Wellposedness and dynamics of two types of reaction--nonlocal diffusion systems under the inhomogeneous spectral fractional Laplacian
Pu Yuan, Paul A. Zegeling
TL;DR
This work develops a unified semigroup-based framework for semilinear reaction--nonlocal diffusion equations with the spectral fractional Laplacian on bounded domains under inhomogeneous Dirichlet data. By employing a harmonic lifting, the authors reduce to a homogeneous problem and leverage analytic contraction semigroups to obtain a Duhamel representation, instantaneous smoothing, and local wellposedness for locally Lipschitz nonlinearities, along with a blow-up alternative and, where possible, global existence. They derive an energy dissipation identity for a bistable prototype and establish an invariant-range property via a fractional weak maximum principle, while for a nonlocal Gray--Scott system they prove positivity preservation and an explicit $L^{\infty}$ invariant set plus an interior $L^2$ bound, all supported by numerical simulations using sine pseudospectral discretization and ETDRK4 time-stepping. The numerical experiments illustrate how the fractional orders shape spreading and pattern formation, in line with the analytical energy and maximum-principle results. The framework clarifies the impact of the spectral operator on bounded domains and lays the groundwork for extensions to other RNDEs and to Dirichlet (integral) fractional Laplacians.
Abstract
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 \[ \partial_t u + ε^2(-Δ)_g^αu = \mathcal{N}(u) \] allowing constant inhomogeneous Dirichlet boundary condition $u|_{\partialΩ}=g$. 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 \(C_0(Ω)\), 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 the impact of fractional orders on pattern formation, consistent with our analytical results.