Fractional diffusion with Neumann boundary conditions: the logistic equation

By Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. 2012

74 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/0a1705e7cbe0ae01b25e10067df9d6f6ba8fc8b8

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In Applied Mathematics

In Computational Theory and Mathematics

In Modeling and Simulation

Citations in the corpus

Regularity results for solutions of mixed local and nonlocal elliptic equations (2022, background): “The analysis of different types of mixed operators motivated by biological questions has also carried out in [19, 26, 27].”
Nonlocal Diffusion and Applications (2015, background): “We observe that processes of this type occur in nature quite often, see in particular the biological observations in [91, 62] and the mathematical discussions in [63, 57, 72, 69].”
A Brezis-Nirenberg type result for mixed local and nonlocal operators (2022, unknown): “…subject of investigation, arising naturally in several ﬁelds, for instance as the super-position of diﬀerent types of stochastic processes such as a classical random walk and a L´evy ﬂight, which has also interesting application in the study of optimal animal foraging strategies, see [20,21,26,28].”