Applications of Mathematics

, Volume 57, Issue 2, pp 143–165 | Cite as

A variational approach to bifurcation in reaction-diffusion systems with Signorini type boundary conditions

  • Jamol I. Baltaev
  • Milan Kučera
  • Martin Väth


We consider a simple reaction-diffusion system exhibiting Turing’s diffusion driven instability if supplemented with classical homogeneous mixed boundary conditions. We consider the case when the Neumann boundary condition is replaced by a unilateral condition of Signorini type on a part of the boundary and show the existence and location of bifurcation of stationary spatially non-homogeneous solutions. The nonsymmetric problem is reformulated as a single variational inequality with a potential operator, and a variational approach is used in a certain non-direct way.


reaction-diffusion system unilateral condition variational inequality local bifurcation variational approach spatial patterns 

MSC 2010

35B32 35K57 35J50 35J57 47J20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. I. Baltaev, M. Kučera: Global bifurcation for quasivariational inequalities of reactiondiffusion type. J. Math. Anal. Appl. 345 (2008), 917–928.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    P. Drábek, M. Kučera, M. Míková: Bifurcation points of reaction-diffusion systems with unilateral conditions. Czech. Math. J. 35 (1985), 639–660.Google Scholar
  3. [3]
    P. Drábek, A. Kufner, F. Nicolosi: Quasilinear Elliptic Equations with Degenerations and Singularities. Walter de Gruyter, Berlin, 1997.zbMATHCrossRefGoogle Scholar
  4. [4]
    L. Edelstein-Keshet: Mathematical Models in Biology. McGraw-Hill, Boston, 1988.zbMATHGoogle Scholar
  5. [5]
    J. Eisner: Critical and bifurcation points of reaction-diffusion systems with conditions given by inclusions. Nonlinear Anal., Theory Methods Appl. 46 (2001), 69–90.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    J. Eisner, M. Kučera: Spatial patterning in reaction-diffusion systems with nonstandard boundary conditions. Fields Institute Communications 25 (2000), 239–256.Google Scholar
  7. [7]
    J. Eisner, M. Kučera, L. Recke: Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem. J. Math. Anal. Appl. 274 (2002), 159–180.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    J. Eisner, M. Kučera, L. Recke: Direction and stability of bifurcating branches for variational inequalities. J. Math. Anal. Appl. 301 (2005), 276–294.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    J. Eisner, M. Kučera, M. Väth: Global bifurcation for a reaction-diffusion system with inclusions. J. Anal. Anwend. 28 (2009), 373–409.zbMATHCrossRefGoogle Scholar
  10. [10]
    S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier, Amsterdam-Oxford-New York, 1980.zbMATHGoogle Scholar
  11. [11]
    D. S. Jones, B.D. Sleeman: Differential Equations and Mathematical Biology. Chapman & Hall/CRC, Boca Raton, 2003.zbMATHGoogle Scholar
  12. [12]
    M. Kučera: Reaction-diffusion systems: Stabilizing effect of conditions described by quasivariational inequalities. Czech. Math. J. 47 (1997), 469–486.zbMATHCrossRefGoogle Scholar
  13. [13]
    M. Kučera, L. Recke, J. Eisner: Smooth bifurcation for variational inequalities and reaction-diffusion systems. Progresses in Analysis (J.G.W. Begehr, R.P. Gilbert, M.W. Wong, eds.). World Scientific, Singapore-New Jersey-London-Hong Kong, 2001, pp. 1125–1133.Google Scholar
  14. [14]
    E. Miersemann: Verzweigungsprobleme für Variationsungleichungen. Math. Nachr. 65 (1975), 187–209. (In German.)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    M. Mimura, Y. Nishiura, M. Yamaguti: Some diffusive prey and predator systems and their bifurcation problems. Ann. New York Acad. Sci. 316 (1979), 490–510.MathSciNetCrossRefGoogle Scholar
  16. [16]
    J.D. Murray: Mathematical Biology, 2nd ed. Springer, Berlin, 1993.zbMATHCrossRefGoogle Scholar
  17. [17]
    Y. Nishiura: Global structure of bifurcating solutions of some reaction-diffusion systems and their stability problem. Proceedings of the 5th Int. Symp. Computing Methods in Applied Sciences and Engineering, Versailles, France, 1981 (R. Glowinski, J. L. Lions, eds.). North-Holland, Amsterdam-New York-Oxford,1982.Google Scholar
  18. [18]
    P. Quittner: Bifurcation points and eigenvalues of inequalities of reaction-diffusion type. J. Reine Angew. Math. 380 (1987), 1–13.MathSciNetzbMATHGoogle Scholar
  19. [19]
    L. Recke, J. Eisner, M. Kučera: Smooth bifurcation for variational inequalities based on the implicit function theorem. J. Math. Anal. Appl. 275 (2002), 615–641.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    J. Smoller: Shock Waves and Reaction-Diffusion Equations. Springer, New York-Heidelberg-Berlin, 1983.zbMATHCrossRefGoogle Scholar
  21. [21]
    E. Zeidler: Nonlinear Functional Analysis and Its Applications, vol. II/A. Springer, New York-Berlin-Heidelberg, 1990.CrossRefGoogle Scholar
  22. [22]
    E. Zeidler: Nonlinear Functional Analysis and Its Applications, vol. III. Springer, New York-Berlin-Heidelberg, 1985.Google Scholar
  23. [23]
    E. Zeidler: Nonlinear Functional Analysis and Its Applications, vol. IV. Springer, New York-Berlin-Heidelberg, 1988.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2012

Authors and Affiliations

  • Jamol I. Baltaev
    • 1
  • Milan Kučera
    • 2
  • Martin Väth
    • 2
    • 3
  1. 1.Department of MathematicsUrgench State UniversityKhorezmUzbekistan
  2. 2.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrague 1Czech Republic
  3. 3.Dept. of Mathematics (WE1)Free University of BerlinBerlinGermany

Personalised recommendations