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

2025/09/19 by Alam, Md Shah
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.15598

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 ℝⁿ, n ≥ 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 ≤ b < ∞. Next, we have used this result to obtain a diffusive limit similar to \citelaurenccot2023nonlocal 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.

Citations

• On a nonlinear nonlocal reaction-diffusion system applied to image restoration
• A nonlocal Gray-Scott model: well-posedness and diffusive limit
• The Dirichlet problem for some nonlocal diffusion equations
• Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory
• The chemical basis of morphogenesis

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