Open AccessOn eigenvalue problems arising from nonlocal diffusion models
Author(s) -
Fang Li,
Jérôme Coville,
Xuefeng Wang
Publication year - 2017
Publication title -
discrete and continuous dynamical systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.289
H-Index - 70
eISSN - 1553-5231
pISSN - 1078-0947
DOI - 10.3934/dcds.2017036
Subject(s) - eigenfunction , eigenvalues and eigenvectors , simple (philosophy) , spectrum (functional analysis) , mathematics , diffusion , pure mathematics , maximum principle , mathematical analysis , combinatorics , physics , quantum mechanics , mathematical optimization , philosophy , epistemology , optimal control
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).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 lambda(p) of the real part of the spectrum in two max-min fashions, and prove that in most cases lambda(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 lambda(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 lambda(p) in the max-min fashion, we supply a complete description of the whole spectrum