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

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.