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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.21422 |
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| _version_ | 1866911405961117696 |
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| author | Yuan, Pu Zegeling, Paul A. |
| author_facet | Yuan, Pu Zegeling, Paul A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21422 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wellposedness and dynamics of two types of reaction--nonlocal diffusion systems under the inhomogeneous spectral fractional Laplacian Yuan, Pu Zegeling, Paul A. Analysis of PDEs 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. |
| title | Wellposedness and dynamics of two types of reaction--nonlocal diffusion systems under the inhomogeneous spectral fractional Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.21422 |