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Global Existence of Solutions of Nonlocal Geirer-Meinhardt Model and Effect of Nonlocal Operator in Pattern Formation

Md Shah Alam

TL;DR

This paper addresses the global existence and pattern-forming behavior of a nonlocal two-component Gierer–Meinhardt system on a bounded domain, where diffusion is mediated by a convolution operator $\Gamma_i$. A semigroup framework is used to establish global well-posedness and positivity, and a kernel-independent $L^b$ energy functional for $2\le b<\infty$ yields uniform bounds that do not depend on the kernel. A diffusive limit is proven, showing convergence to the classical local GM system as the nonlocal kernel concentrates, with the limiting diffusion scaled by $M=\int_{\mathbb{R}^n}|z|^2\psi(|z|)\,dz$. Numerical simulations illustrate how nonlocal diffusion shapes pattern formation and corroborate the theoretical link between nonlocal and local diffusion.

Abstract

We study the global existence of solutions to a class of nonlocal Geirer-Meinhardt system. This is a two component reaction-diffusion model on a bounded domain in $\mathbb{R}^n$, $n \ge 1$, with nonlocal diffusion given by a nonlocal convolution operator. We have used semigroup theory and derive estimate to guarantee global existence. Then we build an $L^b$ functional to bound our solution independent of the nonlocal convolution kernel for $2 \le b < \infty$. Next, we have used this result to obtain a diffusive limit similar to \cite{laurenccot2023nonlocal} for our model. We also numerically simulate our model to show the formation of patterns by this model and compare the results with the patterns with the traditional local/classical Geirer-Meinhardt model.

Global Existence of Solutions of Nonlocal Geirer-Meinhardt Model and Effect of Nonlocal Operator in Pattern Formation

TL;DR

This paper addresses the global existence and pattern-forming behavior of a nonlocal two-component Gierer–Meinhardt system on a bounded domain, where diffusion is mediated by a convolution operator . A semigroup framework is used to establish global well-posedness and positivity, and a kernel-independent energy functional for yields uniform bounds that do not depend on the kernel. A diffusive limit is proven, showing convergence to the classical local GM system as the nonlocal kernel concentrates, with the limiting diffusion scaled by . Numerical simulations illustrate how nonlocal diffusion shapes pattern formation and corroborate the theoretical link between nonlocal and local diffusion.

Abstract

We study the global existence of solutions to a class of nonlocal Geirer-Meinhardt system. This is a two component reaction-diffusion model on a bounded domain in , , with nonlocal diffusion given by a nonlocal convolution operator. We have used semigroup theory and derive estimate to guarantee global existence. Then we build an functional to bound our solution independent of the nonlocal convolution kernel for . Next, we have used this result to obtain a diffusive limit similar to \cite{laurenccot2023nonlocal} for our model. We also numerically simulate our model to show the formation of patterns by this model and compare the results with the patterns with the traditional local/classical Geirer-Meinhardt model.

Paper Structure

This paper contains 6 sections, 9 theorems, 100 equations, 10 figures.

Table of Contents

  1. Introduction and Main Result
  2. Introduction
  3. Statement of the Main Result
  4. Proof of Main Theorems
  5. Pattern Formation and Numerical Findings
  6. Conclusion and Future Directions

Key Result

Theorem 1

If $(eq:1.3)$ and $(eq:1.4)$ hold, $(u_0,v_0) \in C(\overline{\Omega},\mathbb{R}^2_+)$, and there exists $\varepsilon>0$ so that $u_0(x),v_0(x) \ge \varepsilon$ for all $x \in \Omega$, then (eq:1.2) has a unique global component-wise positive solution $(u(x,t),v(x,t)) \in C^{(0,1)}(\overline{\Omega}

Figures (10)

  • Figure 1: Patterns of activator, $u(x,t)$ in local Geirer-Meinhardt model.
  • Figure 2: Patterns of activator, $u(x,t)$ in nonlocal Geirer-Meinhardt model
  • Figure 3: Patterns of inhibitor, $v(x,t)$ in local Geirer Meinhardt model.
  • Figure 4: Patterns of inhibitor, $v(x,t)$ in nonlocal Geirer Meinhardt model
  • Figure 5: Patterns for activator, $u(x,t)$ in nonlocal Geirer-Meinhardt model with standard deviation 0.5
  • ...and 5 more figures

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6