# Fractional diffusion with Neumann boundary conditions: the logistic equation

@article{Montefusco2012FractionalDW,
title={Fractional diffusion with Neumann boundary conditions: the logistic equation},
author={Eugenio Montefusco and Benedetta Pellacci and Gianmaria Verzini},
journal={arXiv: Analysis of PDEs},
year={2012}
}
• Published 2 August 2012
• Mathematics
• arXiv: Analysis of PDEs
Motivated by experimental studies on the anomalous diffusion of biological populations, we introduce a nonlocal differential operator which can be interpreted as the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Moreover, we study related linear and nonlinear problems exploiting a local realization of such operator as performed in [X. Cabre' and J. Tan. Positive solutions of nonlinear problems involving the square root of the Laplacian…
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## References

SHOWING 1-10 OF 36 REFERENCES
The Neumann problem for nonlocal nonlinear diffusion equations
• Mathematics
• 2008
Abstract.We study nonlocal diffusion models of the form $$(\gamma(u))_t (t, x) = \int_{\Omega} J(x-y)(u(t, y) - u(t, x))\, dy.$$ Here Ω is a bounded smooth domain andγ is a maximal monotone graph in
Boundary blow up solutions for fractional elliptic equations
• Mathematics
Asymptot. Anal.
• 2012
This work obtains existence and boundary behavior of solution under different hypothesis on f and g, and proves uniqueness of positive solutions.
Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian
• Mathematics
• 2008
We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent
An Extension Problem Related to the Fractional Laplacian
• Mathematics
• 2007
The operator square root of the Laplacian (− ▵)1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the
Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations
• Mathematics
• 2010
We investigate stable solutions of elliptic equations of the type where n ≥ 2, s ∈ (0, 1), λ ≥0 and f is any smooth positive superlinear function. The operator (− Δ) s stands for the fractional
The periodic patch model for population dynamics with fractional diffusion
• Mathematics
• 2010
Fractional diffusions arise in the study of models from population dynamics. In this paper, we derive a class of integro-differential reaction-diffusion equations from simple principles. We then
Extension Problem and Harnack's Inequality for Some Fractional Operators
• Mathematics
• 2009
The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional