# American Institute of Mathematical Sciences

February  2017, 37(2): 879-903. doi: 10.3934/dcds.2017036

## On eigenvalue problems arising from nonlocal diffusion models

 1 Center for PDE, East China Normal University, 500 Dongchuan Road, Minhang 200241, Shanghai, China 2 Biostatistique et Processus Spatiaux, INRA, 84000, Avignon, France 3 Department of Mathematics, Southern University of Science and Technology, 1088 Xueyuan Road, Nanshan 518055, Shenzhen, China

FL is supported by NSF of China (No. 11431005), NSF of Shanghai (No. 16ZR1409600).JC is supported by the French ANR through the ANR JCJC project MODEVOL: ANR-13-JS01-0009 and the ANR project NONLOCAL: ANR-13-JS01-0009.XFW is supported by NSF of China (No. 11671190).

Received  June 2015 Revised  September 2015 Published  November 2016

We aim at saying as much as possible about the spectra of three classes of linear diffusion operators involving nonlocal terms. In all but one cases, we characterize the minimum $λ_p$ of the real part of the spectrum in two max-min fashions, and prove that in most cases $λ_p$ is an eigenvalue with a corresponding positive eigenfunction, and is algebraically simple and isolated; we also prove that the maximum principle holds if and only if $λ_p>0$ (in most cases) or $≥ 0$ (in one case). We prove these results by an elementary method based on the strong maximum principle, rather than resorting to Krein-Rutman theory as did in the previous papers. In one case when it is impossible to characterize $λ_p$ in the max-min fashion, we supply a complete description of the whole spectrum.

Citation: Fang Li, Jerome Coville, Xuefeng Wang. On eigenvalue problems arising from nonlocal diffusion models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 879-903. doi: 10.3934/dcds.2017036
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