Fractional diffusion with Neumann boundary conditions: the logistic equation, Discrete Contin

Motivated by experimental studies on the anomalous diffusion of bi-ological 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 realiza-tion of such operator as performed in [7] 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.