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

Abstract: We analyze the Gray--Scott reaction--diffusion system on $\Omega\subset\mathbb{R}<sup>n$ ($n\ge 1$) with mixed diffusion combining local and nonlocal operators. Using semigroup methods and duality estimates, we prove global existence of componentwise 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.