Stability and Spectral Comparison of a Reaction–Diffusion System with Mass Conservation

  • Evangelos Latos
  • Yoshihisa MoritaEmail author
  • Takashi Suzuki


We study the global-in-time behavior of solutions to a reaction–diffusion system with mass conservation, as proposed in the study of cell polarity, particularly, the second model of the work by Otsuji et al. (PLoS Comput Biol 3:e108, 2007). First, we show the existence of a Lyapunov function and confirm the global-in-time existence of the solution with compact orbit. Then we study the stability and instability of stationary solutions by using the semi-unfolding-minimality property and the spectral comparison. As a result the dynamics near the stationary solutions is qualitatively characterized by a variational function.


Reaction diffusion system Mass conservation Cell polarity Global-in-time behavior Lyapunov function Spectral comparison 

Mathematics Subject Classification

35K457 92C37.1 



The authors would like to thank the referee for careful reading of the manuscript and the valuable comments for the improvement of the first version. The first author was supported from DFG Project CH 955/3-1. The second author was partially supported by the Grand-in-Aid for Scientific Research (A) No. 26247013, (B) No. 26287025 and Challenging Exploratory Research No. 24654044, Japan Society for the Promotion of Science. The third author was partially supported by JST-CREST and JSPS Core to Core Project.


  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.
    Chen, C.-N., Jimbo, S., Morita, Y.: Spectral comparison and gradient-like property in the FitzHugh-Nagmo type equations. Nonlinearity 28, 1003–1016 (2015)CrossRefzbMATHGoogle Scholar
  4. 4.
    Davis, E.B.: Spectral Theory and Differential Operators. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  5. 5.
    Fix, G.J.: Phase field methods for free boundary problems. In: Fasano, A., Primicerio, M. (eds.) Free Boundary Problems: Theory and Applications, pp. 580–589. Pitman, London (1983)Google Scholar
  6. 6.
    Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence (1989)Google Scholar
  7. 7.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Math, vol. 840. Springer, Berlin (1981)Google Scholar
  8. 8.
    Ishihara, S., Otsuji, M., Mochizuki, A.: Transient and steady state of mass-conserved reaction–diffusion systems. Phys. Rev. E 75, 015203(R) (2007)CrossRefGoogle Scholar
  9. 9.
    Ito, A., Suzuki, T.: Asymptotic behavior of the solution to the non-isothermal phase field equation. Nonlinear Anal. 64, 2454–2479 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ito, A., Suzuki, T.: Asymptotic behavior of the solution to the non-isothermal phase separation. Nonlinear Anal. 68, 1825–1843 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jimbo, S., Morita, Y.: Lyapunov function and spectrum comparison for a reaction–diffusion system with mass conservation. J. Differ. Equ. 255, 1657–1683 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kouachi, S.: Existence of global solutions to reaction–diffusion systems via a Lyapunov functional. Erect. J. Differ. Equ. 2001–68, 1–10 (2001)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Latos, E., Suzuki, T.: Global dynamics of a reaction–diffusion system with mass conservation. J. Math. Anal. Appl. 411, 107–118 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mori, Y., Jilkine, A., Edelstein-Keshet, L.: Wave-pinning and cell polarity from bistable reaction–diffusion system. Biophys. J. 94, 3684–3697 (2008)CrossRefGoogle Scholar
  15. 15.
    Morita, Y.: Spectrum comparison for a conserved reaction–diffusion system with a variational property. J. Appl. Anal. Comp. 2, 57–71 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    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, e108 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Pawłov, I., Suzuki, T., Tasaki, S.: Stationary solutions to a strain-gradient type thermovisocelastic system. Differ. Integral Equ. 25, 289–340 (2012)Google Scholar
  19. 19.
    Pham, K., Chauviere, A., Hatzikirou, H., Li, X., Byrne, H.M., Cristini, V., Lowengrub, J.: Density-dependent quiescence in glioma invasion: instability in a simple reaction–diffusion model for the migration/proliferation dichotomy. J. Biol. Dyn. 6, 54–71 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rothe, F.: Global Solutions of Reaction–Diffusion Equations. Lecture Notes in Math, vol. 1072. Springer, Berlin (1984)Google Scholar
  21. 21.
    Suzuki, T.: Mean Field Theories and Dual Variation. Atlantis Press, Amsterdam (2008)zbMATHGoogle Scholar
  22. 22.
    Suzuki, T., Tasaki, S.: Stationary Fix–Caginalp equation with non-local term. Nonlinear Anal. 71, 1329–1349 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Suzuki, T., Tasaki, S.: Stationary solutions to a thermoelastic system on shape memory materials. Nonlinearity 23, 2623–2656 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Suzuki, T., Yoshikawa, S.: Stability of the steady state for the Falk model system of shape memory alloys. Math. Meth. Appl. Sci. 30, 2233–2245 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Suzuki, T., Yoshikawa, S.: Stability of the steady state for multi-dimensional thermoelastic systems of shape memory alloys. Discrete Contin. Dyn. Syst. S 5, 209–217 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Turing, A.M.: The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.University of MannheimMannheimGermany
  2. 2.Department of Applied Mathematics and InformaticsRyukoku UniversitySeta OtsuJapan
  3. 3.Center for Mathematical Modeling and Data ScienceOsaka UniversityMachikaneyamachoJapan

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