Wellposedness and dynamics of two types of reaction--nonlocal diffusion systems under the inhomogeneous spectral fractional Laplacian

Reactio-nonlocal diffusion equations model nonlocal transport and anomalous diffusion by replacing the Laplacian with a fractional power, capturing diffusion mechanisms beyond Brownian motion. We primarily study the semilinear problem \[ \partial_t u + \epsilon^2(-\Delta)_g^\alpha u = \mathcal{N}(u) \] allowing constant inhomogeneous Dirichlet boundary condition $u|_{\partial\Omega}=g$. To handle the boundary constraint, we use a harmonic lifting to reformulate the problem as an equivalent homogeneous system with a shifted nonlinearity. Working in \(C_0(\Omega)\), analytic contraction semigroup theory yields the Duhamel formula and quantitative smoothing, implying local wellposedness for locally Lipschitz reactions and a blow-up alternative. The semigroup viewpoint also provides $L^\infty$-contractivity and positivity preservation, which drive pointwise maximum principles and stability bounds. Furthermore, we analyze two prototypes. For the bistable RNDE, we derive an energy dissipation identity and, using a fractional weak maximum principle, obtain an invariant-range property that confines solutions between the two stable steady states. For the nonlocal Gray-Scott system with possibly different fractional diffusion orders, we prove that solutions preserve positivity. Moreover, we identify an explicit \(L^\infty\) invariant set ensuring global boundedness, and derive an eigenfunction-weighted interior \(L^2\) bound. Finally, we perform numerical simulations using a sine pseudospectral discretization and ETDRK4 time-stepping, which illustrate the impact of fractional orders on pattern formation, consistently with our analytical results.