Fractional diffusion with Neumann boundary conditions: the logistic equation

TL;DR

This paper introduces a nonlocal operator modeling anomalous biological diffusion with Neumann boundary conditions, and analyzes related logistic equations to establish existence and uniqueness of positive solutions.

Contribution

It extends the spectral square root of the Laplacian to Neumann boundary conditions and applies variational and bifurcation methods to nonlinear logistic problems.

Findings

Established existence of positive solutions

Proved uniqueness under certain conditions

Applied variational and bifurcation techniques

Abstract

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. Adv. Math. 2010] for Dirichlet homogeneous data. In particular we tackle a class of nonautonomous nonlinearities of logistic type, proving some existence and uniqueness results for positive solutions by means of variational methods and bifurcation theory.