Nonlocal eigenvalue problems arising in a generalized phase-field-type system

  • Shuichi Jimbo
  • Yoshihisa MoritaEmail author
Original Paper Area 1


We deal with a generalized phase-field-type system that arises as a transformed system of reaction-diffusion equations with a conservation law. We consider the stationary problem which is reduced to a scalar elliptic equation with a nonlocal term, and study the linearized eigenvalue problem. We first prove by the spectral comparison argument that the number of unstable eigenvalues for the problem coincides with the one of the linearized eigenvalue problem for the original system. We next show a limiting behavior of eigenvalues for the scalar problem as the coefficient of the nonlocal term goes to infinity.


Phase-field-type system Spectral comparison Linearized eigenvalue problem Nonlocal eigenvalue problem Mini-max principle 

Mathematics Subject Classification

35B35 35J20 35K57 



This research was partially supported by JSPS KAKENHI Grant Number, 26287025. The first author was also partially supported by JSPS KAKENHI Grant Number 25400153 and the second author was by JSPS KAKENHI Grant Number, 26247013, and JST, CREST. The main part of this research was complete when the second author was visiting EPFL. He takes an opportunity to express their warm hospitality during his visit and special thanks to Professor Hoai-Minh Nguyen. The authors also would like to thank the referees for their valuable comments to improve the manuscript.


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Copyright information

© The JJIAM Publishing Committee and Springer Japan KK 2017

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan
  2. 2.Department of Applied Mathematics and InformaticsRyukoku UniversitySeta OtsuJapan

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