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

TL;DR

This paper establishes well-posedness, stability, and boundedness results for reaction-nonlocal diffusion systems involving the spectral fractional Laplacian, and explores their dynamics through analytical and numerical methods.

Contribution

It introduces a unified framework for analyzing reaction-nonlocal diffusion equations with inhomogeneous boundary conditions, deriving key properties and applying them to prototype systems.

Findings

Solutions are locally well-posed with maximum principles.

Invariant bounds ensure global existence and boundedness.

Numerical simulations illustrate fractional order effects on patterns.

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 \[ \partial_t u + \epsilon^2(-\Delta)_g^\alpha u = \mathcal{N}(u) \] 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 the impact of fractional orders on pattern formation, consistent with our analytical results.