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Global Existence and Boundedness of Gray-Scott Model with Local and Nonlocal Diffusion

Md Shah Alam

TL;DR

The paper addresses global existence and uniform boundedness for a Gray–Scott reaction-diffusion system in which the activator $u$ diffuses via a nonlocal convolution operator $\Gamma$ and the inhibitor $v$ diffuses by the local Laplacian. The authors employ semigroup theory and duality arguments to establish a maximal, then global, solution that remains nonnegative and uniformly bounded in the sup norm for all time, leveraging quasi-positivity and a bootstrapping scheme from $L^1$ to higher $L^p$ spaces. A key contribution is extending global existence results to a mixed diffusion setting in any spatial dimension $n\ge1$, clarifying the well-posedness of the nonlocal–local diffusion combination, and providing computational demonstrations of pattern formation under this framework. The work offers a rigorous mathematical foundation for studying pattern formation under nonlocal transport, with potential implications for chemical, biological, and material systems where anomalous diffusion plays a role.

Abstract

In this paper, we study the global existence of component-wise nonnegative solutions of the Gray-Scott model in $Ω\subset \mathbb{R}^n$, $n \ge 1$, with a mixture of both local and nonlocal diffusion operators. We use semigroup theory with duality arguments to establish the global existence and boundedness of solutions of our model.

Global Existence and Boundedness of Gray-Scott Model with Local and Nonlocal Diffusion

TL;DR

The paper addresses global existence and uniform boundedness for a Gray–Scott reaction-diffusion system in which the activator diffuses via a nonlocal convolution operator and the inhibitor diffuses by the local Laplacian. The authors employ semigroup theory and duality arguments to establish a maximal, then global, solution that remains nonnegative and uniformly bounded in the sup norm for all time, leveraging quasi-positivity and a bootstrapping scheme from to higher spaces. A key contribution is extending global existence results to a mixed diffusion setting in any spatial dimension , clarifying the well-posedness of the nonlocal–local diffusion combination, and providing computational demonstrations of pattern formation under this framework. The work offers a rigorous mathematical foundation for studying pattern formation under nonlocal transport, with potential implications for chemical, biological, and material systems where anomalous diffusion plays a role.

Abstract

In this paper, we study the global existence of component-wise nonnegative solutions of the Gray-Scott model in , , with a mixture of both local and nonlocal diffusion operators. We use semigroup theory with duality arguments to establish the global existence and boundedness of solutions of our model.

Paper Structure

This paper contains 8 sections, 1 theorem, 34 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Introduction
  3. Statement of the Main Result
  4. Proof of Main Result
  5. Patterns
  6. Computational domain and lattice
  7. Visualization
  8. Conclusion

Key Result

Theorem 1

Assume that eq:1.2 and eq:1.3 hold, and let $u_0 \in C(\overline{\Omega}, \mathbb{R}_+)$ and $v_0 \in C^2(\overline{\Omega}, \mathbb{R}_+)$ satisfy $\dfrac{\partial v_0}{\partial \eta} = 0$ on $\partial \Omega$. Then there exists a unique global component-wise nonnegative solution $(u(x,t), v(x,t))$

Figures (2)

  • Figure 1: Gray-Scott patterns with local diffusion.
  • Figure 2: Gray-Scott patterns with local and nonlocal diffusion.

Theorems & Definitions (1)

  • Theorem 1