Fractional diffusion with Neumann boundary conditions: The logistic equation

Eugenio Montefusco; Benedetta Pellacci; Gianmaria Verzini

doi:10.3934/dcdsb.2013.18.2175

Article Contents

2013, Volume 18, Issue 8: 2175-2202. Doi: 10.3934/dcdsb.2013.18.2175

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Fractional diffusion with Neumann boundary conditions: The logistic equation

1.
Dipartimento di Matematica, Sapienza Università di Roma, Piazzale A. Moro 5, 00185 Roma
2.
Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli Parthenope, Centro Direzionale Isola C4, 80143 Napoli
3.
Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano

Received: January 31, 2013

Revised: April 30, 2013

Published: September 2013

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Abstract

Motivated by experimental studies on the anomalous diffusion of biological populations, we study the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Such operator arises in the continuous limit for long jumps random walks with reflecting barriers. Existence and uniqueness results for positive solutions are proved in the case of indefinite nonlinearities of logistic type by means of bifurcation theory.

Keywords:

Spectral fractional Laplacian,
eigenvalue problems for nonlocal operators,
bifurcation theory.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35J65, 92D25, 35B32.

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