate that fractional diffusion qualitatively reshapes pattern formation compared to the purely local model, supported by numerical simulations across varying fractional order . The work provides a rigorous analytical framework for nonlocal diffusion in reaction-diffusion systems and informs modeling of pattern formation where different species diffuse at different (local vs nonlocal) rates.

Abstract

We analyze the Gray-Scott reaction--diffusion system on () with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of component-wise nonnegative solutions and establish uniform-in-time bounds. Numerical simulations illustrate pattern formation and highlight qualitative differences between the purely local and mixed-diffusion models.

Paper Structure

This paper contains 6 sections, 1 theorem, 35 equations, 4 figures.

Table of Contents

  1. Introduction and Main Results
  2. Introduction
  3. Statement of the Main Results
  4. Proof of Main Results
  5. Pattern Formation
  6. Conclusion

Key Result

Theorem 1

If $(u_0,v_0) \in L^2(\Omega,\mathbb{R}^2_+)$ with $\frac{\partial}{\partial \eta} v_0=0$ on $\partial \Omega=0$ then there exists a unique global component-wise nonnegative solution $(u(x,t),v(x,t))$ to (eq:1.1) and both $u$ and $v$ remain uniformly bounded in the sup norm for all time..

Figures (4)

  • Figure 1: Pattern of local Gray-Scott model.
  • Figure 2: Pattern of mixed Gray-Scott model with $s=0.25$.
  • Figure 3: Pattern of mixed Gray-Scott model with $s=0.50$.
  • Figure 4: Pattern of mixed Gray-Scott model with $s=0.75$.

Theorems & Definitions (1)

  • Theorem 1