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

  • Shuichi Jimbo
  • Yoshihisa MoritaEmail author
Original Paper Area 1
  • 107 Downloads

Abstract

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.

Keywords

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

Mathematics Subject Classification

35B35 35J20 35K57 

Notes

Acknowledgements

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.

References

  1. 1.
    Bates, P.W., Fife, P.C.: Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening. Physica D 43, 335–348 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caginalp, G.: An analysis of a phase field model of a free boundary. Arch. Ration. Mech. Anal. 92, 205–245 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)CrossRefGoogle Scholar
  4. 4.
    Courant, R., Hilbert, D.: Method of Mathematical Physics, vol. I. Wiley Interscience, New York (1953)Google Scholar
  5. 5.
    Chen, C.-N., Jimbo, S., Morita, Y.: Spectral comparison and gradient-like property in the FitzHugh-Nagumo type equations. Nonlinearity 28, 1003–1016 (2015)CrossRefzbMATHGoogle Scholar
  6. 6.
    Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Fife, P.C.: Models for phase separation and their mathematics. Electron. J. Diff. Equ. 2000(48), 1–26 (2000)Google Scholar
  8. 8.
    Fix, G.J.: Phase filed methods for free boundary problems. In: Fasano, A., Primicero, M. (Eds) Free Boundary Problems: Theory and Applications. Pitman, London, pp 580–589 (1983)Google Scholar
  9. 9.
    Gurtin, M.E., Matano, H.: On the structure of equilibrium phase transitions within the gradient theory of fluids. Q. Appl. Math. 156, 301–317 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, New York (1981)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jimbo, S., Morita, Y.: Lyapunov function and spectrum comparison for a reaction-diffusion system with mass conservation. J. Diff. Equ. 255, 1657–1683 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1995)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kosugi, S., Morita, Y., Yotsutani, S.: Stationary solutions to the one-dimensional Cahn-Hilliard equation: proof by the complete elliptic integrals. Discrete Contin. Dyn. Syst. 19, 609–629 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Latos, E., Morita,Y., Suzuki, T.: Stability and spectral comparison of a reaction-diffusion system with mass conservation (preprint) Google Scholar
  15. 15.
    Latos, E., Suzuki, T.: Global dynamics of a reaction-diffusion system with mass conservation. J. Math. Anal. Appl. 411, 107–118 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Miyamoto, Y.: Stability of a boundary spike layer for the Gierer-Meinhardt system. Eur. J. Appl. Math. 16, 467–491 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Miyamoto, Y.: An instability criterion for activator-inhibitor systems in a two-dimensional ball. J. Diff. Equ. 229, 494–508 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mizhohata, S.: The Theory of Partial Differential Equations. Cambridge University Press, Cambridge (1979)Google Scholar
  19. 19.
    Morita, Y.: Spectrum comparison for a conserved reaction-diffusion system with a variational property. J. Appl. Anal. Comput. 2, 57–71 (2012)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Morita, Y., Ogawa, T.: Stability and bifurcation of nonconstant solutions to a reaction-diffusion system with conservation of mass. Nonlinearity 23, 1387–1411 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ni, W.M., Takagi, I., Yanagida, E.: Stability of least energy patterns of the shadow system for an activator? Inhibitor model. Jpn. J. Ind. Appl. Math. 18, 259–272 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nishiura, Y.: Coexistence of infinitely many stable solutions to reaction-diffusion systems in the singular limit. Dyn. Rep. 3, 25–103 (1994)CrossRefzbMATHGoogle Scholar
  23. 23.
    Nishiura, Y.: Global structure of bifurcating solutions of some reaction-diffusion systems. SIAM J. Math. Anal. 13, 555–593 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Nishiura, Y., Fujii, H.: Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J. Math. Anal. 18, 1726–1770 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nishiura, Y., Mimura, M., Ikeda, H., Fujii, H.: Singular limit analysis of stability of traveling wave solutions in bistable reaction-diffusion systems. SIAM J. Math. Anal. 21, 85–122 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Novick-Cohen, A.: On the viscous Chan-Hilliard equation. In: Ball, J.M. (ed.) Matherial Instabilities in Continuum Mechanics and Related Mathematical Problems, pp. 329–342. Clarendon, Oxford (1988)Google Scholar
  27. 27.
    Otsuji, M., Ishihara, S., Co, C., Kaibuchi, K., Mochizuki, A., Kuroda, S.: A mass conserved reaction-diffusion system captures properties of cell polarity. PLoS Comput. Biol. 3, 1040–1054 (2007)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Ohnishi, I., Nishiura, Y.: Spectral comparison between the second and the fourth order equations of conservative type with non-local terms. Jpn. J. Ind. Appl. Math. 15, 253–262 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Rothe, F.: Global Solutions of Reaction-Diffusion Systems. Lecture Notes in Mathematics, vol. 1072. Springer, Berlin (1984)Google Scholar
  30. 30.
    Suzuki, T., Tasaki, S.: Stationary Fix-Caginalp equation with non-local term. Nonlinear Anal. 71, 1329–1349 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wei, J.: On single interior spike solutions of the Gierer-Meinhardt system: uniqueness and spectrum estimates. Eur. J. Appl. Math. 10, 353–378 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wei, J., Winter, M.: Mathematical Aspects of Pattern Formation in Biological Systems. Springer, London (2014)CrossRefzbMATHGoogle Scholar
  33. 33.
    Wei, J., Zhang, L.: On a nonlocal eigenvalue problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30, 41–61 (2001)MathSciNetzbMATHGoogle Scholar

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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