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On eigenvalue problems arising from nonlocal diffusion models

  • Author Bio: E-mail address: fli@cpde.ecnu.edu.cn; E-mail address: jerome.coville@inra.fr; E-mail address: wangxf@sustc.edu.cn
  • Xuefeng Wang, wangxf@sustc.edu.cn
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  • 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.

    Mathematics Subject Classification: Primary:45C05, 45K05;Secondary:35P15.


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