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

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